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 he 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, II. Dyadics are shown in boldface with an overbar: E,A Frequency dependent quantities are indicated by a tilde, whereas time dependent tities are written without additional indication; thus we write e(r, o)and e(r, t) quantities, such as impedance, are used in the frequency domain to interrelate m pectra: 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, E denotes a frequency domain dyadic distingU carefully between phasor and frequency domain quantities. The variable o is used for the frequency variable of the Fourier spectrum, while a is used to indicate the constant equency 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: E(r) However, there is often a simple relationship between the two, such as E=e(d 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,<). The sets of coordinate variables in rectangular, cylindrical, and spherical coordinates are denoted by (x,y,z),(0,中,z),(r,6,φ espectively.(In the spherical system o is the azimuthal angle and 0 is the polar angle. We freely use the del"operator notation V for gradient, curl, divergence, Laplacia The SI (MKS) system of units is employed throughout the book @2001 by CRC Press LLC
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 Introductory treatments of electromagnetics often stress the role of the field in force ransmission: 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 Coulombs 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 dielec tric)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 flur), which was thought to determine the amplitude of the effect observed in Faraday s experiments on electromagnetic induction. Faradays ideas presented a new world view: electromagnetic phenomena occur in the egion surrounding charged bodies, and can be described in terms of the laws governing the "field"of his lines of force. nalogies were made to the stresses and strains in material objects, and it appeared that Faraday's force lines created equivalent electromagnetic @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 Faradays ideas, Gauss restated Coulombs law in terms of flux lines, and Maxwell extended the idea to time changing fields through his concept of displacement In the 1860s Maxwell created what Einstein called "the most important invention ince Newtons 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 Newtons 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 inter act with each other, but rather with the(very real) electromagnetic field an interaction described by a supplementary equation(the Lorentz force la ter understand the interactions in terms of mechanical concepts, Maxwell properties of stress and energy to the field Using constructs that we now call the electric and magnetic fields and potential 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(Ohms law and Kirchoff's laws)from the field expressions. His experiments with high-frequency fields verified Maxwells 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 trans- verse 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 convey 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 energ 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 @2001 by CRC Press LLC
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
1.2.2 Formalization of field theory Before we can invoke physical laws, we must find a way to describe the state of the ystem we intend to study. We generally begin by identifying a set of state variable that can depict the physical nature of the system. In a mechanical theory such as Newtons law of gravitation, the state of a system of point masses is expressed in terms of the instantaneous positions and momenta of the individual particles. Hence 6N state variables are needed to describe the state of a system of N particles, each particle having three position coordinates and three momentum components. The time evolution of the system state is determined by a supplementary force function(e. g, gravitation attraction), the initial state(initial conditions), and Newtons second law F=dP/dr Descriptions using finite sets of state variables are appropriate for action-at-a-distance interpretations of physical laws such as Newtons law of gravitation or the interaction of charged particles. If Coulomb,s law were taken as the force law in a mechanical description of electromagnetics, the state of a system of particles could be described completely in terms of their positions, momenta, and charges. Of course, charged particle interaction is not this simple. An attempt to augment Coulombs force law with Amperes force law would not account for kinetic energy loss via radiation. Hence we abandon the mechanical viewpoint in favor of the field viewpoint, selecting a different set of state variables. The essence of field theory is to regard electromagnetic phenomena as affecting all of space. We shall find that we can describe the field in terms of the four vector quantities E, D, B, and H. Because these fields exist by definition at each point in space and each time t, a finite set of state variables cannot describe the system Here then is an important distinction between field theories and mechanical theories the state of a field at any instant can only be described by an infinite number of state variables. Mathematically we describe fields in terms of functions of continuous variables however, we must be careful not to confuse all quantities described as"fields"with those fields innate to