where each of the quantities P, L, M, Q may be dyadics in the usual sense, or dyadic operators containing space or time derivatives or integrals, or some nonlinear operations on the fields. We may write these expressions as a single matrix equation E H where the6×6 matrix [C]= P M Q This most general relationship between fields is the property of a bianisotropic material We may wonder why d is not related to (E, B, h),e to (D, B),etc. The reason that since the field pairs(e, B)and(D, H)convert identically under a lorentz transfor- mation, a constitutive relation that maps fields as in(2. 18)is form invariant, as are the Maxwell-Minkowski equations. That is, although the constitutive parameters may vary numerically between observers moving at different velocities, the form of the relationship Many authors choose to relate(, B)to (E, H), often because the expressions are simpler and can be more easily applied to specific problems. For instance, in a linear isotropic material (as shown below) D is directly proportional to E and B is directly proportional to H. To provide the appropriate expression for the constitutive relations re need only remap(2.18). This gives s·H B=sE+乒·H, 2] where the new constitutive parameters E, 4, S, i can be easily found from the original constitutive parameters P, L, M, Q. We do note, however, that in the form(2. 19 )-(2.20) the Lorentz invariance of the constitutive equations is not obvious In the following paragraphs we shall characterize some of the most common materials according to these classifications. With this approach effects such as temporal or spatial dispersion are not part of the classification process, but arise from the nature of the constitutive parameters. Hence we shall not dwell on the particulars of the constitutive parameters, but shall concentrate on the form of the constitutive relations. Constitutive relations for fields in free space. In a vacuum the fields are related by the simple constitutive equation H B (223) The quantities uo and Eo are, respectively, the free-space permeability and permittivity constants. It is convenient to use three numerical quantities to describe the electromag- netic properties of free space--Ao, E0, and the speed of light c- and interrelate them through the equation C=1/(u0∈0)/2 @2001 by CRC Press LLC
where each of the quantities P¯,L¯ , M¯ , Q¯ may be dyadics in the usual sense, or dyadic operators containing space or time derivatives or integrals, or some nonlinear operations on the fields. We may write these expressions as a single matrix equation cD H = [C¯ ] E cB (2.18) where the 6 × 6 matrix [C¯ ] = P¯ L¯ M¯ Q¯ . This most general relationship between fields is the property of a bianisotropic material. We may wonder why D is not related to (E,B, H), E to (D,B), etc. The reason is that since the field pairs (E,B) and (D, H) convert identically under a Lorentz transformation, a constitutive relation that maps fields as in (2.18) is form invariant, as are the Maxwell–Minkowski equations. That is, although the constitutive parameters may vary numerically between observers moving at different velocities, the form of the relationship given by (2.18) is maintained. Many authors choose to relate (D,B) to (E, H), often because the expressions are simpler and can be more easily applied to specific problems. For instance, in a linear, isotropic material (as shown below) D is directly proportional to E and B is directly proportional to H. To provide the appropriate expression for the constitutive relations, we need only remap (2.18). This gives D = ¯ · E + ξ¯ · H, (2.19) B = ζ¯ · E + µ¯ · H, (2.20) or D B = C¯ E H E H , (2.21) where the new constitutive parameters ¯, ξ¯, ζ¯, µ¯ can be easily found from the original constitutive parameters P¯,L¯ , M¯ , Q¯ . We do note, however, that in the form (2.19)–(2.20) the Lorentz invariance of the constitutive equations is not obvious. In the following paragraphs we shall characterize some of the most common materials according to these classifications. With this approach effects such as temporal or spatial dispersion are not part of the classification process, but arise from the nature of the constitutive parameters. Hence we shall not dwell on the particulars of the constitutive parameters, but shall concentrate on the form of the constitutive relations. Constitutive relations for fields in free space. In a vacuum the fields are related by the simple constitutive equations D = 0E, (2.22) H = 1 µ0 B. (2.23) The quantities µ0 and 0 are, respectively, the free-space permeability and permittivity constants. It is convenient to use three numerical quantities to describe the electromagnetic properties of free space — µ0, 0, and the speed of light c — and interrelate them through the equation c = 1/(µ0 0) 1/2
