《电磁学》电子书（英文版）Chapter 2 Maxwell’s theory of electromagnetism

Chapter 2 Maxwells theory of electromagnetism 2.1 The postulate In 1864, James Clerk Maxwell proposed one of the history of science. In a famous memoir to the royal B02 ul theories in the ented nir equations summarizing all known laws on electricity and magnetism. This was more than a mere cataloging of the laws of nature. By postulating the need for an additional term to make the set of equations self-consistent, Maxwell was able to put forth what is still considered a complete theory of macroscopic electromagnetism. The beauty of Maxwell's equations led Boltzmann to ask, Was it a god who wrote these lines.? Since that time authors have struggled to find the best way to present Maxwells theory. Although it is possible to study electromagnetics from an"empirical-inductive viewpoint(roughly following the historical order of development beginning with static fields), it is only by postulating the complete theory that we can do justice to Maxwell,s vision. His concept of the existence of an electromagnetic "field"(as introduced by Faraday) is fundamental to this theory, and has become one of the most significant principles of modern scie We find controversy even over the best way to present Maxwells equations. Maxwell worked at a time before vector notation was completely in place, and thus chose to use scalar variables and equations to represent the fields. Certainly the true beauty of Maxwell's equations emerges when they are written in vector form, and the use of tensors reduces the equations to their underlying physical simplicity. We shall use vector otation in this book because of its wide acceptance by engineers, but we still must decide whether it is more appropriate to present the vector equations in integral or point On one side of this debate, the brilliant mathematician David Hilbert felt that the fundamental natural laws should be posited as axioms, each best described in terms of integral equations [154]. This idea has been championed by Truesdell and Toupin [199. On the other side, we may quote from the great physicist Arnold Sommerfeld "The general development of Maxwells theory must proceed from its differential form; for special problems the integral form may, however, be more advantageous"(185, p 23). Special relativity flows naturally from the point forms, with fields easily converted between moving reference frames. For stationary media, it seems to us that the only difference between the two approaches arises in how we handle discontinuities in sources and materials. If we choose to use the point forms of Maxwells equations, then we must also postulate the boundary conditions at surfaces of discontinuity. This is pointed out @2001 by CRC Press LLC

Chapter 2 Maxwell’s theory of electromagnetism 2.1 The postulate In 1864, James Clerk Maxwell proposed one of the most successful theories in the history of science. In a famous memoir to the Royal Society [125] he presented nine equations summarizing all known laws on electricity and magnetism. This was more than a mere cataloging of the laws of nature. By postulating the need for an additional term to make the set of equations self-consistent, Maxwell was able to put forth what is still considered a complete theory of macroscopic electromagnetism. The beauty of Maxwell’s equations led Boltzmann to ask, “Was it a god who wrote these lines ... ?” [185]. Since that time authors have struggled to find the best way to present Maxwell’s theory. Although it is possible to study electromagnetics from an “empirical–inductive” viewpoint (roughly following the historical order of development beginning with static fields), it is only by postulating the complete theory that we can do justice to Maxwell’s vision. His concept of the existence of an electromagnetic “field” (as introduced by Faraday) is fundamental to this theory, and has become one of the most significant principles of modern science. We find controversy even over the best way to present Maxwell’s equations. Maxwell worked at a time before vector notation was completely in place, and thus chose to use scalar variables and equations to represent the fields. Certainly the true beauty of Maxwell’s equations emerges when they are written in vector form, and the use of tensors reduces the equations to their underlying physical simplicity. We shall use vector notation in this book because of its wide acceptance by engineers, but we still must decide whether it is more appropriate to present the vector equations in integral or point form. On one side of this debate, the brilliant mathematician David Hilbert felt that the fundamental natural laws should be posited as axioms, each best described in terms of integral equations [154]. This idea has been championed by Truesdell and Toupin [199]. On the other side, we may quote from the great physicist Arnold Sommerfeld: “The general development of Maxwell’s theory must proceed from its differential form; for special problems the integral form may, however, be more advantageous” ([185], p. 23). Special relativity flows naturally from the point forms, with fields easily converted between moving reference frames. For stationary media, it seems to us that the only difference between the two approaches arises in how we handle discontinuities in sources and materials. If we choose to use the point forms of Maxwell’s equations, then we must also postulate the boundary conditions at surfaces of discontinuity. This is pointed out