a scientific field theory. For instance, we may refer to a temperature "field"in the sense that we can describe temperature as a function of space and time However, we do not mean by this that temperature obeys a set of physical laws analogous to those obeyed by the electromagnetic field What special character, then, can we ascribe to the electromagnetic field that has meaning beyond that given by its mathematical implications? In this book, E, D, B and H are integral parts of a field-theory description of electromagnetics. In any field theory we need two types of fields: a mediating field generated by a source, and a field describing the source itself. In free-space electromagnetics the mediating field consists of E and B, while the source field is the distribution of charge or current. An important consideration is that the source field must be independent of the mediating field that it"sources. "Additionally, fields are generally regarded as unobservable: they can only be measured indirectly through interactions with observable quantities. We need a link to mechanics to observe e and B: we might measure the change in kinetic energy of a particle as it interacts with the field through the lorentz force. The Lorentz force becomes the force function in the mechanical interaction that uniquely determines the (observable)mechanical state of the particle A field is associated with a set of field equations and a set of constitutive relations. The field equations describe, through partial derivative operations, both the spatial distribu- tion and temporal evolution of the field. The constitutive relations describe the effect TAttempts have been made to formulate electromagnetic theory purely in action-at-a-distance terms but this viewpoint has not been generally adopted 69 @2001 by CRC Press LLC
1.2.2 Formalization of field theory Before we can invoke physical laws, we must find a way to describe the state of the system we intend to study. We generally begin by identifying a set of state variables that can depict the physical nature of the system. In a mechanical theory such as Newton’s law of gravitation, the state of a system of point masses is expressed in terms of the instantaneous positions and momenta of the individual particles. Hence 6N state variables are needed to describe the state of a system of N particles, each particle having three position coordinates and three momentum components. The time evolution of the system state is determined by a supplementary force function (e.g., gravitational attraction), the initial state (initial conditions), and Newton’s second law F = dP/dt. Descriptions using finite sets of state variables are appropriate for action-at-a-distance interpretations of physical laws such as Newton’s law of gravitation or the interaction of charged particles. If Coulomb’s law were taken as the force law in a mechanical description of electromagnetics, the state of a system of particles could be described completely in terms of their positions, momenta, and charges. Of course, charged particle interaction is not this simple. An attempt to augment Coulomb’s force law with Ampere’s force law would not account for kinetic energy loss via radiation. Hence we abandon1 the mechanical viewpoint in favor of the field viewpoint, selecting a different set of state variables. The essence of field theory is to regard electromagnetic phenomena as affecting all of space. We shall find that we can describe the field in terms of the four vector quantities E, D, B, and H. Because these fields exist by definition at each point in space and each time t, a finite set of state variables cannot describe the system. Here then is an important distinction between field theories and mechanical theories: the state of a field at any instant can only be described by an infinite number of state variables. Mathematically we describe fields in terms of functions of continuous variables; however, we must be careful not to confuse all quantities described as “fields” with those fields innate to a scientific field theory. For instance, we may refer to a temperature “field” in the sense that we can describe temperature as a function of space and time. However, we do not mean by this that temperature obeys a set of physical laws analogous to those obeyed by the electromagnetic field. What special character, then, can we ascribe to the electromagnetic field that has meaning beyond that given by its mathematical implications? In this book, E, D, B, and H are integral parts of a field-theory description of electromagnetics. In any field theory we need two types of fields:a mediating field generated by a source, and a field describing the source itself. In free-space electromagnetics the mediating field consists of E and B, while the source field is the distribution of charge or current. An important consideration is that the source field must be independent of the mediating field that it “sources.” Additionally, fields are generally regarded as unobservable:they can only be measured indirectly through interactions with observable quantities. We need a link to mechanics to observe E and B:we might measure the change in kinetic energy of a particle as it interacts with the field through the Lorentz force. The Lorentz force becomes the force function in the mechanical interaction that uniquely determines the (observable) mechanical state of the particle. A field is associated with a set of field equations and a set of constitutive relations. The field equations describe, through partial derivative operations, both the spatial distribution and temporal evolution of the field. The constitutive relations describe the effect 1Attempts have been made to formulate electromagnetic theory purely in action-at-a-distance terms, but this viewpoint has not been generally adopted [69]