Historically it has been the practice to define uo, measure c, and compute Eo. In SI units 4x×10-H/ c=2998×108m/s, With the two constitutive equations we have enough information to put Maxwell equations into definite form. Traditionally(2.22)and(2.23)are substituted into(2.1)- (22) V×E= 2.24 V×B=μoJ+p0∈0 (225) These are two vector equations in two vector unknowns(equivalently, six scalar equations in six scalar unknowns) In terms of the general constitutive relation(2.18), we find that free space is isoti with where no=(uo/Eo) /2 is called the intrinsic impedance of free space. This emphasizes the fact that free space has, along with c, only a single empirical constant associated with it(i. e, Eo or no). Since no derivative or integral operators appear in the constitutive relations, free space is nondispersive Constitutive relations in a linear isotropic material. In a linear isotropic mate rial there is proportionality between D and E and between B and H. The constants of proportionality are the permittivity E and the permeability u. If the material is nor pensive, the constitutive relations take the form E B=uH, where e and u may depend on position for inhomogeneous materials. Often the permit wity and permeability are referenced to the permittivity and permeability of free space according t 1=以r10 Here the dimensionless quantities Er and ur are called, respectively, the relative permit tivity and relative permeability When dealing with the Maxwell-Boffi equations(8 2.4) the difference between the material and free space values of D and H becomes important. Thus for linear isotropic materials we often write the constitutive relations as D=∈0E+60xeE, (226) B=uoH uoXmH where the dimensionless quantities xe =Er-1 and xm=ur-1 are called, respectively the electric and magnetic susceptibilities of the material. In terms of (2.18)we have = @2001 by CRC Press LLC
Historically it has been the practice to define µ0, measure c, and compute 0. In SI units µ0 = 4π × 10−7 H/m, c = 2.998 × 108 m/s, 0 = 8.854 × 10−12 F/m. With the two constitutive equations we have enough information to put Maxwell’s equations into definite form. Traditionally (2.22) and (2.23) are substituted into (2.1)– (2.2) to give ∇ × E = −∂B ∂t , (2.24) ∇ × B = µ0J + µ0 0 ∂E ∂t . (2.25) These are two vector equations in two vector unknowns (equivalently, six scalar equations in six scalar unknowns). In terms of the general constitutive relation (2.18), we find that free space is isotropic with P¯ = Q¯ = 1 η0 ¯ I, L¯ = M¯ = 0, where η0 = (µ0/ 0)1/2 is called the intrinsic impedance of free space. This emphasizes the fact that free space has, along with c, only a single empirical constant associated with it (i.e., 0 or η0). Since no derivative or integral operators appear in the constitutive relations, free space is nondispersive. Constitutive relations in a linear isotropic material. In a linear isotropic material there is proportionality between D and E and between B and H. The constants of proportionality are the permittivity and the permeability µ. If the material is nondispersive, the constitutive relations take the form D = E, B = µH, where and µ may depend on position for inhomogeneous materials. Often the permittivity and permeability are referenced to the permittivity and permeability of free space according to = r 0, µ = µrµ0. Here the dimensionless quantities r and µr are called, respectively, the relative permittivity and relative permeability. When dealing with the Maxwell–Boffi equations (§ 2.4) the difference between the material and free space values of D and H becomes important. Thus for linear isotropic materials we often write the constitutive relations as D = 0E + 0χeE, (2.26) B = µ0H + µ0χmH, (2.27) where the dimensionless quantities χe = r − 1 and χm = µr − 1 are called, respectively, the electric and magnetic susceptibilities of the material. In terms of (2.18) we have P¯ = r η0 ¯ I, Q¯ = 1 η0µr ¯ I, L¯ = M¯ = 0
Generally a material will have either its electric or magnetic properties dominant. If ur= I and Er+ l then the material is generally called a perfect dielectric or a perfect insulator, and is said to be an electric material. If Er l and ur# 1, the material is said to be a magnetic material. A linear isotropic material may also have conduction properties. In a conducting material, a constitutive relation is generally used to describe the mechanical interaction of field and charge by relating the electric field to a secondary electric current. For a nondispersive isotropic material, the current is aligned with, and proportional to, the electric field; there are no temporal operators in the constitutive relation, which is simply J=σE. (228) This is known as Ohm's law. Here o is the conductivity of the material. If ur a 1 and o is very small, the material is generally called a good dielectric. If o is very large, the material is generally called a good conductor. The conditions by which we say the conductivity is"small"or " large"are usually established using the frequency response of the material. Materials that are good dielectrics over broad ranges of frequency include various glasses and plastics such as fused quartz, polyethylene and teflon. Materials that are good conductors over broad ranges of frequency include For dispersive linear isotropic materials, the constitutive parameters become nonsta- tionary(time dependent), and the constitutive relations involve time operators.