《电磁学》参考书籍英文电子书（Edward J. Rothwell Michael J. Cloud）Chapter 2 Maxwell’s theory of electromagnetism

2. 2 The well-posed nature of the postulate It is important to investigate whether Maxwell's equations, along with the point form of the continuity equation, suffice as a useful theory of electromagnetics. Certainly we must agree that a theory is" as long as it is defined as such by the scientists and engineers who employ it. In practice a theory is considered useful if it predicts accurately the behavior of nature under given circumstances, and even a theory that often fails may be useful if it is the best available. We choose here to take a more narrow view and investigate whether the theory is "well-posed A mathematical model for a physical problem is said to be well-posed, or correctly set, if three conditions hold 1. the model has at least one solution(eristence) 2. the model has at most one solution (uniqueness) 3. the solution is continuously dependent on the data supplied The importance of the first condition is obvious: if the electromagnetic model has no solution, it will be of little use to scientists and engineers. The importance of the second condition is equally obvious: if we apply two different solution methods to the same model and get two different answers, the model will not be very helpful in analysis or design work. The third point is more subtle; it is often extended in a practical sense the following statement 3. Small changes in the data supplied produce equally small changes in the solution That is, the solution is not sensitive to errors in the data. To make sense of this we must decide which quantity is specified (the independent quantity) and which remains to be calculated (the dependent quantity). Commonly the source field(charge) is taken as the independent quantity, and the mediating(electromagnetic) field is computed from it; in such cases it can be shown that Maxwells equations are well-posed. Taking the electromagnetic field to be the independent quantity, we can produce situations in which the computed quantity(charge or current) changes wildly with small changes in th specified fields. These situations(called inverse problems )are of great importance in remote sensing, where the field is measured and the properties of the object probed are thereby deduced At this point we shall concentrate on the "forward"problem of specifying the source field(charge) and computing the mediating field(the electromagnetic field). In this case we may question whether the first of the three conditions (existence) holds. We have twelve unknown quantities(the scalar components of the four vector fields), but only eight equations to describe them(from the scalar components of the two fundamental Maxwell equations and the two scalar auxiliary equations). With fewer equations than uNknowns we cannot be sure that a solution exists, and we refer to Maxwell's equations as being indefinite. To overcome this problem we must specify more information in the form of constitutive relations among the field quantities E, B, D, H, and J. Wher these are properly formulated, the number of unknowns and the number of equations are equal and Maxwells equations are in definite form. If we provide more equations than unknowns, the solution may be non-unique. When we model the electromagnetic properties of materials we must supply precisely the right amount of information in the nstitutive relat or our postulate will not be well-posed @2001 by CRC Press LLC

2.2 The well-posed nature of the postulate It is important to investigate whether Maxwell’s equations, along with the point form of the continuity equation, suffice as a useful theory of electromagnetics. Certainly we must agree that a theory is “useful” as long as it is defined as such by the scientists and engineers who employ it. In practice a theory is considered useful if it predicts accurately the behavior of nature under given circumstances, and even a theory that often fails may be useful if it is the best available. We choose here to take a more narrow view and investigate whether the theory is “well-posed.” A mathematical model for a physical problem is said to be well-posed, or correctly set, if three conditions hold: 1. the model has at least one solution (existence); 2. the model has at most one solution (uniqueness); 3. the solution is continuously dependent on the data supplied. The importance of the first condition is obvious: if the electromagnetic model has no solution, it will be of little use to scientists and engineers. The importance of the second condition is equally obvious: if we apply two different solution methods to the same model and get two different answers, the model will not be very helpful in analysis or design work. The third point is more subtle; it is often extended in a practical sense to the following statement: 3 . Small changes in the data supplied produce equally small changes