clearly by Tai [ 192, who also notes that if the integral forms are used, then their validity across regions of discontinuity should be stated as part of the postulate We have decided to use the point form in this text. In doing so we follow a long history begun by Hertz in 1890[85 when he wrote down Maxwell's differential equations as a set of axioms, recognizing the equations as the launching point for the theory of electromagnetism. Also, by postulating Maxwell's equations in point form we can take full advantage of modern developments in the theory of partial differential equations; in particular, the idea of a "well-posed" theory determines what sort of information must be specified to make the postulate useful. We must also decide which form of Maxwells differential equations to use as the basis of our postulate. There are several competing forms, each differing on the manner in which materials are considered. The oldest and most widely used form was suggeste by Minkowski in 1908 [130. In the Minkowski form the differential equations contain no mention of the materials supporting the fields; all information about material media is relegated to the constitutive relationships. This places simplicity of the differential equations above intuitive understanding of the behavior of fields in materials. We choose the Maxwell-Minkowski form as the basis of our postulate, primarily for ease of ma- nipulation. But we also recognize the value of other versions of Maxwells equations We shall present the basic ideas behind the Boffi form, which places some information about materials into the differential equations(although constitutive relationships are still required). Missing, however, is any information regarding the velocity of a moving medium. By using the polarization and magnetization vectors P and M rather than the fields D and H, it is sometimes easier to visualize the meaning of the field vectors and to understand(or predict) the nature of the constitutive relations. The Chu and Amperian forms of Maxwells equations have been promoted as useful alternatives to the Minkowski and Boffi forms. These include explicit information about the velocity of a moving material, and differ somewhat from the boffi form in the physical interpretation of the electric and magnetic properties of matter. Although each of these models matter in terms of charged particles immersed in free space, magnetization in the Boffi and Amperian forms arises from electric current loops, while the Chu form employs nagnetic dipoles. In all three forms polarization is modeled using electric dipoles. For a detailed discussion of the Chu and Amperian forms, the reader should consult the work of Kong [101, Tai 193, Penfield and Haus [145, or Fano, Chu and Adler [70 o Importantly, all of these various forms of Maxwell's equations produce the same values the physical fields(at least external to the material where the fields are measurable) We must include several other constituents, besides the field equations, to make the ostulate complete. To form a complete field theory we need a source field, a mediating field, and a set of field differential equations. This allows us to mathematically describe the relationship between effect(the mediating field) and cause(the source field).In a well-posed postulate we must also include a set of constitutive relationships and a specification of some field relationship over a bounding surface and at an initial time. If the electromagnetic field is to have physical meaning, we must link it to some observable quantity such as force. Finally, to allow the solution of problems involving mathematical discontinuities we must specify certain boundary, or jump, " conditions. 2.1.1 The Maxwell-Minkowski equations In Maxwell,s macroscopic theory of electromagnetics, the source field consists of the ector field J(r, t)(the current density) and the scalar field p(r, t)(the charge density) @2001 by CRC Press LLC

