We have obtained the second form of each of these expressions using the property (4.3) for the transform of a real function, and by using the change of variables =-o Multiplying the two forms of the expressions and adding half of each, we find that du x27/27[aE(a),D(o)-joEo),(]e-1m-o.(457) Now let us consider a dispersive isotropic medium described by the constitutive rela- tions D=2E, B= kH. Since the imaginary parts of 2 and A are associated with power dissipation in the medium, we shall approximate 2 and A as purely real. Then(4.57) Substitution from(4.55) now gives 2T Ljoe(o)-jo2()] [·F(a-a)F(a3-a)+上.EF(a+o)F(a+)+ +EEF(a-∞)F(a+∞)+至EF(a+ao)F(a-a)小e1o-) Let o-o wherever the term F( + oo)appears, and a'-- wherever the term F(a+a)appears. Since F(o)= F(o)and E(O)=E(o), we find that dwe 1 F(a-∞0)F(a EEUoe(a)-jae(a)je1-0+E·EUjoE(o)-joe(o)l1-)+ +E.E[o(o)+jol()eooy+E[-jo(ao)-jio(o)je-1o+oy (4.58) For small a the spectra are concentrated near o= oo or o'= oo. For terms involving the difference in the permittivities we can expand g(o)=oE(o)in a Taylor series about oo to obtain the approximation WE(O)No0E(oo)+(o-oo)g(oo) he doe(o)] This is not required for terms involving a sum of permittivities since these will not tend to cancel. For such terms we merely substitute or o'= ao. With these(4.58) becomes F(u-c0)F(a)-a0) 上·E*g(ab)j(o-)l1-o0+E.E*g'(a)j(a-a)e(-)+ +E·E(a0)j(a+o)e++E*E*(am)-j(o+a)]e-1o+ 2001 by CRC Press LLC
We have obtained the second form of each of these expressions using the property (4.3) for the transform of a real function, and by using the change of variables ω = −ω. Multiplying the two forms of the expressions and adding half of each, we find that ∂we ∂t = 1 2 ∞ −∞ dω 2π ∞ −∞ dω 2π jωE˜ ∗(ω ) · D˜ (ω) − jω E˜(ω) · D˜ ∗(ω ) e− j(ω −ω)t . (4.57) Now let us consider a dispersive isotropic medium described by the constitutive relations D˜ = ˜E˜ , B˜ = µ˜ H˜ . Since the imaginary parts of ˜ and µ˜ are associated with power dissipation in the medium, we shall approximate ˜ and µ˜ as purely real. Then (4.57) becomes ∂we ∂t = 1 2 ∞ −∞ dω 2π ∞ −∞ dω 2π E˜ ∗(ω ) · E˜(ω) jω(ω) ˜ − jω (ω˜ ) e− j(ω −ω)t . Substitution from (4.55) now gives ∂we ∂t = 1 8 ∞ −∞ dω 2π ∞ −∞ dω 2π jω(ω) ˜ − jω (ω˜ ) · · Eˇ · Eˇ ∗F˜ (ω − ω0)F˜ (ω − ω0) + Eˇ · Eˇ ∗F˜ (ω + ω0)F˜ (ω + ω0)+ + Eˇ · Eˇ F˜ (ω − ω0)F˜ (ω + ω0) + Eˇ ∗ · Eˇ ∗F˜ (ω + ω0)F˜ (ω − ω0) e− j(ω −ω)t . Let ω → −ω wherever the term F˜ (ω + ω0) appears, and ω → −ω wherever the term F˜ (ω + ω0) appears. Since F˜ (−ω) = F˜ (ω) and (˜ −ω) = (ω) ˜ , we find that ∂we ∂t = 1 8 ∞ −∞ dω 2π ∞ −∞ dω 2π F˜ (ω − ω0)F˜ (ω − ω0) · · Eˇ · Eˇ ∗[jω(ω) ˜ − jω (ω˜ )]e j(ω−ω )t + Eˇ · Eˇ ∗[jω (ω˜ ) − jω(ω) ˜ ]e j(ω −ω)t + + Eˇ · Eˇ [jω(ω) ˜ + jω (ω˜ )]e j(ω+ω )t + Eˇ ∗ · Eˇ ∗[− jω(ω) ˜ − jω (ω˜ )]e− j(ω+ω )t . (4.58) For small α the spectra are concentrated near ω = ω0 or ω = ω0. For terms involving the difference in the permittivities we can expand g(ω) = ω(ω) ˜ in a Taylor series about ω0 to obtain the approximation ω(ω) ˜ ≈ ω0(ω˜ 0) + (ω − ω0)g (ω0) where g (ω0) = ∂[ω(ω) ˜ ] ∂ω ω=ω0 . This is not required for terms involving a sum of permittivities since these will not tend to cancel. For such terms we merely substitute ω = ω0 or ω = ω0. With these (4.58) becomes ∂we ∂t = 1 8 ∞ −∞ dω 2π ∞ −∞ dω 2π F˜ (ω − ω0)F˜ (ω − ω0) · · Eˇ · Eˇ ∗g (ω0)[j(ω − ω )]e j(ω−ω )t + Eˇ · Eˇ ∗g (ω0)[j(ω − ω)]e j(ω −ω)t + + Eˇ · Eˇ (ω˜ 0)[j(ω + ω )]e j(ω+ω )t + Eˇ ∗ · Eˇ ∗(ω˜ 0)[− j(ω + ω )]e− j(ω+ω )t .