of the supporting medium on the fields and are dependent upon the physical state of the medium. The state may include macroscopic effects, such as mechanical stress and thermodynamic temperature, as well as the microscopic, quantum-mechanical properties of matter The value of the field at any position and time in a bounded region V is then determined Uniquely by specifying the sources within V, the initial state of the fields within V, and the value of the field or finitely many of its derivatives on the surface bounding V. If differing physical characteristics, or across discontinuous sources, then jump conditon.. the boundary surface also defines a surface of discontinuity between adjacent regions of may be used to relate the fields on either side of the surface The variety of forms of field equations is restricted by many physical principles in- cluding reference-frame invariance, conservation, causality, symmetry, and simplicity Causality prevents the field at time t =0 from being influenced by events occurring at subsequent times t >0. Of course, we prefer that a field equation be mathematically robust and well-posed to permit solutions that are unique and stable. Many of these ideas are well illustrated by a consideration of electrostatics. We can describe the electrostatic field through a mediating scalar field p(x, y, z) known as the electrostatic potential. The spatial distribution of the field is governed by Poisson equation a2a2Φa2∮ ax2 av where p=p(x, y, z) is the source charge density. No temporal derivatives appear, and th spatial derivatives determine the spatial behavior of the field. The function p represents the spatially-averaged distribution of charge that acts as the source term for the field p Note that p incorporates no information about p. To uniquely specify the field at any point, we must still specify its behavior over a boundary surface. We could, for instance specify p on five of the six faces of a cube and the normal derivative a p/ an on the emaining face. Finally, we cannot directly observe the static potential field, but we can observe its interaction with a particle. We relate the static potential field theory to the realm of mechanics via the electrostatic force F=qE acting on a particle of charge q In future chapters we shall present a classical field theory for macroscopic electroma netics. In that case the mediating field quantities are E, D, B, and H, and the source field is the current density J 1. 3 The sources of the electromagnetic field Electric charge is an intriguing natural entity. Human awareness of charge and its effects dates back to at least 600 BC, when the greek philosopher Thales of Miletus observed that rubbing a piece of amber could enable the amber to attract bits of straw Although charging by friction is probably still the most common and familiar manifes- tation of electric charge, systematic experimentation has revealed much more about the eHavior of charge and its role in the physical universe. There are two kinds of charge, to ch benja jamin Franklin assigned the respective names positive and negative. Franklin observed that charges of opposite kind attract and charges of the same kind repel. He also found that an increase in one kind of charge is accompanied by an increase in the @2001 by CRC Press LLC
of the supporting medium on the fields and are dependent upon the physical state of the medium. The state may include macroscopic effects, such as mechanical stress and thermodynamic temperature, as well as the microscopic, quantum-mechanical properties of matter. The value of the field at any position and time in a bounded region V is then determined uniquely by specifying the sources within V, the initial state of the fields within V, and the value of the field or finitely many of its derivatives on the surface bounding V. If the boundary surface also defines a surface of discontinuity between adjacent regions of differing physical characteristics, or across discontinuous sources, then jump conditions may be used to relate the fields on either side of the surface. The variety of forms of field equations is restricted by many physical principles including reference-frame invariance, conservation, causality, symmetry, and simplicity. Causality prevents the field at time t = 0 from being influenced by events occurring at subsequent times t > 0. Of course, we prefer that a field equation be mathematically robust and well-posed to permit solutions that are unique and stable. Many of these ideas are well illustrated by a consideration of electrostatics. We can describe the electrostatic field through a mediating scalar field (x, y,z) known as the electrostatic potential. The spatial distribution of the field is governed by Poisson’s equation ∂2 ∂x 2 + ∂2 ∂y2 + ∂2 ∂z2 = − ρ 0 , θ where ρ = ρ(x, y,z) is the source charge density. No temporal derivatives appear, and the spatial derivatives determine the spatial behavior of the field. The function ρ represents the spatially-averaged distribution of charge that acts as the source term for the field . Note that ρ incorporates no information about . To uniquely specify the field at any point, we must still specify its behavior over a boundary surface. We could, for instance, specify on