Kong, J.A."Electromagnetic Fields The Electrical Engineering Handbook Ed. Richard C. Dorf Boca raton crc Press llc. 2000
Kong, J.A. “Electromagnetic Fields” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
35 Electromagnetic Fields 35.1 Maxwell Equations 35.2 Constitutive Relations Anisotropic and Bianisotropic Media. Biisotropic Jin Au Kong Media.Constitutive matrices Massachusetts Institute 35.3 Wave Equations and Wave Solutions Wave solution· Wave Vector k· Wavenumbers k 35.1 Maxwell Equations The fundamental equations of electromagnetic theory were established by James Clerk Maxwell in 1873. In three-dimensional vector notation, the Maxwell equations are V×E(r,t)+B(f,t) (35.1) vXHG,D-ar DG, D)=/, D) (35.2) V·B(,t)=0 (35 v·D(F,t)=p(,t) (354) where E, B, H, D,J, and p are real functions of position and time E(r,t)=electric field strength (volts/m) B(r, t)=magnetic flux density (v H(F, t)=magnetic field strength (amperes/m) D(r, t)=electric displacement (coulombs/m2) IF, t)=electric current density(ampere p(f, t)=electric charge density(coulombs/m) This chapter is an abridged version of Chapter 1 in Electromagnetic Wave Theory (. A. Kong), New York: Wiley Interscience, 1990. c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 35 Electromagnetic Fields 35.1 Maxwell Equations 35.2 Constitutive Relations Anisotropic and Bianisotropic Media • Biisotropic Media • Constitutive Matrices 35.3 Wave Equations and Wave Solutions Wave Solution • Wave Vector k • Wavenumbers k 35.1 Maxwell Equations1 The fundamental equations of electromagnetic theory were established by James Clerk Maxwell in 1873. In three-dimensional vector notation, the Maxwell equations are (35.1) (35.2) (35.3) (35.4) where , , , , , and r are real functions of position and time. 1 This chapter is an abridged version of Chapter 1 in Electromagnetic Wave Theory (J. A. Kong), New York: WileyInterscience, 1990. —¥ + = Ert t (,) (,) Brt ¶ ¶ 0 —¥ = Hrt t ( , )– ( , ) ( , ) Drt Jrt ¶ ¶ —× = Brt (,) 0 —× = Drt rt (,) (,) r E – B – H – D – J Er t Br t Hr t Dr t Jr t r t , , , , ) , ) , ) ( ) = ( ) = ( ) = ( ) = ( ) = ( ) = electric field strength (volts/m) magnetic flux density (webers/m ) magnetic field strength (amperes/m) electric displacement (coulombs/m electric current density (amperes/m electric charge density (coulombs/m 2 2 2 3 r Jin Au Kong Massachusetts Institute of Technology
Equation(35. 1)is Faraday's induction law Equation (35.2)is the generalized Ampere's circuit law Equations (35.3)and(35.4)are Gauss' laws for magnetic and electric fields. Taking the divergence of (35. 2)and intro- ducing(35.4), we find that v·J(F,t)+p(7,t)=0 (35.5 at This is the conservation law for electric charge and current densities. Regarding (35.5) as a fundamental quation, we can use it to derive(35.4)by taking the divergence of (35.2).Equation(35.3 )can also be derived by taking the divergence of(35. 1)which gives a(v- B(, n))at=0 or that V. B(T, t)is a constant independent of time. Such a constant, if not zero, then implies the existence of magnetic monopoles similar to free electric charges. Since magnetic monopoles have not been found to exist, this constant must be zero and we arrive at (35.3) 35.2 Constitutive relations The Maxwell equations are fundamental laws governing the behavior of electromagnetic fields in free spad and in media. We have so far made no reference to the various material properties that provide connections to other disciplines of physics, such as plasma physics, continuum mechanics, solid-state physics, fluid dynamics, statistical physics, thermodynamics, biophysics, etc, all of which interact in one way or another with electro- magnetic fields. We did not even mention the Lorentz force law, which constitutes a direct link to mechanics It is time to state how we are going to account for this vast"outside "world. From the electromagnetic way point of view, we shall be interested in how electromagnetic fields behave in the presence of media, whether the wave is diffracted, refracted, or scattered. Whatever happens to a medium, whether it is moved or deformed of secondary interest. Thus we shall characterize material media by the so-called constitutive relations that can be classified according to the various properties of the media The necessity of using constitutive relations to supplement the Maxwell equations is clear from the following mathematical observations. In most problems we shall assume that sources of electromagnetic fields are given Thus and p are known and they satisfy the conservation law (35.5). Let us examine the Maxwell equations and see if there are enough equations for the number of unknown quantities. There are a total of 12 scalar unknowns for the four field vectors E, H, B, and D. As we have learned, Eqs. (35.3)and(35.4)are not dependent equations; they can be derived from Eqs.(35. 1),(35. 2), and (35.5). The independent equations are Eqs.(35. 1)and (35. 2), which constitute six scalar equations. Thus we need six more scalar equations. These are the constitutive relations The constitutive relations for an isotropic medium can be written simply as D=EE where e permittivity (356a) B=uH where u permittivity By isotropy we mean that the field vector E is parallel to D and the field vector His parallel to B In free space void of any matter, u=μand∈=∈ μ=4×107 henry/ meter ∈n=8.85×10-12 farad/ meter Inside a material medium, the permittivity e is determined by the electrical properties of the medium and the permeability u by the magnetic properties of the medium. c 2000 by CRC Press LLC
