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 on's law of gr tation the s of the instanta nta of the individual pa rticles Hence 6N state iables eeded to describe the state of a s m of N pa rticl each particle hay nd thre The tim volutio tate is determined b e function attraction).the initial state (initial tions),and Newh on's nd law F=dP/dt Des using finite sets of state riable phy apropriatefo vitati r actio 0 the int a the icles ould be this radi oin sel ting all that ntities E.D.B 11 141 ust be care confuse all quantities described a with thos scientific field theory For instance,we may reter to a temperatur eld" in the However. What special character,then,can we ascribe to the electror agnetic 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 fiel 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 importan 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 Attempts have been made to formulate electrom etic 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 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 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 (x,y.z)known as the electrostatic potential.The spatial distribution of the field is governed by Poisson's equation field.The arg as the source To uniquely specify 1e point,we must still specify its ary surface e coul specifyΦon five of the six face cub and the norma derivativ Φ/dn on th remaining face Finally,we cannot y oD erve the static potentia but we can observe its interaction with a particle.We relate th static potential ry to tn realm of me hanics via the electrostati in future chapters we shall present a classical field theory for macrosc orce F= narge d opic electromag 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 electrie charge systematic exnerimentation has revealed much more about the in2pca出 observed that charges of opr site kind attract and charges of the same kind repel.He also found that an increase in one kind of charge is acc ease 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 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. e begtigation of the propertics of the trmtifield with detaild 1.3.1 Macroscopic electromagnetics Weare intersted primarily in those electro classical es (charge and cu limit n scop ons nte electrical ma minat in es,and fields ons can s where quantum effects are agnetics to determine the macroscopic electromagneti cha nuou e.The quantization of atomic charge 1 dprincipls plysics (verted tot).The of owh te/3 an establishe to great accuracy: e=1.60217733×10-19 Coulombs(C). However,the discrete nature of cha sily incorporated into everyday engine ng co The well de by particl only by quantun re is lt hope that earn to des using suct cepts. we th oscopic idea and re th he discretization ernative s to use atomic theories of matter to estimate scope o be limited Like are of matter as if the vere continuou d as valid ecause it is verif by experime This applicab corresponds to dimensions on a laboratory scale,implying a very wide range of validity for 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 ove physical scales that include vast numbers of charges.Common dev larger than individual particles,"average" spaces between charges,and this allows us to view a source as a continuous smear 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 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(10-14 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 f(r)dv =1. An example is the Gaussian distribution f(r)=(Ta2)-3Pe-rla where a is the approximate radial extent of averaging.The spatial average of a micro- scopic quantity F(r.t)is given by (F(rD)=F(r-rt)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 p oves 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 mav 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°c,)=∑q6r-r). where ri(t)is the position of the charge qi at time t.Substitution into(1.1)gives pc,)=(pc,t》=gfc-rt) (1.2) the oscillati averaging,the time variations of microscopic fields are not present in the aggregate(macroscopic charge motion). With the definition of macroscopic charge density given by (1.2),we can determine the total charge (t)in any macroscopic volume region V using 2()=p(r.1)dv. (1.3) We have 0-avg. r年B In this case m= 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(r)/dt, the charge aggregate has volume current density JPc,)=q08r-r0). Spatial averaging gives the macroscopic volume current density Jc,)=°c,0》=∑qfr-r0) (1.4 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)