five of the six faces of a cube and the normal derivative ∂/∂n on the remaining face. Finally, we cannot directly observe the static potential field, but we can observe its interaction with a particle. We relate the static potential field theory to the realm of mechanics via the electrostatic force F = qE acting on a particle of charge q. In future chapters we shall present a classical field theory for macroscopic electromagnetics. In that case the mediating field quantities are E, D, B, and H, and the source field is the current density J. 1.3 The sources of the electromagnetic field Electric charge is an intriguing natural entity. Human awareness of charge and its effects dates back to at least 600 BC, when the Greek philosopher Thales of Miletus observed that rubbing a piece of amber could enable the amber to attract bits of straw. Although charging by friction is probably still the most common and familiar manifestation of electric charge, systematic experimentation has revealed much more about the behavior of charge and its role in the physical universe. There are two kinds of charge, to which Benjamin Franklin assigned the respective names positive and negative. Franklin observed that charges of opposite kind attract and charges of the same kind repel. He also found that an increase in one kind of charge is accompanied by an increase in the
other, and so first described the principle of charge conservation. Twentieth century physics has added dramatically to the understanding of charge 1. Electric charge is a fundamental property of matter, as is mass or dimension 2. Charge is quantized: there exists a smallest quantity (quantum) of charge that can be associated with matter. No smaller amount has been observed, and large amounts always occur in integral multiples of this quantity. 3. The charge quantum is associated with the smallest subatomic particles, and these particles interact through electrical forces. In fact, matter is organized and arranged through electrical interactions; for example, our perception of physical contact is merely the macroscopic manifestation of countless charges in our fingertips pushing against charges in the things we touch 4. Electric charge is an invariant: the value of charge on a particle does not depend on the speed of the particle. In contrast, the mass of a particle increases with speed 5. Charge acts as the source of an electromagnetic field; the field is an entity that can carry energy and momentum away from the charge via propagating waves We begin our investigation of the properties of the electromagnetic field with a detailed examination of its source 1.3.1 Macroscopic electromagnetics We are interested primarily in those electromagnetic effects that can be predicted by classical techniques using continuous sources(charge and current densities ). Although macroscopic electromagnetics is limited in scope, it is useful in many situations en countered by engineers. These include, for example, the determination of currents and voltages in lumped circuits, torques exerted by electrical machines, and fields radiated by antennas. Macroscopic predictions can fall short in cases where quantum effects are im- portant: e. g, with devices such as tunnel diodes. Even so, quantum mechanics can often be coupled with classical electromagnetics to determine the macroscopic electromagnetic properties of important materials Electric charge is not of a continuous nature. The quantization of atomic charge te for electrons and protons, te/3 and +2e/ 3 for quarks - is one of the most precisely established principles in physics(verified to 1 part in 10-). The value of e itself is known to great accuracy e=1.60217733 x 10-19 Coulombs(C) However, the discrete nature of charge is not easily incorporated into everyday engineer ing concerns. The strange world of the individual charge - characterized by particle spin, molecular moments, and thermal vibrations is well described only by quantum theory. There is little hope that we can learn to describe electrical machines using such concepts. Must we therefore retreat to the macroscopic idea and ignore the discretization of charge completely? A viable alternative is to use atomic theories of matter to estimate the useful scope of macroscopic electromagnetics Remember, we are completely free to postulate a theory of nature whose scope may be limited. Like continuum mechanics, which treats distributions of matter as if they were continuous, macroscopic electromagnetics is regarded as valid because it is verified by experiment over a certain range of conditions. This applicability range generally corresponds to dimensions on a laboratory scale, implying a very wide range of validit or engineers @2001 by CRC Press LLC
