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 Media ? Constitutive Matrices 35.3 Wave Equations and Wave Solutions Wave Solution?Wave Vector k ? Wavenumbers k 35.1 Maxwell Equations 1 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: Wiley- Interscience, 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 rt , , , ,) ( ) = ( ) = ( ) = ( ) = ( ) = ( ) = 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 ? 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 intro- ducing (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 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 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 = m o and e = e o , m o = 4p ′ 10 –7 henry/meter e o ? 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. ?× + =Jrt t rt(,) (,) ? ? r 0 B – r – B – r – J E – H – B – D – DE=eewhere = permittivity BH=m mwhere = permittivity E – D – H – B – ? 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 + m o . A medium is diamagnetic if m ￡ m o and paramagnetic if m 3 m o . 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 · w h e r e = 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 trans- formation 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, e x = e y = e z 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 e z > e; it is negative uniaxial if e z < e. In orthorhombic, monoclinic, and triclinic crystals, all three crystallographic axes are unequal. We have e x e y e z , 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 = = é ? ê ê ê ù ? ú ú ú x y z 00 00 00 e e e e = = é ? ê ê ê ù ? ú ú ú 00 00 00 z ? 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 x 2 /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 DEH=× +×e x BEH=× +×z e DEH BEH zz zz = é ? ê ê ê ê ù ? ú ú ú ú ×+ é ? ê ê ê ê ù ? ú ú ú ú × = é ? ê ê ê ê ù ? ú ú ú ú ×+ é ? ê ê ê ê ù ? ú ú ú ú × e e e 00 00 00 00 00 00 00 00 00 00 00 00 x x x x x x m m m DEH=+e x BEH=+mx ? 2000 by CRC Press LLC (35.13a) (35.13b) where c is the chiral parameter. Media characterized by the constitutive relations, Eqs. (35.12) and (35.13), are biisotropic media. Media in motion were the first bianisotropic media to receive attention in electromagnetic theory. In 1888, Roentgen discovered that a moving dielectric becomes magnetized when it is placed in an electric field. In 1905, Wilson showed that a moving dielectric in a uniform magnetic field becomes electrically polarized. Almost any medium becomes bianisotropic when it is in motion. The bianisotropic description of material has fundamental importance from the point of view of relativity. The principle of relativity postulates that all physical laws of nature must be characterized by mathematical equations that are form-invariant from one observer to the other. For electromagnetic theory, the Maxwell equations are form-invariant with respect to all observers, although the numerical values of the field quantities may vary from one observer to another. The constitutive relations are form-invariant when they are written in bianisotropic form. Constitutive Matrices Constitutive relations in the most general form can be written as (35.14a) (35.14b) where c = 3 ′ 10 8 m/s is the velocity of light in vacuum, and , , , and are all 3 ′ 3 matrices. Their elements are called constitutive parameters. In the definition of the constitutive relations, the constitutive matrices an d relate electric and magnetic fields. When an d are not identically zero, the medium is bianisotropic. When there is no coupling between electric and magnetic fields, = = 0 and the medium is anisotropic. For an anisotropic medium, if = ce and =Q = (1/cm) with denoting the 3 ′ 3 unit matrix, the medium is isotropic. The reason that we write constitutive relations in the present form is based on relativistic considerations. First, the fields and c form a single tensor in four-dimensional space, and so do c and . Second, constitutive relations written in the form Eq. (35.14) are Lorentz-covariant. Equation (35.14) can be rewritten in the form (35.15a) and is a 6 ′ 6 constitutive matrix: (35.15b) which has the dimension of admittance. DE H t = e – c ? ? BH E t =+mc ? ? cD P E L cB=×+× HMEQcB=×+× P Q L M L M L M L M P I I I E – B – D H cD H C E cB é ? ê ù ? ú =× é ? ê ù ? ú C C PL MQ = é ? ê ê ù ? ú ú ? 2000 by CRC Press LLC ELECTROPHOTOGRAPHY Chester F. Carlson Patented October 6, 1942 #2,297,691 ? 2000 by CRC Press LLC The constitutive matrix may be functions of space–time coordinates, thermodynamical and continuum- mechanical variables, or electromagnetic field strengths. According to the functional dependence of , we can classify the various media as (1) inhomogeneous if is a function of space coordinates, (2) nonstationary if is a function of time, (3) time-dispersive if is a function of time derivatives, (4) spatial-dispersive if is a function of spatial derivatives, (5) nonlinear if is a function of the electromagnetic field, and so forth. In the general case may be a function of integral-differential operators and coupled to fundamental equations of other physical disciplines. C C C C C C C C n excerpt from Chester Carl- son’s patent application: A feature of the present invention resides in the use of photoelec- tric or photoconductive materials for photographic purposes. In its preferred form the invention involves the use of materials which are insulators in the dark but become partial conductors when illuminated. These materials respond to light, being slightly conduc- tive whenever they are