clearly by Tai [192], who also notes that if the integral forms are used, then their validity across regions of discontinuity should be stated as part of the postulate. We have decided to use the point form in this text. In doing so we follow a long history begun by Hertz in 1890 [85] when he wrote down Maxwell’s differential equations as a set of axioms, recognizing the equations as the launching point for the theory of electromagnetism. Also, by postulating Maxwell’s equations in point form we can take full advantage of modern developments in the theory of partial differential equations; in particular, the idea of a “well-posed” theory determines what sort of information must be specified to make the postulate useful. We must also decide which form of Maxwell’s differential equations to use as the basis of our postulate. There are several competing forms, each differing on the manner in which materials are considered. The oldest and most widely used form was suggested by Minkowski in 1908 [130]. In the Minkowski form the differential equations contain no mention of the materials supporting the fields; all information about material media is relegated to the constitutive relationships. This places simplicity of the differential equations above intuitive understanding of the behavior of fields in materials. We choose the Maxwell–Minkowski form as the basis of our postulate, primarily for ease of ma￾nipulation. But we also recognize the value of other versions of Maxwell’s equations. We shall present the basic ideas behind the Boffi form, which places some information about materials into the differential equations (although constitutive relationships are still required). Missing, however, is any information regarding the velocity of a moving medium. By using the polarization and magnetization vectors P and M rather than the fields D and H, it is sometimes easier to visualize the meaning of the field vectors and to understand (or predict) the nature of the constitutive relations. The Chu and Amperian forms of Maxwell’s equations have been promoted as useful alternatives to the Minkowski and Boffi forms. These include explicit information about the velocity of a moving material, and differ somewhat from the Boffi form in the physical interpretation of the electric and magnetic properties of matter. Although each of these models matter in terms of charged particles immersed in free space, magnetization in the Boffi and Amperian forms arises from electric current loops, while the Chu form employs magnetic dipoles. In all three forms polarization is modeled using electric dipoles. For a detailed discussion of the Chu and Amperian forms, the reader should consult the work of Kong [101], Tai [193], Penfield and Haus [145], or Fano, Chu and Adler [70]. Importantly, all of these various forms of Maxwell’s equations produce the same values of the physical fields (at least external to the material where the fields are measurable). We must include several other constituents, besides the field equations, to make the postulate complete. To form a complete field theory we need a source field, a mediating field, and a set of field differential equations. This allows us to mathematically describe the relationship between effect (the mediating field) and cause (the source field). In a well-posed postulate we must also include a set of constitutive relationships and a specification of some field relationship over a bounding surface and at an initial time. If the electromagnetic field is to have physical meaning, we must link it to some observable quantity such as force. Finally, to allow the solution of problems involving mathematical discontinuities we must specify certain boundary, or “jump,” conditions. 2.1.1 The Maxwell–Minkowski equations In Maxwell’s macroscopic theory of electromagnetics, the source field consists of the vector field J(r, t) (the current density) and the scalar field ρ(r, t) (the charge density)

In Minkowski's form of Maxwell's equations, the mediating field is the electromagnetic field consisting of the set of four vector fields e(r, t), D(r, 1), B(r, t), and H(r, t). The field equations are the four partial differential equations referred to as the Marwell-Minkowski V×E(r,t) B(r, t) VXH(r,t)=J(r, t)+oD(r, t), V·D(r,t)=p(r,t) (23) along with the continuity equation J(r, t) P(r, t) Here(2.1)is called Faradays law,(2.2)is called Ampere's law, (2.3)is called Gauss's w, and(2. 4)is called the magnetic Gauss's law. For brevity we shall often leave the dependence on r and t implicit, and refer to the Maxwell-Minkowski equations as simply the "Maxwell equations, or"Maxwells equations Equations(2. 1)-(2.5), the point forms of the field equations, describe the relation ships between the fields and their sources at each point in space where the fields are continuously differentiable(i.e, the derivatives exist and are continuous). Such points are called ordinary points. We shall not attempt to define the fields at other points, but instead seek conditions relating the fields across surfaces containing these point Normally this is necessary on surfaces across which either sources or material parameters e discontinuous The electromagnetic fields carry SI units as follows: E is measured in Volts per meter (V/m), B is measured in Teslas(T), H is measured in Amperes per meter(A/m), and D is measured in Coulombs per square meter(C/m"). In older texts we find the units of B given as Webers per square meter(Wb/m2)to reflect the role of B as a flux vector; in that case the Weber(wb=T-m)is regarded as a unit of magnetic flux The interdependence of Maxwells equations. It is often claimed that the diver gence equations(2.3)and(2. 4)may be derived from the curl equations(2.1)and(2.2) While this is true, it is not proper to say that only the two curl equations are required to describe Maxwells theory. This is because an additional physical assumption, not present in the two curl equations, is required to complete the derivation. Either the divergence equations must be specified, or the values of certain constants that fix the initial conditions on the fields must be specified. It is customary to specify the divergence equations and include them with the curl equations to form the complete set we now call To identify the interdependence we take the divergence of(2. 1)to get V·(×E)=V (V·B)=0 @2001 by CRC Press LLC