By integration a、0-0-0 -A\ lee ex)+E* E*(ob)e-i(otoy, t E (a-d)r Our last step is to compute the time-average value of We and let a-0. Applying 4.56)we find u)= 2E·E E”E+E·到()inc(+o where sinc(r)is defined in(A9 )and we note that sinc(-x)= sinc(x). Finally we let a-0 and use(.53)to replace F(o) by a 8-function Upon integration these 8-functions set = oo and w= og. Since sinc(0)= l and sinc(2)=0, the time-average stored electric energy density becomes simply 1÷,2a[oe] WWe==e 4.59 Simila lv2aloi] (Wm)==H This approach can also be applied to anisotropic materials to give ER E dloi] H See Collin 39 for details. For the case of a lossless, nondispersive material where the constitutive parameters are frequency independent, we can use(4.49) and(A76)to simplify this and obtain (va}=-E*E·E=-E·D”, (4.62) H in the anisotropic case and 1配=E (4.64) (m)=H2=H.B” (4.65) in the isotropic case. Here E, D, B, H are all phasor fields as defined by (4.54) 2001 by CRC Press LLC
By integration we(t) = 1 8 ∞ −∞ dω 2π ∞ −∞ dω 2π F˜ (ω − ω0)F˜ (ω − ω0) · · Eˇ · Eˇ ∗g (ω0)e j(ω−ω )t + Eˇ · Eˇ ∗g (ω0)e j(ω −ω)t + + Eˇ · Eˇ (ω˜ 0)e j(ω+ω )t + Eˇ ∗ · Eˇ ∗(ω˜ 0)e− j(ω+ω )t . Our last step is to compute the time-average value of we and let α → 0. Applying (4.56) we find we = 1 8 ∞ −∞ dω 2π ∞ −∞ dω 2π F˜ (ω − ω0)F˜ (ω − ω0) · · 2Eˇ · Eˇ ∗g (ω0)sinc [ω − ω ] π ω0 + Eˇ ∗ · Eˇ ∗ + Eˇ · Eˇ (ω˜ 0)sinc [ω + ω ] π ω0 where sinc(x) is defined in (A.9) and we note that sinc(−x) = sinc(x). Finally we let α → 0 and use (4.53) to replace F˜ (ω) by a δ-function. Upon integration these δ-functions set ω = ω0 and ω = ω0. Since sinc(0) = 1 and sinc(2π) = 0, the time-average stored electric energy density becomes simply we = 1 4 |Eˇ | 2 ∂[ω˜] ∂ω ω=ω0 . (4.59) Similarly, wm = 1 4 |Hˇ | 2 ∂[ωµ˜ ] ∂ω ω=ω0 . This approach can also be applied to anisotropic materials to give we = 1 4 Eˇ ∗ · ∂[ω˜¯] ∂ω ω=ω0 · Eˇ , (4.60) wm = 1 4 Hˇ ∗ · ∂[ωµ˜¯ ] ∂ω ω=ω0 · Hˇ . (4.61) See Collin [39] for details. For the case of a lossless, nondispersive material where the constitutive parameters are frequency independent, we can use (4.49) and (A.76) to simplify this and obtain we = 1 4 Eˇ ∗ · ¯ · Eˇ = 1 4 Eˇ · Dˇ ∗, (4.62) wm = 1 4 Hˇ ∗ · µ¯ · Hˇ = 1 4 Hˇ · Bˇ ∗, (4.63) in the anisotropic case and we = 1 4 |Eˇ | 2 = 1 4 Eˇ · Dˇ ∗, (4.64) wm = 1 4 µ|Hˇ | 2 = 1 4 Hˇ · Bˇ ∗, (4.65) in the isotropic case. Here Eˇ , Dˇ , Bˇ , Hˇ are all phasor fields as defined by (4.54).