(Note that the name dispersive describes the tendency for pulsed electromagnetic waves to spread out, or disperse, in materials of this type. If we assume that the relationships given by(2.26),(2.27), and (2.28)retain their product form in the frequency domain, then by the convolution theorem we have in the time domain the constitutive relations D(r,t)=EoE(r, t)+ Xe(r, t-tE(r, t')dr (229) B(r, t=o(H(r, 1)+/m(r,t-tH(r,t,o)dr (230) J(r, t) σ(r,t-t)E(r,t')d (231) These expressions were first introduced by Volterra in 1912 199. We see that for a linear dispersive material of this type the constitutive operators are time integrals, and that the behavior of D(r)depends not only on the value of e at time t, but on its values at all past times. Thus, in dispersive materials there is a"time lag" between the effect of the applied field and the po tion or magnetizatio quency domain. t al dispersion is associated with complex values of the constitutive pa rameters, which, to describe a causal relationship cannot be constant with frequency The nonzero imaginary component is identified with the dissipation of electromagnetic energy as heat. Causality is implied by the upper limit being t in the convolution inte- grals, which indicates that D(t)cannot depend on future values of E(t) assumption leads to a relationship between the real and imaginary parts of the frequency domain constitutive parameters as described through the Kronig-Kramers equations Constitutive relations for fields in perfect conductors. In a perfect electric con ductor(PEC)or a perfect magnetic conductor(PMC) the fields are exactly specified as @2001 by CRC Press LLC
Generally a material will have either its electric or magnetic properties dominant. If µr = 1 and r = 1 then the material is generally called a perfect dielectric or a perfect insulator, and is said to be an electric material. If r = 1 and µr = 1, the material is said to be a magnetic material. A linear isotropic material may also have conduction properties. In a conducting material, a constitutive relation is generally used to describe the mechanical interaction of field and charge by relating the electric field to a secondary electric current. For a nondispersive isotropic material, the current is aligned with, and proportional to, the electric field; there are no temporal operators in the constitutive relation, which is simply J = σE. (2.28) This is known as Ohm’s law. Here σ is the conductivity of the material. If µr ≈ 1 and σ is very small, the material is generally called a good dielectric. If σ is very large, the material is generally called a good conductor. The conditions by which we say the conductivity is “small” or “large” are usually established using the frequency response of the material. Materials that are good dielectrics over broad ranges of frequency include various glasses and plastics such as fused quartz, polyethylene, and teflon. Materials that are good conductors over broad ranges of frequency include common metals such as gold, silver, and copper. For dispersive linear isotropic materials, the constitutive parameters become nonstationary (time dependent), and the constitutive relations involve time operators. (Note that the name dispersive describes the tendency for pulsed electromagnetic waves to spread out, or disperse, in materials of this type.) If we assume that the relationships given by (2.26), (2.27), and (2.28) retain their product form in the frequency domain, then by the convolution theorem we have in the time domain the constitutive relations D(r, t) = 0 E(r, t) + t −∞ χe(r, t − t )E(r, t ) dt , (2.29) B(r, t) = µ0 H(r, t) + t −∞ χm(r, t − t )H(r, t ) dt , (2.30) J(r, t) = t −∞ σ(r, t − t )E(r, t ) dt . (2.31) These expressions were first introduced by Volterra in 1912[199]. We see that for a linear dispersive material of this type the constitutive operators are time integrals, and that the behavior of D(t) depends not only on the value of E at time t, but on its values at all past times. Thus, in dispersive materials there is a “time lag” between the effect of the applied field and the polarization or magnetization that results. In the frequency domain, temporal dispersion is associated with complex values of the constitutive parameters, which, to describe a causal relationship, cannot be constant with frequency. The nonzero imaginary component is identified with the dissipation of electromagnetic energy as heat. Causality is implied by the upper limit being t in the convolution integrals, which indicates that D(t) cannot depend on future values of E(t). This assumption leads to a relationship between the real and imaginary parts of the frequency domain constitutive parameters as described through the Kronig–Kramers equations. Constitutive relations for fields in perfect conductors. In a perfect electric conductor (PEC) or a perfect magnetic conductor (PMC) the fields are exactly specified as