In Minkowski’s form of Maxwell’s equations, the mediating field is the electromagnetic field consisting of the set of four vector fields E(r, t), D(r, t), B(r, t), and H(r, t). The field equations are the four partial differential equations referred to as the Maxwell–Minkowski equations ∇ × E(r, t) = − ∂ ∂t B(r, t), (2.1) ∇ × H(r, t) = J(r, t) + ∂ ∂t D(r, t), (2.2) ∇ · D(r, t) = ρ(r, t), (2.3) ∇ · B(r, t) = 0, (2.4) along with the continuity equation ∇ · J(r, t) = − ∂ ∂t ρ(r, t). (2.5) Here (2.1) is called Faraday’s law, (2.2) is called Ampere’s law, (2.3) is called Gauss’s law, and (2.4) is called the magnetic Gauss’s law. For brevity we shall often leave the dependence on r and t implicit, and refer to the Maxwell–Minkowski equations as simply the “Maxwell equations,” or “Maxwell’s equations.” Equations (2.1)–(2.5), the point forms of the field equations, describe the relation￾ships between the fields and their sources at each point in space where the fields are continuously differentiable (i.e., the derivatives exist and are continuous). Such points are called ordinary points. We shall not attempt to define the fields at other points, but instead seek conditions relating the fields across surfaces containing these points. Normally this is necessary on surfaces across which either sources or material parameters are discontinuous. The electromagnetic fields carry SI units as follows: E is measured in Volts per meter (V/m), B is measured in Teslas (T), H is measured in Amperes per meter (A/m), and D is measured in Coulombs per square meter (C/m2). In older texts we find the units of B given as Webers per square meter (Wb/m2) to reflect the role of B as a flux vector; in that case the Weber (Wb = T·m2) is regarded as a unit of magnetic flux. The interdependence of Maxwell’s equations. It is often claimed that the diver￾gence equations (2.3) and (2.4) may be derived from the curl equations (2.1) and (2.2). While this is true, it is not proper to say that only the two curl equations are required to describe Maxwell’s theory. This is because an additional physical assumption, not present in the two curl equations, is required to complete the derivation. Either the divergence equations must be specified, or the values of certain constants that fix the initial conditions on the fields must be specified. It is customary to specify the divergence equations and include them with the curl equations to form the complete set we now call “Maxwell’s equations.” To identify the interdependence we take the divergence of (2.1) to get ∇ · (∇ × E) =∇· −∂B ∂t , hence ∂ ∂t (∇ · B) = 0

by(B 49). This requires that V. B be constant with time, say V. B(r, t)= CB(r) The constant CB must be specified as part of the postulate of Maxwells theory, and the choice we make is subject to experimental validation. We postulate that CB(r=0, which leads us to(2. 4). Note that if we can identify a time prior to which B(r, t)=0 then CB(r) must vanish. For this reason, CB(r)=0 and(2.4)are often called the"initial conditions"for Faraday's law [159. Next we take the divergence of (2.2) to find that v·(×H=V·J+-(V·D Using(2.5)and(B49), we obtain and thus p-V. D must be some temporal constant Cp(r). Again, we must postulate the value of C as part of the Maxwell theory. We choose Cp(r)=0 and thus obtain Gauss's law(2.3). If we can identify a time prior to which both D and p are everywhere ual to zero, then CD(r)must vanish. Hence CD(r)=0 and (2.3) may be regarded as "initial conditions"for Ampere's law. Combining the two sets of initial conditions we find that the curl equations imply the divergence equations as long as we can find a time prior to which all of the fields e, D, B, h and the sources j and p are equal to zero ( since all the fields are related through the curl equations, and the charge and current are elated through the continuity equation). Conversely, the empirical evidence supporting the two divergence equations implies that such a time should exist Throughout this book we shall refer to the two curl equations as the "fundamental Maxwell equations, and to the two divergence equations as the "auxiliary"equations The fundamental equations describe the relationships between the fields while, as we have seen, the auxiliary equations provide a sort of initial condition. This does not imply that the auxiliary equations are of lesser importance; indeed, they are required to establish uniqueness of the fields, to derive the wave equations for the fields, and to properly describe static fields. Field vector terminology. Various terms are used for the field vectors, sometimes harkening back to the descriptions used by Maxwell himself, and often based on the physical nature of the fields. We are attracted to Sommerfeld,'s separation of the fields into entities of intensity(E, B)and entities of quantity(D, H). In this system E is called the electric field strength, B the magnetic field strength, D the electric excitation, and H the magnetic excitation [ 185. Maxwell separated the fields into a set(e, h) of vectors that appear within line integrals to give work-related quantities, and a set (B, D) of vectors that appear within surface integrals to give flux-related quantities: we shall see this clearly when considering the integral forms of Maxwells equations. By this system authors such as Jones 97] and Ramo, Whinnery, and Van Duzer [153 call e the electric intensity, H the magnetic intensity, b the magnetic fiur density, and d the electric fur Maxwell himself designated names for each of the vector quantities. In his classic paper A Dynamical Theory of the Electromagnetic Field, [178 Maxwell referred to the quantity we now designate e as the electromotive force, the quantity d as the elec- tric displacement (with a time rate of change given by his now famous "displacement current"), the quantity H as the magnetic force, and the quantity B as the magnetic @2001 by CRC Press LLC