other, and so first described the principle of charge conservation. Twentieth century physics has added dramatically to the understanding of charge: 1. Electric charge is a fundamental property of matter, as is mass or dimension. 2. Charge is quantized:there exists a smallest quantity (quantum) of charge that can be associated with matter. No smaller amount has been observed, and larger amounts always occur in integral multiples of this quantity. 3. The charge quantum is associated with the smallest subatomic particles, and these particles interact through electrical forces. In fact, matter is organized and arranged through electrical interactions; for example, our perception of physical contact is merely the macroscopic manifestation of countless charges in our fingertips pushing against charges in the things we touch. 4. Electric charge is an invariant:the value of charge on a particle does not depend on the speed of the particle. In contrast, the mass of a particle increases with speed. 5. Charge acts as the source of an electromagnetic field; the field is an entity that can carry energy and momentum away from the charge via propagating waves. We begin our investigation of the properties of the electromagnetic field with a detailed examination of its source. 1.3.1 Macroscopic electromagnetics We are interested primarily in those electromagnetic effects that can be predicted by classical techniques using continuous sources (charge and current densities). Although macroscopic electromagnetics is limited in scope, it is useful in many situations encountered by engineers. These include, for example, the determination of currents and voltages in lumped circuits, torques exerted by electrical machines, and fields radiated by antennas. Macroscopic predictions can fall short in cases where quantum effects are important:e.g., with devices such as tunnel diodes. Even so, quantum mechanics can often be coupled with classical electromagnetics to determine the macroscopic electromagnetic properties of important materials. Electric charge is not of a continuous nature. The quantization of atomic charge — ±e for electrons and protons, ±e/3 and ±2e/3 for quarks — is one of the most precisely established principles in physics (verified to 1 part in 1021). The value of e itself is known to great accuracy: e = 1.60217733 × 10−19 Coulombs (C). However, the discrete nature of charge is not easily incorporated into everyday engineering concerns. The strange world of the individual charge — characterized by particle spin, molecular moments, and thermal vibrations — is well described only by quantum theory. There is little hope that we can learn to describe electrical machines using such concepts. Must we therefore retreat to the macroscopic idea and ignore the discretization of charge completely? A viable alternative is to use atomic theories of matter to estimate the useful scope of macroscopic electromagnetics. Remember, we are completely free to postulate a theory of nature whose scope may be limited. Like continuum mechanics, which treats distributions of matter as if they were continuous, macroscopic electromagnetics is regarded as valid because it is verified by experiment over a certain range of conditions. This applicability range generally corresponds to dimensions on a laboratory scale, implying a very wide range of validity for engineers
Macroscopic effects as averaged microscopic effects. Macroscopic electromag. netics can hold in a world of discrete charges because applications usually occur over physical scales that include vast numbers of charges. Common devices, generally much larger than individual particles, " average"the rapidly varying fields that exist in the spaces between charges, and this allows us to view a source as a continuous"smear"of charge. To determine the range of scales over which the macroscopic viewpoint is valid. re must compare averaged values of microscopic fields to the macroscopic fields we mea- sure in the lab. But if the effects of the individual charges are describable only in terms of quantum notions, this task will be daunting at best. A simple compromise, which produces useful results, is to extend the macroscopic theory right down to the micro- scopic level and regard discrete charges as"point "entities that produce electromagnetic fields according to Maxwell's equations. Then, in terms of scales much larger than the classical radius of an electron( 10-4 m), the expected rapid fluctuations of the fields in the spaces between charges is predicted. Finally, we ask: over what spatial scale must we average the effects of the fields and the sources in order to obtain agreement with the macroscopic equations? In the spatial averaging approach a convenient weighting function f(r) is chosen, and normalized so that(r)dv=l An example is the Gaussian distribution f(r)=(xra2)-32e-r2 where a is the approximate radial extent of averaging. The spatial average of a micro- scopic quantity F(r, t) is given by (F(r, t))= F(r-r,of(r)dv (1.1) The scale of validity of the macroscopic model can be found by determining the averaging radius a that produces good agreement between the averaged microscopic fields and the macroscopic fields The macroscopic volume charge density. At this point we do not distinguish between the"free"charge that is unattached to a molecular structure and the charge found near the surface of a conductor. Nor do we consider the dipole nature of polarizable materials or the microscopic motion associated with molecular magnetic moment or the magnetic moment of free charge. For the consideration of free-space electromagnetics, we assume charge exhibits either three degrees of freedom(volume charge), two degrees of freedom(surface charge), or one degree of freedom(line charge) In typical matter, the microscopic fields vary spatially over dimensions of 10-0 m or less, and temporally over periods(determined by atomic motion) of 10-13 s or less At the surface of a material such as a good conductor where charge often concentrates averaging with a radius on the order of 10-10 m may be required to resolve the rapid variation in the distribution of individual charged particles. However, within a solid or liquid material, or within a free-charge distribution characteristic of a dense gas or an electron beam, a radius of 10-8m proves useful, containing typically 10 particles.A diffuse gas, on the other hand, may have a particle density so low that the averaging radius takes on laboratory dimensions, and in such a case the microscopic theory must be employed even at macroscopic dimensions Once the averaging radius has been determined, the value of the charge density may be found via(1.1). The volume density of charge for an assortment of point sources can @2001 by CRC Press LLC