4.5.3 The energy theorem A convenient expression for the time-average stored energies (4.60)and(4.61)is found y manipulating the frequency-domain Maxwell equations. Beginning with the complex conjugates of the two frequency-domain curl equations for anisotropic media E V×H*=J*-joe·E*, we differentiate with respect to frequency dlo] H+ jop ah (4.66) aH aJ ER-joE E (4.67) These terms also appear as a part of the expansion ah dER IV×E]-E +豆.V do 8o.[ X HI where we have used(B.44). Substituting from(4.66)-(4.67)and eliminating V xE and V x H by Maxwells equations we have 4.ExONs des 4(E.g.dE. m-:)+r aHaH aH+ an1.E:+.,r)-1(.+J Let us assume that the sources and fields are narrow band, centered on oo, and that o lies within a transparency range so that within the band the material may be considered lossless. Invoking from(4.49) the facts that E=e and A=A', we find that the first two terms on the right are zero. Integrating over a volume and taking the complex conjugate of both sides we obtain E+H' p 丁* Evaluating each of the terms at o=ao and using(4.60)-(4. 61) we have S(E ah dE H j[(We)+(Wm) 1「(g dE (4.68) 2001 by CRC Press LLC
4.5.3 The energy theorem A convenient expression for the time-average stored energies (4.60) and (4.61) is found by manipulating the frequency-domain Maxwell equations. Beginning with the complex conjugates of the two frequency-domain curl equations for anisotropic media, ∇ × E˜ ∗ = jωµ˜¯ ∗ · H˜ ∗, ∇ × H˜ ∗ = J˜∗ − jω˜¯ ∗ · E˜ ∗, we differentiate with respect to frequency: ∇ × ∂E˜ ∗ ∂ω = j ∂[ωµ˜¯ ∗ ] ∂ω · H˜ ∗ + jωµ˜¯ ∗ · ∂H˜ ∗ ∂ω , (4.66) ∇ × ∂H˜ ∗ ∂ω = ∂J˜∗ ∂ω − j ∂[ω˜¯ ∗ ] ∂ω · E˜ ∗ − jω˜¯ ∗ · ∂E˜ ∗ ∂ω . (4.67) These terms also appear as a part of the expansion ∇ · E˜ × ∂H˜ ∗ ∂ω + ∂E˜ ∗ ∂ω × H˜ = ∂H˜ ∗ ∂ω · [∇ × E˜ ] − E˜ ·∇× ∂H˜ ∗ ∂ω + H˜ ·∇× ∂E˜ ∗ ∂ω − ∂E˜ ∗ ∂ω · [∇ × H˜ ] where we have used (B.44). Substituting from (4.66)–(4.67) and eliminating ∇ × E˜ and ∇ × H˜ by Maxwell’s equations we have 1 4 ∇ · E˜ × ∂H˜ ∗ ∂ω + ∂E˜ ∗ ∂ω × H˜ = j 1 4 ω E˜ · ˜¯ ∗ · ∂E˜ ∗ ∂ω − ∂E˜ ∗ ∂ω · ˜¯ · E˜ + j 1 4 ω H˜ · µ˜¯ ∗ · ∂H˜ ∗ ∂ω − ∂H˜ ∗ ∂ω · µ˜¯ · H˜ + + j 1 4 E˜ · ∂[ω˜¯ ∗ ] ∂ω · E˜ ∗ + H˜ · ∂[ωµ˜¯ ∗ ] ∂ω · H˜ ∗ − 1 4 E˜ · ∂J˜∗ ∂ω + J˜ · ∂E˜ ∗ ∂ω . Let us assume that the sources and fields are narrowband, centered on ω0, and that ω0 lies within a transparency range so that within the band the material may be considered lossless. Invoking from (4.49) the facts that ˜¯ = ˜¯ † and µ˜¯ = µ˜¯ † , we find that the first two terms on the right are zero. Integrating over a volume and taking the complex conjugate of both sides we obtain 1 4 S E˜ ∗ × ∂H˜ ∂ω + ∂E˜ ∂ω × H˜ ∗ · dS = − j 1 4 V E˜ ∗ · ∂[ω˜¯] ∂ω · E˜ + H˜ ∗ · ∂[ωµ˜¯ ] ∂ω · H˜ dV − 1 4 V E˜ ∗ · ∂J˜ ∂ω + J˜∗ · ∂E˜ ∂ω dV. Evaluating each of the terms at ω = ω0 and using (4.60)–(4.61) we have 1 4 S E˜ ∗ × ∂H˜ ∂ω + ∂E˜ ∂ω × H˜ ∗ ω=ω0 · dS = − j [We+Wm] − 1 4 V E˜ ∗ · ∂J˜ ∂ω + J˜∗ · ∂E˜ ∂ω ω=ω0 dV (4.68)
where(We)+(Wm)is the total time-average electromagnetic energy stored in the volume region V. This is known as the energy theorem. We shall use it in 8 4.11.3 to determine the velocity of energy transport for a pla e wave 4.6 Some simple models for constitutive parameters phenomena. Although we recognize that matter is composed of microscopic constituents Thus far our discussion of electromagnetic fields has been restricted to macroscoj ve have chosen to describe materials using constitutive relationships whose parameters, such as permittivity, conductivity, and permeability, are viewed in the macroscopic sense By performing experiments on the laboratory scale we can measure the constitutive parameters to the precision required for engineering applications At some point it becomes useful to establish models of the macroscopic behavior of materials based on microscopic considerations, formulating expressions for the consti tutive parameters using atomic descriptors such as number density, atomic charge, and molecular dipole moment. These models allow us to predict the behavior of broad classes of materials, such as dielectrics and conductors, over wide ranges of frequency and field strength. Accurate models for the behavior of materials under the influence of electromagnetic fields must account for many complicated effects, including those best described by quar tum mechanics. However, many simple models can be obtained using classical