© 2000 by CRC Press LLC Equation (35.1) is Faraday’s induction law. Equation (35.2) is the generalized Ampere’s circuit law. Equations (35.3) and (35.4) are Gauss’ laws for magnetic and electric fields. Taking the divergence of (35.2) and introducing (35.4), we find that (35.5) This is the conservation law for electric charge and current densities. Regarding (35.5) as a fundamental equation, we can use it to derive (35.4) by taking the divergence of (35.2). Equation (35.3) can also be derived by taking the divergence of (35.1) which gives ¶(— · ( ,t))/¶t = 0 or that — · ( ,t) is a constant independent of time. Such a constant, if not zero, then implies the existence of magnetic monopoles similar to free electric charges. Since magnetic monopoles have not been found to exist, this constant must be zero and we arrive at (35.3). 35.2 Constitutive Relations The Maxwell equations are fundamental laws governing the behavior of electromagnetic fields in free space and in media. We have so far made no reference to the various material properties that provide connections to other disciplines of physics, such as plasma physics, continuum mechanics, solid-state physics, fluid dynamics, statistical physics, thermodynamics, biophysics, etc., all of which interact in one way or another with electromagnetic fields. We did not even mention the Lorentz force law, which constitutes a direct link to mechanics. It is time to state how we are going to account for this vast “outside” world. From the electromagnetic wave point of view, we shall be interested in how electromagnetic fields behave in the presence of media, whether the wave is diffracted, refracted, or scattered. Whatever happens to a medium, whether it is moved or deformed, is of secondary interest. Thus we shall characterize material media by the so-called constitutive relations that can be classified according to the various properties of the media. The necessity of using constitutive relations to supplement the Maxwell equations is clear from the following mathematical observations. In most problems we shall assume that sources of electromagnetic fields are given. Thus and r are known and they satisfy the conservation law (35.5). Let us examine the Maxwell equations and see if there are enough equations for the number of unknown quantities. There are a total of 12 scalar unknowns for the four field vectors , , , and . As we have learned, Eqs. (35.3) and (35.4) are not independent equations; they can be derived from Eqs. (35.1), (35.2), and (35.5). The independent equations are Eqs. (35.1) and (35.2), which constitute six scalar equations. Thus we need six more scalar equations. These are the constitutive relations. The constitutive relations for an isotropic medium can be written simply as (35.6a) (35.6b) By isotropy we mean that the field vector is parallel to and the field vector is parallel to . In free space void of any matter, m = mo and e = eo , mo = 4p ¥ 10–7 henry/meter eo ª 8.85 ¥ 10–12 farad/meter Inside a material medium, the permittivity e is determined by the electrical properties of the medium and the permeability m by the magnetic properties of the medium. — × J r t + = t ( , ) (r,t) ¶ ¶ r 0 B – r – B – r – J E – H – B – D – D = eE where e = permittivity B = mH where m = permittivity E – D – H – B –
dielectric material can be described by a free-space part and a part that is due to the material alone. The material part can be characterized by a polarization vector P such that d=E, E+ P. The polarization symbolizes the electric dipole moment per unit volume of the dielectric material. In the presence of an external electric field, the polarization vector may be caused by induced dipole moments, alignment of the permanent dipole moments of the medium, or migration of ionic charges magnetic material can also be described by a free-space part and a part characterized by a magnetization vector M such that b=μ。