in the solution. That is, the solution is not sensitive to errors in the data. To make sense of this we must decide which quantity is specified (the independent quantity) and which remains to be calculated (the dependent quantity). Commonly the source field (charge) is taken as the independent quantity, and the mediating (electromagnetic) field is computed from it; in such cases it can be shown that Maxwell’s equations are well-posed. Taking the electromagnetic field to be the independent quantity, we can produce situations in which the computed quantity (charge or current) changes wildly with small changes in the specified fields. These situations (called inverse problems) are of great importance in remote sensing, where the field is measured and the properties of the object probed are thereby deduced. At this point we shall concentrate on the “forward” problem of specifying the source field (charge) and computing the mediating field (the electromagnetic field). In this case we may question whether the first of the three conditions (existence) holds. We have twelve unknown quantities (the scalar components of the four vector fields), but only eight equations to describe them (from the scalar components of the two fundamental Maxwell equations and the two scalar auxiliary equations). With fewer equations than unknowns we cannot be sure that a solution exists, and we refer to Maxwell’s equations as being indefinite. To overcome this problem we must specify more information in the form of constitutive relations among the field quantities E, B, D, H, and J. When these are properly formulated, the number of unknowns and the number of equations are equal and Maxwell’s equations are in definite form. If we provide more equations than unknowns, the solution may be non-unique. When we model the electromagnetic properties of materials we must supply precisely the right amount of information in the constitutive relations, or our postulate will not be well-posed

Once Maxwells equations are in definite form, standard methods for partial differential a nutshell, the system(2. 1)-(2.2)of hyperbolic differential equations is well-posed if and only if we specify e and H throughout a volume region V at some time instant and alse specify, at all subsequent times 1. the tangential component of E over all of the boundary surface S,or 2. the tangential component of H over all of S, or 3. the tangential component of e over part of S, and the tangential component of H over the remainder of s Proof of all three of the conditions of well-posedness is quite tedious, but a simplified uniqueness proof is often given in textbooks on electromagnetics. The procedure used by Stratton [187 is reproduced below. The interested reader should refer to Hansen [81 for a discussion of the existence of solutions to Maxwells equations 2.2.1 Uniqueness of solutions to Maxwell equations Consider a simply connected region of space V bounded by a surface S, where both V and S contain only ordinary points. The fields within V are associated with a current distribution J, which may be internal to V(entirely or in part). By the initial conditions that imply the auxiliary Maxwells equations, we know there is a time, say t=0, prior to which the current is zero for all time, and thus by causality the fields throughout v are identically zero for all times t<0. We next assume that the fields are specified hroughout V at some time to >0, and seek conditions under which they are determine uniquely for all t >to Let the field set(E1, D, Bl, Hi) be a solution to Maxwells equations(2.1)-(2.2) associated with the current J(along with an appropriate set of constitutive relations) and let(E,, D,, B,, H,) be a second solution associated with j. To determine the con- ditions for uniqueness of the fields, we look for a situation that results in El= e2 B1= B2, and so on. The electromagnetic fields must obey aB V×E1 V×H1=J+ V×E Subtracting d(B1-B2) V×(H1-H2)= d(D (2.10) hence defining Eo =El-e2, Bo= B1-B2, and so on, we have Eo·(V×Ho)=Eo H·(×E0)=-、aB (212) @2001 by CRC Press LLC

Once Maxwell’s equations are in definite form, standard methods for partial differential equations can be used to determine whether the electromagnetic model is well-posed. In a nutshell, the system (2.1)–(2.2) of hyperbolic differential equations is well-posed if and only if we specify E and H throughout a volume region V at some time instant and also specify, at all subsequent times, 1. the tangential component of E over all of the boundary surface S, or 2. the tangential component of H over all of S, or 3. the tangential component of E over part of S, and the tangential component of H over the remainder of S. Proof of all three of the conditions of well-posedness is quite tedious, but a simplified uniqueness proof is often given in textbooks on electromagnetics. The procedure used by Stratton [187] is reproduced below. The interested reader should refer to Hansen [81] for a discussion of the existence of solutions to Maxwell’s equations. 