by (B.49). This requires that ∇ · B be constant with time, say ∇ · B(r, t) = CB(r). The constant CB must be specified as part of the postulate of Maxwell’s theory, and the choice we make is subject to experimental validation. We postulate that CB(r) = 0, which leads us to (2.4). Note that if we can identify a time prior to which B(r, t) ≡ 0, then CB(r) must vanish. For this reason, CB(r) = 0 and (2.4) are often called the “initial conditions” for Faraday’s law [159]. Next we take the divergence of (2.2) to find that ∇ · (∇ × H) =∇· J + ∂ ∂t (∇ · D). Using (2.5) and (B.49), we obtain ∂ ∂t (ρ −∇· D) = 0 and thus ρ −∇· D must be some temporal constant CD(r). Again, we must postulate the value of CD as part of the Maxwell theory. We choose CD(r) = 0 and thus obtain Gauss’s law (2.3). If we can identify a time prior to which both D and ρ are everywhere equal to zero, then CD(r) must vanish. Hence CD(r) = 0 and (2.3) may be regarded as “initial conditions” for Ampere’s law. Combining the two sets of initial conditions, we find that the curl equations imply the divergence equations as long as we can find a time prior to which all of the fields E, D,B, H and the sources J and ρ are equal to zero (since all the fields are related through the curl equations, and the charge and current are related through the continuity equation). Conversely, the empirical evidence supporting the two divergence equations implies that such a time should exist. Throughout this book we shall refer to the two curl equations as the “fundamental” Maxwell equations, and to the two divergence equations as the “auxiliary” equations. The fundamental equations describe the relationships between the fields while, as we have seen, the auxiliary equations provide a sort of initial condition. This does not imply that the auxiliary equations are of lesser importance; indeed, they are required to establish uniqueness of the fields, to derive the wave equations for the fields, and to properly describe static fields. Field vector terminology. Various terms are used for the field vectors, sometimes harkening back to the descriptions used by Maxwell himself, and often based on the physical nature of the fields. We are attracted to Sommerfeld’s separation of the fields into entities of intensity (E,B) and entities of quantity (D, H). In this system E is called the electric field strength, B the magnetic field strength, D the electric excitation, and H the magnetic excitation [185]. Maxwell separated the fields into a set (E, H) of vectors that appear within line integrals to give work-related quantities, and a set (B, D) of vectors that appear within surface integrals to give flux-related quantities; we shall see this clearly when considering the integral forms of Maxwell’s equations. By this system, authors such as Jones [97] and Ramo, Whinnery, and Van Duzer [153] call E the electric intensity, H the magnetic intensity, B the magnetic flux density, and D the electric flux density. Maxwell himself designated names for each of the vector quantities. In his classic paper “A Dynamical Theory of the Electromagnetic Field,” [178] Maxwell referred to the quantity we now designate E as the electromotive force, the quantity D as the elec￾tric displacement (with a time rate of change given by his now famous “displacement current”), the quantity H as the magnetic force, and the quantity B as the magnetic