mechanics and field theory. We shall investigate several of the most useful of these, and in the process try to gain a feeling for the relationship between the field applied to a material and the resulting polarization or magnetization of the underlying atomic structure. For simplicity we shall consider only homogeneous materials. The fundamental atomic descriptor of"number density, N, is thus taken to be independent of position and time The result may be more generally applicable since we may think of an inhomogeneous material in terms of the spatial variation of constitutive parameters originally deter mined assuming homogeneity. However, we shall not attempt to study the microscopic conditions that give rise to inhomogeneities 4.6.1 Complex permittivity of a non-magnetized plasma a plasma is an ionized gas in which the charged particles are free to move under the influence of an applied field and through particle-particle interactions. A plasma differs from other materials in that there is no atomic lattice restricting the motion of the particles. However, even in a gas the interactions between the particles and the fields give rise to a polarization effect, causing the permittivity of the gas to differ from that of free space. In addition, exposing the gas to an external field will cause a secondary urrent to flow as a result of the lorentz force on the particles. As the moving particles collide with one another they relinquish their momentum, an effect describable in terms of a conductivity. In this section we shall perform a simple analysis to determine the complex permittivity of a non-magnetized plasma To make our analysis tractable, we shall make several assumptions. 1. We assume that the plasma is neutral: i.e., that the free electrons and positive ions are of equal number and distributed in like manner. If the particles are sufficientl 2001 by CRC Press LLC
where We+Wm is the total time-average electromagnetic energy stored in the volume region V. This is known as the energy theorem. We shall use it in § 4.11.3 to determine the velocity of energy transport for a plane wave. 4.6 Some simple models for constitutive parameters Thus far our discussion of electromagnetic fields has been restricted to macroscopic phenomena. Although we recognize that matter is composed of microscopic constituents, we have chosen to describe materials using constitutive relationships whose parameters, such as permittivity, conductivity, and permeability, are viewed in the macroscopic sense. By performing experiments on the laboratory scale we can measure the constitutive parameters to the precision required for engineering applications. At some point it becomes useful to establish models of the macroscopic behavior of materials based on microscopic considerations, formulating expressions for the constitutive parameters using atomic descriptors such as number density, atomic charge, and molecular dipole moment. These models allow us to predict the behavior of broad classes of materials, such as dielectrics and conductors, over wide ranges of frequency and field strength. Accurate models for the behavior of materials under the influence of electromagnetic fields must account for many complicated effects, including those best described by quantum mechanics. However, many simple models can be obtained using classical mechanics and field theory. We shall investigate several of the most useful of these, and in the process try to gain a feeling for the relationship between the field applied to a material and the resulting polarization or magnetization of the underlying atomic structure. For simplicity we shall consider only homogeneous materials. The fundamental atomic descriptor of “number density,” N, is thus taken to be independent of position and time. The result may be more generally applicable since we may think of an inhomogeneous material in terms of the spatial variation of constitutive parameters originally determined assuming homogeneity. However, we shall not attempt to study the microscopic conditions that give rise to inhomogeneities. 