2.2.1 Uniqueness of solutions to Maxwell’sequations Consider a simply connected region of space V bounded by a surface S, where both V and S contain only ordinary points. The fields within V are associated with a current distribution J, which may be internal to V (entirely or in part). By the initial conditions that imply the auxiliary Maxwell’s equations, we know there is a time, say t = 0, prior to which the current is zero for all time, and thus by causality the fields throughout V are identically zero for all times t < 0. We next assume that the fields are specified throughout V at some time t0 > 0, and seek conditions under which they are determined uniquely for all t > t0. Let the field set (E1, D1,B1, H1) be a solution to Maxwell’s equations (2.1)–(2.2) associated with the current J (along with an appropriate set of constitutive relations), and let (E2, D2,B2, H2) be a second solution associated with J. To determine the con￾ditions for uniqueness of the fields, we look for a situation that results in E1 = E2, B1 = B2, and so on. The electromagnetic fields must obey ∇ × E1 = −∂B1 ∂t , ∇ × H1 = J + ∂D1 ∂t , ∇ × E2 = −∂B2 ∂t , ∇ × H2 = J + ∂D2 ∂t . Subtracting, we have ∇ × (E1 − E2) = −∂(B1 − B2) ∂t , (2.9) ∇ × (H1 − H2) = ∂(D1 − D2) ∂t , (2.10) hence defining E0 = E1 − E2, B0 = B1 − B2, and so on, we have E0 · (∇ × H0) = E0 · ∂D0 ∂t , (2.11) H0 · (∇ × E0) = −H0 · ∂B0 ∂t . (2.12)

Subtracting again, we have Eo·(V×H0)-Ho·(V×Eo)=H v·(Eo×Ho)=Eo + Ho by(B. 44). Integrating both sides throughout V and using the divergence theorem on the left-hand side, we get +H0 dv Breaking S into two arbitrary portions and using(B6), we obtain E0·(×Ho)dS-/Ho(xEo)dS dv Now if nx Eo=0 or n x Ho=0 over all of S, or some combination of these conditions holds over all of s. then ar+Ho dBo dv=o (213) This expression implies a relationship between Eo, Do, Bo, and Ho. Since V is arbitrary, we see that one possibility is simply to have Do and Bo constant with time. However since the fields are identically zero for t <0, if they are constant for all time then those constant values must be zero. Another possibility is to have one of each pair (Eo, Do) and(Ho, Bo) equal to zero. Then, by(2.9) and(2.10), Eo =0 implies Bo=0, and Do =0 implies Ho =0. Thus El= E, BI= B2, and so on, and the solution is unique throughout V. However, we cannot in general rule out more complicated relationships The number of possibilities depends on the additional constraints on the relationship between Eo, Do, Bo, and Ho that we must supply to describe the material supporting the field -i. e, the constitutive relationships. For a simple medium described by the time-constant permittivity e and permeability u,(13)becomes aHo 0, 1 a 2 d (∈Eo·Eo+puHo·Ho)dV=0. the integrand is always positive or zero(and not constant with time, as mentioned ) the only possible conclusion is that Eo and Ho must both be zero, and thus the fields are unique. When establishing more complicated constitutive relations, we must be careful to en- sure that they lead to a unique solution, and that the condition for uniqueness is un- derstood. In the case above, the assumption n x Eos=0 implies that the tangential components of E and Eg are identical over S- that is, we must give specific values of these quantities on S to ensure uniqueness. A similar statement holds for the condition f x Hols =0. Requiring that constitutive relations lead to a unique solution is known @2001 by CRC Press LLC

Subtracting again, we have E0 · (∇ × H0) − H0 · (∇ × E0) = H0 · ∂B0 ∂t + E0 · ∂D0 ∂t , hence −∇ · (E0 × H0) = E0 · ∂D0 ∂t + H0 · ∂B0 ∂t by (B.44). Integrating both sides throughout V and using the divergence theorem on the left-hand side, we get −  S (E0 × H0) · dS =  V E0 · ∂D0 ∂t + H0 · ∂B0 ∂t dV. Breaking S into two arbitrary portions and using (B.6), we obtain  S1 E0 · (nˆ × H0) d S −  S2 H0 · (nˆ × E0) d S =  V E0 · ∂D0 ∂t + H0 · ∂B0 ∂t dV. Now if nˆ × E0 = 0 or nˆ × H0 = 0 over all of S, or some combination of these conditions holds over all of S, then  V E0 · ∂D0 ∂t + H0 · ∂B0 ∂t dV = 0. (2.13) This expression implies a relationship between E0, D0, B0, and H0. Since V is arbitrary, we see that one possibility is simply to have D0 and B0 constant with time. However, since the fields are identically zero for t < 0, if they are constant for all time then those constant values must be zero. Another possibility is to have one of each pair (E0, D0) and (H0,B0) equal to zero. Then, by (2.9) and (2.10), E0 = 0 implies B0 = 0, and D0 = 0 implies H0 = 0. Thus E1 = E2, B1 = B2, and so on, and the solution is unique throughout V. However, we cannot in general rule out more complicated relationships. The number of possibilities depends on the additional constraints on the relationship between E0, D0, B0, and H0 that we must supply to describe the material supporting the