Macroscopic effects as averaged microscopic effects. Macroscopic electromagnetics can hold in a world of discrete charges because applications usually occur over physical scales that include vast numbers of charges. Common devices, generally much larger than individual particles, “average” the rapidly varying fields that exist in the spaces between charges, and this allows us to view a source as a continuous “smear” of charge. To determine the range of scales over which the macroscopic viewpoint is valid, we must compare averaged values of microscopic fields to the macroscopic fields we measure in the lab. But if the effects of the individual charges are describable only in terms of quantum notions, this task will be daunting at best. A simple compromise, which produces useful results, is to extend the macroscopic theory right down to the microscopic level and regard discrete charges as “point” entities that produce electromagnetic fields according to Maxwell’s equations. Then, in terms of scales much larger than the classical radius of an electron (≈ 10−14 m), the expected rapid fluctuations of the fields in the spaces between charges is predicted. Finally, we ask:over what spatial scale must we average the effects of the fields and the sources in order to obtain agreement with the macroscopic equations? In the spatial averaging approach a convenient weighting function f (r) is chosen, and is normalized so that f (r) dV = 1. An example is the Gaussian distribution f (r) = (πa2 ) −3/2 e−r 2/a2 , where a is the approximate radial extent of averaging. The spatial average of a microscopic quantity F(r, t) is given by F(r, t) = F(r − r , t) f (r ) dV . (1.1) The scale of validity of the macroscopic model can be found by determining the averaging radius a that produces good agreement between the averaged microscopic fields and the macroscopic fields. The macroscopic volume charge density. At this point we do not distinguish between the “free” charge that is unattached to a molecular structure and the charge found near the surface of a conductor. Nor do we consider the dipole nature of polarizable materials or the microscopic motion associated with molecular magnetic moment or the magnetic moment of free charge. For the consideration of free-space electromagnetics, we assume charge exhibits either three degrees of freedom (volume charge), two degrees of freedom (surface charge), or one degree of freedom (line charge). In typical matter, the microscopic fields vary spatially over dimensions of 10−10 m or less, and temporally over periods (determined by atomic motion) of 10−13 s or less. At the surface of a material such as a good conductor where charge often concentrates, averaging with a radius on the order of 10−10 m may be required to resolve the rapid variation in the distribution of individual charged particles. However, within a solid or liquid material, or within a free-charge distribution characteristic of a dense gas or an electron beam, a radius of 10−8 m proves useful, containing typically 106 particles. A diffuse gas, on the other hand, may have a particle density so low that the averaging radius takes on laboratory dimensions, and in such a case the microscopic theory must be employed even at macroscopic dimensions. Once the averaging radius has been determined, the value of the charge density may be found via (1.1). The volume density of charge for an assortment of point sources can
be written in terms of the three-dimensional dirac delta as p(r,)=∑ here ri (t)is the position of the charge qi at time 1. Substitution into(1.1)gives p(r,1)=p°r,1)=∑qf(r-r1(t) (1.2) as the averaged charge density appropriate for use in a macroscopic field theory. Because the oscillations of the atomic particles are statistically uncorrelated over the distances used in spatial averaging, the time variations of microscopic fields are not present in th macroscopic fields and temporal averaging is unnecessary. In(1.2) the time dependence of the spatially-averaged charge density is due entirely to bulk motion of the charge aggregate(macroscopic charge motion) With the definition of macroscopic charge density given by(1.2), we can determine the total charge Q(t) in any macroscopic volume region V using e()= p(r, t)dv (1.3) We have 0=∑%-;o)y=∑ o Here we ignore the small discrepancy produced by charges lying within distance a of e boundary of v. It is common to employ a box b having volume AV (r)=1/△V,r∈B E B In this case A∑ r-r()∈B The size of b is chosen with the same considerations as to atomic scale as was the averaging radius a. Discontinuities at the edges of the box introduce some difficulties concerning charges that move in and out of the box because of molecular motion The macroscopic volume current density. Electric charge in motion is referred to as electric current. Charge motion can be associated with external forces and with microscopic fluctuations in position. Assuming charge qi has velocity v;(t)=dri (t)/dt the charge aggregate has volume current density T(r, t) qv()8(r-r;(t) Spatial averaging gives the macroscopic volume current densit J(r,)=o(r, d)=2qivi(f(r-r() @2001 by CRC Press LLC