the null field. E=D=B=H=O By Ampere's and Faraday 's laws we must also have J= Jm =0; hence, by the continuity equation, p=pm=0 In addition to the null field, we have the condition that the tangential electric field on the surface of a PEC must be zero. Similarly, the tangential magnetic field on the surface of a PMC must be zero. This implies(s 2.8.3) that an electric surface current may exist on the surface of a PEC but not on the surface of a PMC, while a magnetic surface current may exist on the surface of a PMC but not on the surface of a PE 心、 A PEC may be regarded as the limit of a conducting material as o→∞. In many actical cases, good conductors such as gold and copper can be assumed to be perfect electric conductors, which greatly simplifies the application of boundary conditions. No physical material is known to behave as a PMC, but the concept is mathematically useful for applying symmetry conditions(in which a PMC is sometimes referred to as a "magnetic wall")and for use in developing equivalence theorems Constitutive relations in a linear anisotropic material. In a linear anisotropic material there are relationships between B and H and between D and e, but the field vectors are not aligned as in the isotropic case. We can thus write D=E·E,B=p·H,J=a·E where 2 is called the permittivity dyadic, A is the permeability dyadic, and a is the conductivity dyadic. In terms of the general constitutive relation(2.18)we have Many different types of materials demonstrate anisotropic behavior, including opt al crystals, magnetized plasmas, and ferrites. Plasmas and ferrites are examples of gyrotropic media. With the proper choice of coordinate system, the frequency-domain permittivity or permeability can be written in matrix form as 0 =-∈12∈110 [=-u12110 (232) 0033 of the matrix entries may be complex. For the special case of a lossless gyrotropic erial, the matrices become hermitian u -jK 0 [A]= jK u 0 (233) 00 where∈,∈3,8,,3, and k are real numbers. Crystals have received particular attention because of their birefringent properties. A birefringent crystal can be characterized by a symmetric permittivity dyadic that has real permittivity parameters in the frequency domain; equivalently, the constitutive relations do not involve constitutive operators. A coordinate system called the principal system, with axes called the principal ares, can always be found so that the permittivity dyadic in that system is diagonal 00 ]=0∈,0 E- @2001 by CRC Press LLC
the null field: E = D = B = H = 0. By Ampere’s and Faraday’s laws we must also have J = Jm = 0; hence, by the continuity equation, ρ = ρm = 0. In addition to the null field, we have the condition that the tangential electric field on the surface of a PEC must be zero. Similarly, the tangential magnetic field on the surface of a PMC must be zero. This implies (§ 2.8.3) that an electric surface current may exist on the surface of a PEC but not on the surface of a PMC, while a magnetic surface current may exist on the surface of a PMC but not on the surface of a PEC. A PEC may be regarded as the limit of a conducting material as σ → ∞. In many practical cases, good conductors such as gold and copper can be assumed to be perfect electric conductors, which greatly simplifies the application of boundary conditions. No physical material is known to behave as a PMC, but the concept is mathematically useful for applying symmetry conditions (in which a PMC is sometimes referred to as a “magnetic wall”) and for use in developing equivalence theorems. Constitutive relations in a linear anisotropic material. In a linear anisotropic material there are relationships between B and H and between D and E, but the field vectors are not aligned as in the isotropic case. We can thus write D = ¯ · E, B = µ¯ · H, J = σ¯ · E, where ¯ is called the permittivity dyadic, µ¯ is the permeability dyadic, and σ¯ is the conductivity dyadic. In terms of the general constitutive relation (2.18) we have P¯ = c¯, Q¯ = µ¯ −1 c , L¯ = M¯ = 0. Many different types of materials demonstrate anisotropic behavior, including optical crystals, magnetized plasmas, and ferrites. Plasmas and ferrites are examples of gyrotropic media. With the proper choice of coordinate system, the frequency-domain permittivity or permeability can be written in matrix form as [˜¯] = 11 12 0 − 12 11 0 0 0 33 , [µ˜¯ ] = µ11 µ12 0 −µ12 µ11 0 0 0 µ33 . (2.32) Each of the matrix entries may be complex. For the special case of a lossless gyrotropic material, the matrices become hermitian: [˜¯] = − jδ 0 jδ 0 0 0 3 , [µ˜¯ ] = µ − jκ 0 jκ µ 0 0 0 µ3 , (2.33) where , 3, δ, µ, µ3, and κ are real numbers. Crystals have received particular attention because of their birefringent properties. A birefringent crystal can be characterized by a symmetric permittivity dyadic that has real permittivity parameters in the frequency domain; equivalently, the constitutive relations do not involve constitutive operators. A coordinate system called the principal system, with axes called the principal axes, can always be found so that the permittivity dyadic in that system is diagonal: [˜¯] = x 0 0 0 y 0 0 0 z