illuminated and again becoming insulating when the light is cut off. They can be called pho- toconductive insulating materials. Working in the patent department at the P.R. Mallory Company in the 1930s, Carlson became frustrated try- ing to obtain copies of patent drawings and specifications. Unlike others attempting chemical photographic methods, he used the principles of electrostatics to produce his first dry copy in 1938. A photoconductive plate connected to an electric charge was exposed to the desired image. The plate retained the electric charge on the dark areas of the image and lost it on the white areas. Dusting the plate with a powder reproduced the image. Xerox Corporation negotiated rights to the process in 1947 and introduced its first office copier in 1968. (Copyright 1995, DewRay Products, Inc. Used with permission.) A 35.3 Wave Equations and Wave Solutions The Maxwell equations in differential form are valid at all times for every point in space. First we shall investigate solutions to the Maxwell equations in regions void of source, namely, in regions where = r = 0. This, of course, does not mean that there is no source anywhere in all space. Sources must exist outside the regions of interest in order to produce fields in these regions. From the source-free Maxwell equations, a wave equation for the electric field can be easily derived for isotropic permittivity e and permeability m (35.16) The Laplacian operator ? 2 in a rectangular coordinate system is The wave Eq. (35.16) is a second-order partial differential equation of space and time coordinates x, y, z, and t. Wave Solution The simplest solution to Eq. (35.16) for the electric field E is (35.17) Substituting Eqs. (35.17) in (35.16) we find that the following equation, called the dispersion relation, which relates w and k, must be satisfied: k 2 = w 2 me (35.18) There are two points of view useful in the study of a space–time varying quantity such as E x (z,t). The first is to examine the time variation at fixed points in space. The second is to examine spatial variation at fixed times, a process that amounts to taking a series of pictures. We first fix our attention to one particular point in space, say z = 0. We then have the electric vector E x (z,t) = E 0 cos wt. Plotted as a function of time, we find that the waveform repeats itself in time as wt = 2mp for any integer m. The period is defined as the time T for which wT = 2p. The frequency f is defined as f = 1/T which gives Since w = 2pf, w is the angular frequency of the wave. To examine wave behavior from the other point of view, we let wt = 0 and plot E x (z,t). The waveform repeats itself in space when kz = 2mp for integer values of m. The wavelength l is defined as the distance for which kl = 2p. Thus l = 2p/k, or J E – ?= 2 2 2 0E t E– m ? ? e ?= + + 2 2 2 2 2 2 2 ? ? ? ? ? ?xyz ExE kz t xEzt x =- ( ) = ( ) ? cos ? , 0 w f = w p2 k = 2p l ? 2000 by CRC Press LLC We call k the wavenumber which is equal to the number of wavelengths in a distance of 2p and has the dimension inverse length. Wave Vector k The solution for the electric field in Eq. (35.17) represents an electromagnetic wave propagating in the ^ z-direction.For a wave propagating in a general direction, we define a wave vector = ? xk x + ? yk y + ? zk z (35.19) It is easily verified that the electric field (,t) = 0 cos(k x x + k y y + k z z – wt) (35.20) is a solution to Eq. (35.16), where 0 is a constant vector. The dispersion relation corresponding to Eq. (35.18) is obtained by substituting Eq. (35.20) in Eq. (35.16) which yields k 2 x + k 2 y + k 2 z = w 2 me We may write the solution, Eq. (35.20), in the following form: (, t) = 0 cos( · – wt) where = ? xx + ? yy + ? xx is a position vector. The wave vector is often referred to simply as the vector. Wavenumbers k The wavenumber k is the magnitude of the wave vector and is of more fundamental importance in electro- magnetic wave theory than both of the more popular concepts of wavelength l and frequency f. In Fig. 35.1, we illustrate the electromagnetic wave spectrum according to the free space wavenumber k = k o = w/c. The corresponding values of frequency and wavelength are f = ck o /2p and l = 2p/k o . It is useful to define a fundamental unit K o such that for free space k o = 1K o = 2p m –1 . Thus k o = A K o corresponds to l = 1/A m and f = 3 ′ 10 8 A Hz. The photon energy in electronvolts is calculated from \w = \ck where \ = 1.05 ′ 10 –34 Joule- sec is Planck’s constant divided by 2p and the electron charge is q = 1.6 ′ 10 –19 C. Thus \w = (2p\c/q)k o ? 1.26 ′ 10 –6 k o and k o = A K o corresponds to 1.26 ′ 10 –6 A eV. Defining Terms Electric field:State of a region in which charged bodies are subject to forces by virtue of their charge, the force acting on a unit positive charge. Magnetic field: State produced by electric charge in motion and evidenced by a force exerted on a moving charge in the field. Magnetic flux:Summation obtained by integrating flux density over an area. Magnetic flux density:Measure of the strength and direction of a magnetic field at a point. E – k E – r E – E – E – r E – k r r k k k ? 2000 by CRC Press LLC Related Topics 39.1 Passive Microwave Devices ? 44.2 The Field Equations Reference J.A. Kong, Electromagnetic Wave Theory, New York: Wiley-Interscience, 1990, chap. 1. Further Information IEEE Transactions on Microwave Theory and Techniques IEEE Transactions on Antennas and Propagation FIGURE 35.1 Electromagnetic wave spectrum. ? 2000 by CRC Press LLC