be written in terms of the three-dimensional Dirac delta as ρo (r, t) = i qi δ(r − ri(t)), where ri(t) is the position of the charge qi at time t. Substitution into (1.1) gives ρ(r, t) = ρo (r, t) = i qi f (r − ri(t)) (1.2) as the averaged charge density appropriate for use in a macroscopic field theory. Because the oscillations of the atomic particles are statistically uncorrelated over the distances used in spatial averaging, the time variations of microscopic fields are not present in the macroscopic fields and temporal averaging is unnecessary. In (1.2) the time dependence of the spatially-averaged charge density is due entirely to bulk motion of the charge aggregate (macroscopic charge motion). With the definition of macroscopic charge density given by (1.2), we can determine the total charge Q(t) in any macroscopic volume region V using Q(t) = V ρ(r, t) dV. (1.3) We have Q(t) = i qi V f (r − ri(t)) dV = ri(t)∈V qi . Here we ignore the small discrepancy produced by charges lying within distance a of the boundary of V. It is common to employ a box B having volume V: f (r) = 1/V, r ∈ B, 0, r ∈/ B. In this case ρ(r, t) = 1 V r−ri(t)∈B qi . The size of B is chosen with the same considerations as to atomic scale as was the averaging radius a. Discontinuities at the edges of the box introduce some difficulties concerning charges that move in and out of the box because of molecular motion. The macroscopic volume current density. Electric charge in motion is referred to as electric current. Charge motion can be associated with external forces and with microscopic fluctuations in position. Assuming charge qi has velocity vi(t) = dri(t)/dt, the charge aggregate has volume current density Jo (r, t) = i qivi(t)δ(r − ri(t)). Spatial averaging gives the macroscopic volume current density J(r, t) = Jo (r, t) = i qivi(t) f (r − ri(t)). (1.4)
Figure 1. 1: Intersection of the averaging function of a point charge with a surface S, as the charge crosses S with velocity v:(a) at some time t= tI, and(b)at t=f2>t1. The averaging function is represented by a sphere of radius a Spatial averaging at time t eliminates currents associated with microscopic motions that are uncorrelated at the scale of the averaging radius (again, we do not consider the magnetic moments of particles). The assumption of a sufficiently large averaging radius leads to J(r, t)=p(r, t)v(r, t) The total Hux /(t) of current through a surface S is gi I(r)=J(r, t)ndS where n is the unit normal to S. Hence, using(4), we have ∑qaco)/rr-ro)ds if f stays approximately constant over the extent of the averaging function and S is not in motion. We see that the integral effectively intersects S with the averaging function su rounding each moving point charge (Figure 1.1). The time derivative of ri. f represents the velocity at which the averaging function is"carried across"the surface. Electric current takes a variety of forms, each described by the relation J= pv. Isolated charged particles(positive and negative)and charged insulated bodies moving through space comprise convection currents. Negatively-charged electrons moving through the positive background lattice within a conductor comprise a conduction current. Empirical evidence suggests that conduction currents are also described by the relation J= oE known as Ohm's law. a third type of current, called electrolytic current, results from the How of positive or negative ions through a fuid 1.3.2 Impressed vS secondary sources In addition to the simple classification given above we may classify currents as primary or secondary, depending on the action that sets the charge in motion @2001 by CRC Press LLC
Figure 1.1:Intersection of the averaging function of a point charge with a surface S, as the charge crosses S with velocity v:(a) at some time t = t1, and (b) at t = t2 > t1. The averaging function is represented by a sphere of radius a. Spatial averaging at time t eliminates currents associated with microscopic motions that are uncorrelated at the scale of the averaging radius (again, we do not consider the magnetic moments of particles). The assumption of a sufficiently large averaging radius leads to J(r, t) = ρ(r, t) v(r, t). (1.5) The total flux I(t) of current through a surface S is given by I(t) = S J(r, t) · nˆ d S where nˆ is the unit normal to S. Hence, using (4), we have I(t) = i qi d dt (ri(t) · nˆ) S f (r − ri(t)) d S if nˆ stays approximately constant over the extent of the averaging function and S is not in motion. We see that the integral effectively intersects S with the averaging function surrounding each moving point charge (Figure 1.1). The time derivative of r i ·nˆ represents the velocity at which the averaging function is “carried across” the surface. Electric current takes a variety of forms, each described by the relation J = ρv. Isolated charged particles (positive and negative) and charged insulated bodies moving through space comprise convection currents. Negatively-charged electrons moving through the positive background lattice within a conductor comprise a conduction current. Empirical evidence suggests that conduction currents are also described by the relation J = σE known as Ohm’s law. A third type of current, called electrolytic current, results from the flow of positive or negative ions through a fluid. 1.3.2 Impressed vs. secondary sources In addition to the simple classification given above we may classify currents as primary or secondary, depending on the action that sets the charge in motion.