The geometrical structure of a crystal determines the relationship between Ex, Ey, and ∈z.If∈x=y<∈a, then the crystal is positive uniaxial(e.g, quartz).Ifex=ey>∈z the crystal is negative uniaxial(e.g, calcite).If∈x≠y≠∈x, the crystal is biarial(e.g, mica). In uniaxial crystals the z-axis is called the optical aris If the anisotropic material is dispersive, we can generalize the convolutional form of the isotropic dispersive media to obtain the constitutive relations D(r,)=EoE(r,t)+e(r, t-t).E(r,t')dr (234) B(r,t) H(r,t1)+元n(r,t-t)·H(r,t')dr (235) I(r,t a(r,t-t)·E(r,t)dt Constitutive relations for biisotropic materials. A biisotropic material is an isotropic magnetoelectric material. Here we have D related to E and B, and H related to E and B, but with no realignment of the fields as in anisotropic(or bianisotropic)mate- rials. Perhaps the simplest example is the Tellegen medium devised by B. D.H. Tellegen in 1948196, having D=∈E+H (237) (238) Tellegen proposed that his hypothetical material be composed of small(but macroscopic ferromagnetic particles suspended in a liquid. This is an example of a synthetic mate- rial, constructed from ordinary materials to have an exotic electromagnetic behavior Other examples include artificial dielectrics made from metallic particles imbedded in lightweight foams 66, and chiral materials made from small metallic helices suspended in resins [112 Chiral materials are also isotropic, and have the constitutive relations ah D=∈E-at (239) B=RH+X ar, where the constitutive parameter x is called the chirality parameter. Note the presence of temporal derivative operators. Alternatively, D=∈CE+BV×E) B=μ(H+B×H), (2.42 by Faradays and Ampere's laws. Chirality is a natural state of symmetry; many natural substances are chiral materials, including DNA and many sugars. The time derivatives in(2.39)-(2.40)produce rotation of the polarization of time harmonic electromagnetic waves propagating in chiral media Constitutive relations in nonlinear media. Nonlinear electromagnetic effects have been studied by scientists and engineers since the beginning of the era of electrical tech- nology. Familiar examples include saturation and hysteresis in ferromagnetic materials @2001 by CRC Press LLC
The geometrical structure of a crystal determines the relationship between x , y , and z. If x = y < z, then the crystal is positive uniaxial (e.g., quartz). If x = y > z, the crystal is negative uniaxial (e.g., calcite). If x = y = z, the crystal is biaxial (e.g., mica). In uniaxial crystals the z-axis is called the optical axis. If the anisotropic material is dispersive, we can generalize the convolutional form of the isotropic dispersive media to obtain the constitutive relations D(r, t) = 0 E(r, t) + t −∞ χ¯ e(r, t − t ) · E(r, t ) dt , (2.34) B(r, t) = µ0 H(r, t) + t −∞ χ¯ m(r, t − t ) · H(r, t ) dt , (2.35) J(r, t) = t −∞ σ¯ (r, t − t ) · E(r, t ) dt . (2.36) Constitutive relations for biisotropic materials. A biisotropic material is an isotropic magnetoelectric material. Here we have D related to E and B, and H related to E and B, but with no realignment of the fields as in anisotropic (or bianisotropic) materials. Perhaps the simplest example is the Tellegen medium devised by B.D.H. Tellegen in 1948 [196], having D = E + ξH, (2.37) B = ξE + µH. (2.38) Tellegen proposed that his hypothetical material be composed of small (but macroscopic) ferromagnetic particles suspended in a liquid. This is an example of a synthetic material, constructed from ordinary materials to have an exotic electromagnetic behavior. Other examples include artificial dielectrics made from metallic particles imbedded in lightweight foams [66], and chiral materials made from small metallic helices suspended in resins [112]. Chiral materials are also biisotropic, and have the constitutive relations D = E − χ ∂H ∂t , (2.39) B = µH + χ ∂E ∂t , (2.40) where the constitutive parameter χ is called the chirality parameter. Note the presence of temporal derivative operators. Alternatively, D = (E + β∇ × E), (2.41) B = µ(H + β∇ × H), (2.42) by Faraday’s and Ampere’s laws. Chirality is a natural state of symmetry; many natural substances are chiral materials, including DNA and many sugars. The time derivatives in (2.39)–(2.40) produce rotation of the polarization of time harmonic electromagnetic waves propagating in chiral media. Constitutive relations in nonlinear media. Nonlinear electromagnetic effects have been studied by scientists and engineers since the beginning of the era of electrical technology. Familiar examples include saturation and hysteresis in ferromagnetic materials