H+μM. A medium is diamagnetic ifμ≤ ul, and paramagnetic ifμ≥pa Diamagnetism is caused by induced magnetic moments that tend to oppose the externally applied magnetic field. Paramagnetism is due to alignment of magnetic moments When placed in an inhomogeneous magnetic field, a diamagnetic material tends to move toward regions of weaker magnetic field and a paramagnetic material toward regions of stronger magnetic field. Ferromagnetism and antiferromagnetism are highly nonlinear effects. Ferromagnetic substances are characterized by spontaneous magnetization below the Curie temperature. The medium also depends on the history of applied fields, and in many instances the magnetization curve forms a hysteresis loop. In an antiferromagnetic material, the spins form sublattices that become spontaneousl magnetized in an antiparallel arrangement below the Neel temperature Anisotropic and Bianisotropic Media The constitutive relations for anisotropic media are usually written as permittivity tensor (35.7a) B=u- H where u= permeability tensor (357b) The field vector E is no longer parallel to D, and the field vector H is no longer parallel to B A medium is electrically anisotropic if it is described by the permittivity tensor E and a scalar permeability u, and magnetically anisotropic if it is described by the permeability tensor u and a scalar permittivity E. Note that a medium can be both electrically and magnetically anisotropic as described by both E and A in Eq (35.7) Crystals are described in general by symmetric permittivity tensors. There always exists a coordinate trans- formation that transforms a symmetric matrix into a diagonal matrix. In this coordinate system, called the 00 (35.8) 00∈ The three coordinate axes are referred to as the principal axes of the crystal. For cubic crystals,eE, E, and they are isotropic. In tetragonal, hexagonal, and rhombohedral crystals, two of the three parameters are equal. Such crystals are uniaxial. Here there is a two-dimensional degeneracy, the principal axis that exhibits this anisotropy is called the optic axis. For a uniaxial crystal with 00 ∈=0∈0 (359) the z axis is the optic axis. The crystal is positive uniaxialifE2>E it is negative uniaxialif E, <E In orthorhombic, monoclinic, and triclinic crystals, all three crystallographic axes are unequal. We hav ∈,an medium is biaxial c 2000 by CRC Press LLC
© 2000 by CRC Press LLC A dielectric material can be described by a free-space part and a part that is due to the material alone. The material part can be characterized by a polarization vector such that = e o + . The polarization symbolizes the electric dipole moment per unit volume of the dielectric material. In the presence of an external electric field, the polarization vector may be caused by induced dipole moments, alignment of the permanent dipole moments of the medium, or migration of ionic charges. A magnetic material can also be described by a free-space part and a part characterized by a magnetization vector such that = m o + mo . A medium is diamagnetic if m £ mo and paramagnetic if m ³ mo . Diamagnetism is caused by induced magnetic moments that tend to oppose the externally applied magnetic field. Paramagnetism is due to alignment of magnetic moments. When placed in an inhomogeneous magnetic field, a diamagnetic material tends to move toward regions of weaker magnetic field and a paramagnetic material toward regions of stronger magnetic field. Ferromagnetism and antiferromagnetism are highly nonlinear effects. Ferromagnetic substances are characterized by spontaneous magnetization below the Curie temperature. The medium also depends on the history of applied fields, and in many instances the magnetization curve forms a hysteresis loop. In an antiferromagnetic material, the spins form sublattices that become spontaneously magnetized in an antiparallel arrangement below the Néel temperature. Anisotropic and Bianisotropic Media The constitutive relations for anisotropic media are usually written as = = e · where = e = permittivity tensor (35.7a) = = m · where = m = permeability tensor (35.7b) The field vector is no longer parallel to , and the field vector is no longer parallel to . A medium is electrically anisotropic if it is described by the permittivity tensor = e and a scalar permeability m, and magnetically anisotropic if it is described by the permeability tensor = m and a scalar permittivity e. Note that a medium can be both electrically and magnetically anisotropic as described by both = e and = m in Eq. (35.7). Crystals are described in general by symmetric permittivity tensors. There always exists a coordinate transformation that transforms a symmetric matrix into a diagonal matrix. In this coordinate system, called the principal system, (35.8) The three coordinate axes are referred to as the principal axes of the crystal. For cubic crystals, ex = ey = ez and they are isotropic. In tetragonal, hexagonal, and rhombohedral crystals, two of the three parameters are equal. Such crystals are uniaxial. Here there is a two-dimensional degeneracy; the principal axis that exhibits this anisotropy is called the optic axis. For a uniaxial crystal with (35.9) the z axis is the optic axis. The crystal is positive uniaxial if ez > e; it is negative uniaxial if ez < e. In orthorhombic, monoclinic, and triclinic crystals, all three crystallographic axes are unequal. We have ex Þ ey Þ ez, and the medium is biaxial. P – D – E – P – P – M – B – H – M – D E – B – H – E – D – H – B – e e e e = = È Î Í Í Í ˘ ˚ ˙ ˙ ˙ x y z 0 0 0 0 0 0 e e e e = = È Î Í Í Í ˘ ˚ ˙ ˙ ˙ 0 0 0 0 0 0 z