It is helpful primary or "impressed"sources, whic dependent of tl fields they source, secondary sources which result from interactions between the sourced fields and Tedium in which the fields exist, most familiar is the conduc- tion current set up in a conducting medium by an externally applied electric field. The impressed source concept is particularly important in circuit theory, where independent voltage sources are modeled as providing primary voltage excitations that are indepen- dent of applied load. In this way they differ from the secondary or " dependent"sources that react to the effect produced by the application of primary sources In applied electromagnetics the primary source may be so distant that return effects resulting from local interaction of its impressed fields can be ignored. Other examples of primary sources include the applied voltage at the input of an antenna, the current on a probe inserted into a waveguide, and the currents producing a power-line field in which a biological body is immersed 1. 3. 3 Surface and line source densities Because they are spatially averaged macroscopic sources and the fields they source cannot have true spatial discont However, it is often convenient to work with sources in one or two dimensions and line source densities are idealizations of actual, continuous macroscopic densities The entity we describe as a surface charge is a continuous volume charge distributed in a thin layer across some surface S. If the thickness of the layer is small compared to laboratory dimensions, it is useful to assign to each point r on the surface a quantity describing the amount of charge contained within a cylinder oriented normal to the surface and having infinitesimal cross section dS. We call this quantity the surface charge density ps(r, 0), and write the volume charge density as p(r, w, t)=Ps(r, t)f(w, A), where w is distance from S in the normal direction and A in some way parameterizes the "thickness"of the charge layer at r. The continuous density function f(x, A)satisfie limf(x,△)=8(x) For instance. we might have f(x,△)e-/42 (1.6) With this definition the total charge contained in a cylinder normal to the surface at r and having cross-sectional area ds is dQ()=[P, (r, t)dS]f(w, A)dw=ps(r, t)dS, and the total charge contained within any cylinder oriented normal to s i Q()=/ps(r, t)ds (1.7) @2001 by CRC Press LLC
It is helpful to separate primary or “impressed” sources, which are independent of the fields they source, from secondary sources which result from interactions between the sourced fields and the medium in which the fields exist. Most familiar is the conduction current set up in a conducting medium by an externally applied electric field. The impressed source concept is particularly important in circuit theory, where independent voltage sources are modeled as providing primary voltage excitations that are independent of applied load. In this way they differ from the secondary or “dependent” sources that react to the effect produced by the application of primary sources. In applied electromagnetics the primary source may be so distant that return effects resulting from local interaction of its impressed fields can be ignored. Other examples of primary sources include the applied voltage at the input of an antenna, the current on a probe inserted into a waveguide, and the currents producing a power-line field in which a biological body is immersed. 1.3.3 Surface and line source densities Because they are spatially averaged effects, macroscopic sources and the fields they source cannot have true spatial discontinuities. However, it is often convenient to work with sources in one or two dimensions. Surface and line source densities are idealizations of actual, continuous macroscopic densities. The entity we describe as a surface charge is a continuous volume charge distributed in a thin layer across some surface S. If the thickness of the layer is small compared to laboratory dimensions, it is useful to assign to each point r on the surface a quantity describing the amount of charge contained within a cylinder oriented normal to the surface and having infinitesimal cross section d S. We call this quantity the surface charge density ρs(r, t), and write the volume charge density as ρ(r,w, t) = ρs(r, t) f (w, ), where w is distance from S in the normal direction and in some way parameterizes the “thickness” of the charge layer at r. The continuous density function f (x, ) satisfies ∞ −∞ f (x, ) dx = 1 and lim →0 f (x, ) = δ(x). For instance, we might have f (x, ) = e−x2/2 √π . (1.6) With this definition the total charge contained in a cylinder normal to the surface at r and having cross-sectional area d S is d Q(t) = ∞ −∞ [ρs(r, t) d S] f (w, ) dw = ρs(r, t) d S, and the total charge contained within any cylinder oriented normal to S is Q(t) = S ρs(r, t) d S. (1.7)