induction(although he described B as a density of lines of magnetic force). Maxwell also included a quantity designated electromagnetic momentum as an integral part of his theory. We now know this as the vector potential a which is not generally included as a part of the electromagnetics postulate. Many authors follow the original terminology of Maxwell, with some slight modifica- tions. For instance, Stratton [187 calls e the electric field intensity, H the magneti field intensity, D the electric displacement, and B the magnetic induction. Jackson 91 calls e the electric field, H the magnetic field, D the displacement, and B the magnetic induction Other authors choose freely among combinations of these terms. For instance, Kong L01] calls E the electric field strength, H the magnetic field strength, B the magnetic fur density, and d the electric displacement. We do not wish to inject further confusion into the issue of nomenclature; still, we find it helpful to use as simple a naming system as possible. We shall refer to E as the electric field, H as the magnetic field, D as the electric flur density and b as the magnetic fur density. When we use the term electromagnetic field we imply the entire set of field vectors(E, D, B, H) used in Maxwell's theory Invariance of Maxwells equations. Maxwells differential equations are valid for any system in uniform relative motion with respect to the laboratory frame of reference in which we normally do our measurements. The field equations describe the relationships between the source and mediating fields within that frame of reference. This property was first proposed for moving material media by Minkowski in 1908(using the term covariance)[130. For this reason, Maxwell's equations expressed in the form(2. 1)-(2.2) are referred to as the Minkowski form. 2.1.2 Connection to mechanics Our postulate must include a connection between the abstract quantities of charge and field and a measurable physical quantity. A convenient means of linking electromagnetics to other classical theories is through mechanics. We postulate that charges experience mechanical forces given by the Lorentz force equation. If a small volume element dV contains a total charge p dv, then the force experienced by that charge when moving at velocity v in an electromagnetic field is dF=pdVE+pvdV×B. As with any postulate, we verify this equation through experiment. Note that we write the lorentz force in terms of charge pdV, rather than charge density p, since charge n invariant quantity under a Lorentz transformation The important links between the electromagnetic fields and energy and momentum must also be postulated. We postulate that the quantity Sm=E×H represents the transport density of electromagnetic power, and that the quantity gm=D×B represents the transport density of electromagnetic momentum. @2001 by CRC Press LLC

induction (although he described B as a density of lines of magnetic force). Maxwell also included a quantity designated electromagnetic momentum as an integral part of his theory. We now know this as the vector potential A which is not generally included as a part of the electromagnetics postulate. Many authors follow the original terminology of Maxwell, with some slight modifica￾tions. For instance, Stratton [187] calls E the electric field intensity, H the magnetic field intensity, D the electric displacement, and B the magnetic induction. Jackson [91] calls E the electric field, H the magnetic field, D the displacement, and B the magnetic induction. Other authors choose freely among combinations of these terms. For instance, Kong [101] calls E the electric field strength, H the magnetic field strength, B the magnetic flux density, and D the electric displacement. We do not wish to inject further confusion into the issue of nomenclature; still, we find it helpful to use as simple a naming system as possible. We shall refer to E as the electric field, H as the magnetic field, D as the electric flux density and B as the magnetic flux density. When we use the term electromagnetic field we imply the entire set of field vectors (E, D,B, H) used in Maxwell’s theory. Invariance of Maxwell’s equations. Maxwell’s differential equations are valid for any system in uniform relative motion with respect to the laboratory frame of reference in which we normally do our measurements. The field equations describe the relationships between the source and mediating fields within that frame of reference. This property was first proposed for moving material media by Minkowski in 1908 (using the term covariance) [130]. For this reason, Maxwell’s equations expressed in the form (2.1)–(2.2) are referred to as the Minkowski form. 2.1.2 Connection to mechanics Our postulate must include a connection between the abstract quantities of charge and field and a measurable physical quantity. A convenient means of linking electromagnetics to other classical theories is through mechanics. We postulate that charges experience mechanical forces given by the Lorentz force equation. If a small volume element dV contains a total charge ρ dV, then the force experienced by that charge when moving at velocity v in an electromagnetic field is dF = ρ dV E + ρv dV × B. (2.6) As with any postulate, we verify this equation through experiment. Note that we write the Lorentz force in terms of charge ρ dV, rather than charge density ρ, since charge is an invariant quantity under a Lorentz transformation. The important links between the electromagnetic fields and energy and momentum must also be postulated. We postulate that the quantity Sem = E × H (2.7) represents the transport density of electromagnetic power, and that the quantity gem = D × B (2.8) represents the transport density of electromagnetic momentum

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 t0, 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, 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)