field — i.e., the constitutive relationships. For a simple medium described by the time-constant permittivity and permeability µ, (13) becomes  V E0 · ∂E0 ∂t + H0 · µ ∂H0 ∂t dV = 0, or 1 2 ∂ ∂t  V ( E0 · E0 + µH0 · H0) dV = 0. Since the integrand is always positive or zero (and not constant with time, as mentioned above), the only possible conclusion is that E0 and H0 must both be zero, and thus the fields are unique. When establishing more complicated constitutive relations, we must be careful to en￾sure that they lead to a unique solution, and that the condition for uniqueness is un￾derstood. In the case above, the assumption nˆ × E0   S = 0 implies that the tangential components of E1 and E2 are identical over S — that is, we must give specific values of these quantities on S to ensure uniqueness. A similar statement holds for the condition nˆ × H0   S = 0. Requiring that constitutive relations lead to a unique solution is known

as just setting, and is one of several factors that must be considered, as discussed in the next section Uniqueness implies that the electromagnetic state of an isolated region of space may be determined without the knowledge of conditions outside the region. If we wish to solve Maxwells equations for that region, we need know only the source density within the region and the values of the tangential fields over the bounding surface. The effects of a complicated external world are thus reduced to the specification of surface fields. This concept has numerous applications to problems in antennas, diffraction, and guided 2.2.2 Constitutive relations We now supply a set of constitutive relations to complete the conditions for well- posedness. We generally split these relations into two sets. The first describes the relationships between the electromagnetic field quantities, and the second describes chanical interaction between the fields and resulting secondary sources. All of these relations depend on the properties of the medium supporting the electromagnetic field Material phenomena are quite diverse, and it is remarkable that the Maxwell-Minkowski equations hold for all phenomena yet discovered. All material effects, from nonlinearity to chirality to temporal dispersion, are described by the constitutive relations. The specification of constitutive relationships is required in many areas of physical science to describe the behavior of "ideal materials " mathematical models of actual materials encountered in nature. For instance, in continuum mechanics the constitutive equations describe the relationship between material motions and stress tensors 209 Truesdell and Toupin [199 give an interesting set of"guiding principles"for the con cerned scientist to use when constructing constitutive relations. These include consider- ation of consistency(with the basic conservation laws of nature), coordinate invariance (independence of coordinate system), isotropy and aeolotropy(dependence on, or inde- pendence of, orientation), just setting(constitutive parameters should lead to a unique solution), dimensional invariance(similarity), material indifference(non-dependence on the observer), and equipresence (inclusion of all relevant physical phenomena in all of the constitutive relations across disciplines) The constitutive relations generally involve a set of constitutive parameters and a set of constitutive operators. The constitutive parameters may be as simple as constants of proportionality between the fields or they may be components in a dyadic relation- ship. The constitutive operators may be linear and integro-differential in nature, or may imply some nonlinear operation on the fields. If the constitutive parameters are spa tially constant within a certain region, we term the medium homogeneous within that region. If the constitutive parameters vary spatially, the medium is inhomogeneous. If the constitutive parameters are constants with time, we term the medium stationary if they are time-changing, the medium is nonstationary. If the constitutive operators involve time derivatives or integrals, the medium is said to be temporally dispersive: if space derivatives or integrals are involved, the medium is spatially dispersive. Examples of all these effects can be found in common materials. It is important to note that the constitutive parameters may depend on other physical properties of the material, such as temperature, mechanical stress, and isomeric state, just as the mechanical constitu tive parameters of a material may depend on the electromagnetic properties(principle of equipresence) Many effects produced by linear constitutive operators, such as those associated with @2001 by CRC Press LLC