or isotropic or anisotropic media, the constitutive relations relate the two electric field vectors and the two magnetic field vectors by either a scalar or a tensor. Such media become polarized when placed in an electric field and become magnetized when placed in a magnetic field. A bianisotropic medium provides the cross oupling between the electric and magnetic fields. The constitutive relations for a bianisotropic medium can be written as D=E·E H E+∈·H (35.10b) When placed in an electric or a magnetic field, a bianisotropic medium becomes both polarized and magnetized. Magnetoelectric materials, theoretically predicted by Dzyaloshinskii and by Landau and Lifshitz, were observed experimentally in 1960 by Astrov in antiferromagnetic chromium oxide. The constitutive relations that Dzyaloshinskii proposed for chromium oxide have the following form D=0∈0·E+050·H 351la) 00 B=050·E+|0u0·H (35.1b) 00μ effect. Rado proved that the effect is not restricted to antiferromagnetics fe an exhibit the magnetoelectric It was then shown by Indenbom and by Birss that 58 magnetic crystal classes magnetic gallium iron oxide is also magnetoelectric Biisotropic Media In 1948, the gyrator was introduced by Tellegen as a new element, in addition to the resistor, the capacitor, the inductor, and the ideal transformer, for describing a network. To realize his new network element, Tellegen conceived of a medium possessing constitutive relations of the form D=∈E+H (35.12a) B= 5E+HH where $/ue is nearly equal to 1. Tellegen considered that the model of the medium had elements possessin permanent electric and magnetic dipoles parallel or antiparallel to each other, so that an applied electric field that aligns the electric dipoles simultaneously aligns the magnetic dipoles, and a magnetic field that aligns the magnetic dipoles simultaneously aligns the electric dipoles. Tellegen also wrote general constitutive relations Eq(35.10)and examined the symmetry properties by energy conservation Chiral media, which include many classes of sugar solutions, amino acids, DNA, and natural substances, have the following con c 2000 by CRC Press LLC
© 2000 by CRC Press LLC For isotropic or anisotropic media, the constitutive relations relate the two electric field vectors and the two magnetic field vectors by either a scalar or a tensor. Such media become polarized when placed in an electric field and become magnetized when placed in a magnetic field. A bianisotropic medium provides the cross coupling between the electric and magnetic fields. The constitutive relations for a bianisotropic medium can be written as (35.10a) (35.10b) When placed in an electric or a magnetic field, a bianisotropic medium becomes both polarized and magnetized. Magnetoelectric materials, theoretically predicted by Dzyaloshinskii and by Landau and Lifshitz, were observed experimentally in 1960 by Astrov in antiferromagnetic chromium oxide. The constitutive relations that Dzyaloshinskii proposed for chromium oxide have the following form: (35.11a) (35.11b) It was then shown by Indenbom and by Birss that 58 magnetic crystal classes can exhibit the magnetoelectric effect. Rado proved that the effect is not restricted to antiferromagnetics; ferromagnetic gallium iron oxide is also magnetoelectric. Biisotropic Media In 1948, the gyrator was introduced by Tellegen as a new element, in addition to the resistor, the capacitor, the inductor, and the ideal transformer, for describing a network. To realize his new network element, Tellegen conceived of a medium possessing constitutive relations of the form (35.12a) (35.12b) where x2 /me is nearly equal to 1. Tellegen considered that the model of the medium had elements possessing permanent electric and magnetic dipoles parallel or antiparallel to each other, so that an applied electric field that aligns the electric dipoles simultaneously aligns the magnetic dipoles, and a magnetic field that aligns the magnetic dipoles simultaneously aligns the electric dipoles. Tellegen also wrote general constitutive relations Eq. (35.10) and examined the symmetry properties by energy conservation. Chiral media, which include many classes of sugar solutions, amino acids, DNA, and natural substances, have the following constitutive relations D EH =× +× e x B EH = × +× z e DEH BEH z z z z = È Î Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ × + È Î Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ × = È Î Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ × + È Î Í Í Í Í ˘ ˚ ˙ ˙ ˙ ˙ × e e e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 x x x x x x m m m D EH = + e x B EH = +m x