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 manipulation. 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 relationships 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 divergence 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 electric 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 modifications. 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