as just setting, and is one of several factors that must be considered, as discussed in the next section. Uniqueness implies that the electromagnetic state of an isolated region of space may be determined without the knowledge of conditions outside the region. If we wish to solve Maxwell’s equations for that region, we need know only the source density within the region and the values of the tangential fields over the bounding surface. The effects of a complicated external world are thus reduced to the specification of surface fields. This concept has numerous applications to problems in antennas, diffraction, and guided waves. 2.2.2 Constitutive relations We now supply a set of constitutive relations to complete the conditions for well￾posedness. We generally split these relations into two sets. The first describes the relationships between the electromagnetic field quantities, and the second describes me￾chanical interaction between the fields and resulting secondary sources. All of these relations depend on the properties of the medium supporting the electromagnetic field. Material phenomena are quite diverse, and it is remarkable that the Maxwell–Minkowski equations hold for all phenomena yet discovered. All material effects, from nonlinearity to chirality to temporal dispersion, are described by the constitutive relations. The specification of constitutive relationships is required in many areas of physical science to describe the behavior of “ideal materials”: mathematical models of actual materials encountered in nature. For instance, in continuum mechanics the constitutive equations describe the relationship between material motions and stress tensors [209]. Truesdell and Toupin [199] give an interesting set of “guiding principles” for the con￾cerned scientist to use when constructing constitutive relations. These include consider￾ation of consistency (with the basic conservation laws of nature), coordinate invariance (independence of coordinate system), isotropy and aeolotropy (dependence on, or inde￾pendence of, orientation), just setting (constitutive parameters should lead to a unique solution), dimensional invariance (similarity), material indifference (non-dependence on the observer), and equipresence (inclusion of all relevant physical phenomena in all of the constitutive relations across disciplines). The constitutive relations generally involve a set of constitutive parameters and a set of constitutive operators. The constitutive parameters may be as simple as constants of proportionality between the fields or they may be components in a dyadic relation￾ship. The constitutive operators may be linear and integro-differential in nature, or may imply some nonlinear operation on the fields. If the constitutive parameters are spa￾tially constant within a certain region, we term the medium homogeneous within that region. If the constitutive parameters vary spatially, the medium is inhomogeneous. If the constitutive parameters are constants with time, we term the medium stationary; if they are time-changing, the medium is nonstationary. If the constitutive operators involve time derivatives or integrals, the medium is said to be temporally dispersive; if space derivatives or integrals are involved, the medium is spatially dispersive. Examples of all these effects can be found in common materials. It is important to note that the constitutive parameters may depend on other physical properties of the material, such as temperature, mechanical stress, and isomeric state, just as the mechanical constitu￾tive parameters of a material may depend on the electromagnetic properties (principle of equipresence). Many effects produced by linear constitutive operators, such as those associated with

temporal dispersion, have been studied primarily in the frequency domain. In this case temporal derivative and integral operations produce complex constitutive parameters. It is becoming equally important to characterize these effects directly in the time domain for use with direct time-domain field solving techniques such as the finite-difference time domain(FDTD)method. We shall cover the very basic properties of dispersive media in this section. A detailed description of frequency-domain fields(and a discussion of complex constitutive parameters)is deferred until later in this book It is difficult to find a simple and consistent means for classifying materials by their ectromagnetic effects. One way is to separate linear and nonlinear materials, then cate- gorize linear materials by the way in which the fields are coupled through the constitutive 1. Isotropic materials are those in which D is related to E, B is related to H the secondary source current J is related to E, with the field direction in each 2. In anisotropic materials the pairings are the same, but the fields in each pair are generally not aligned 3. In biisotropic materials(such as chiral media)the fields D and B depend on both E and H, but with no realignment of E or H; for instance, D is given by the addition of a scalar times E plus a second scalar times H. Thus the contributions o d involve no changes to the directions of E and H 4. Bianisotropic materials exhibit the most general behavior: D and H depend on both E and B, with an arbitrary realignment of either or both of these fields In 1888, Roentgen showed experimentally that a material isotropic in its own station ry reference frame exhibits bianisotropic properties when observed from a moving frame Only recently have materials bianisotropic in their own rest frame been discovered. In 1894 Curie predicted that in a stationary material, based on symmetry, an electric field might produce magnetic effects and a magnetic field might produce electric effects. These effects, coined magnetoelectric by Landau and Lifshitz in 1957, were sought unsuccess- fully by many experimentalists during the first half of the twentieth century. In 1959 the Soviet scientist I. E. Dzyaloshinskii predicted that, theoretically, the antiferromagnetic material chromium oxide(Cr2O3)should display magnetoelectric effects. The magneto- electric effect was finally observed soon after by D N. Astrov in a single crystal of Cr2O3 using a 10 kHz electric field. Since then the effect has been observed in many different materials. Recently, highly exotic materials with useful electromagnetic properties have been proposed and studied in depth, including chiroplasmas and chiroferrites 211. As the technology of materials synthesis advances, a host of new and intriguing media will The most general forms of the constitutive relations between the fields may be written in symbolic form as D=DE, B (214) HE, BI That is, D and H have some mathematically descriptive relationship to E and B. The specific forms of the relationships may be written in terms of dyadics as [102 cD=P·E+L H=M·E+Q·(cB)

temporal dispersion, have been studied primarily in the frequency domain. In this case temporal derivative and integral operations produce complex constitutive parameters. It is becoming equally important to characterize these effects directly in the time domain for use with direct time-domain field solving techniques such as the finite-difference time￾domain (FDTD) method. We shall cover the very basic properties of dispersive media in this section. A detailed description of frequency-domain fields (and a discussion of complex constitutive parameters) is deferred until later in this book. It is difficult to find a simple and consistent means for classifying materials by their electromagnetic effects. One way is to separate linear and nonlinear materials, then cate￾gorize linear materials by the way in which the fields are coupled through the constitutive relations: 1. Isotropic materials are those in which D is related to E, B is related to H, and the secondary source current J is related to E, with the field direction in each pair aligned. 2 . In anisotropic materials the pairings are the same, but the fields in each pair are generally not aligned. 3. In biisotropic materials (such as chiral media) the fields D and B depend on both E and H, but with no realignment of E or H; for instance, D is given by the addition of a scalar times E plus a second scalar times H. Thus the contributions to D involve no changes to the directions of E and H. 4. Bianisotropic materials exhibit the most general behavior: D and H depend on both E and B, with an arbitrary realignment of either or both of these fields. In 1888, Roentgen showed experimentally that a material isotropic in its own station￾ary reference frame exhibits bianisotropic properties when observed from a moving frame. Only recently have materials bianisotropic in their own rest frame been discovered. In 1894 Curie predicted that in a stationary material, based on symmetry, an electric field might produce magnetic effects and a magnetic field might produce electric effects. These effects, coined magnetoelectric by Landau and Lifshitz in 1957, were sought unsuccess￾fully by many experimentalists during the first half of the twentieth century. In 1959 the Soviet scientist I.E. Dzyaloshinskii predicted that, theoretically, the antiferromagnetic material chromium oxide (Cr2O3) should display magnetoelectric effects. The magneto￾electric effect was finally observed soon after by D.N. Astrov in a single crystal of Cr2O3 using a 10 kHz electric field. Since then the effect has been observed in many different materials. Recently, highly exotic materials with useful electromagnetic properties have been proposed and studied in depth, including chiroplasmas and chiroferrites [211]. As the technology of materials synthesis advances, a host of new and intriguing media will certainly be created. The most general forms of the constitutive relations between the fields may be written in symbolic form as D = D[E,B], (2.14) H = H[E,B]. (2.15) That is, D and H have some mathematically descriptive relationship to E and B. The specific forms of the relationships may be written in terms of dyadics as [102] cD = P¯ · E + L¯ · (cB), (2.16) H = M¯ · E + Q¯ · (cB), (2.17)