4.6.1 Complex permittivity of a non-magnetized plasma A plasma is an ionized gas in which the charged particles are free to move under the influence of an applied field and through particle-particle interactions. A plasma differs from other materials in that there is no atomic lattice restricting the motion of the particles. However, even in a gas the interactions between the particles and the fields give rise to a polarization effect, causing the permittivity of the gas to differ from that of free space. In addition, exposing the gas to an external field will cause a secondary current to flow as a result of the Lorentz force on the particles. As the moving particles collide with one another they relinquish their momentum, an effect describable in terms of a conductivity. In this section we shall perform a simple analysis to determine the complex permittivity of a non-magnetized plasma. To make our analysis tractable, we shall make several assumptions. 1. We assume that the plasma is neutral: i.e., that the free electrons and positive ions are of equal number and distributed in like manner. If the particles are sufficiently
dense to be considered in the macroscopic sense, then there is no net field produced by the gas and thus no electromagnetic interaction between the particles. We also assume that the plasma is homogeneous and that the number density of the electrons N(number of electrons per m )is independent of time and position. In contrast to this are electron beams, whose properties differ significantly from neutral plasmas because of bunching of electrons by the applied field [148] 2. We ignore the motion of the positive ions in the computation of the secondary current. since the ratio of the mass of an ion to that of an electron is at least as large as the ratio of a proton to an electron(mp/me 1837) and thus the ions accelerate much more slowly 3. We assume that the applied field is that of an electromagnetic wave. In 82.10.6 we found that for a wave in free space the ratio of magnetic to electric field is H/E|=√∈0/μ0, so that VHO Thus, in the lorentz force equation we may approximate the force on an electron F=-qE+v×B)≈-qE as long as U c. Here qe is the unsigned charge on an electron, ge = 1.6021 X 10-19C. Note that when an external static magnetic field accompanies the field of the wave, as is the case in the earths ionosphere for example, we cannot ignore the magnetic component of the lorentz force. This case will be considered in 8 4.6.2 4. We assume that the mechanical interactions between particles can be described using a collision frequency v, which describes the rate at which a directed plasma velocity becomes random in the absence of external forces With these assumptions we can write the equation of motion for the plasma medium. Let v(r, t) represent the macroscopic velocity of the plasma medium. Then, by Newtons second law, the force acting at each point on the medium is balanced by the time-rate of change in momentum at that point. Because of collisions, the total change in momentum cribed by r qE(,)=2(r,D) +vp(r, t) o(r, t)=Nmev(r, t) is the volume density of momentum. Note that if there is no externally-applied electro- magnetic force, then(4.69) becomes dp(r, t) (r,t)=0. ence and we see that v describes the rate at which the electron velocities move toward a random state, producing a macroscopic plasma velocity v of zero 2001 by CRC Press LLC
dense to be considered in the macroscopic sense, then there is no net field produced by the gas and thus no electromagnetic interaction between the particles. We also assume that the plasma is homogeneous and that the number density of the electrons N (number of electrons per m3) is independent of time and position. In contrast to this are electron beams, whose properties differ significantly from neutral plasmas because of bunching of electrons by the applied field [148]. 2. We ignore the motion of the positive ions in the computation of the secondary current, since the ratio of the mass of an ion to that of an electron is at least as large as the ratio of a proton to an electron (mp/me = 1837) and thus the ions accelerate much more slowly. 3. We assume that the applied field is that of an electromagnetic wave. In § 2.10.6 we found that for a wave in free space the ratio of magnetic to electric field is |H|/|E| = √0/µ0, so that |B| |E| = µ0 0 µ0 = √µ00 = 1 c . Thus, in the Lorentz force equation we may approximate the force on an electron as F = −qe(E + v × B) ≈ −qeE as long as v c. Here qe is the unsigned charge on an electron, qe = 1.6021 × 10−19 C. Note that when an external static magnetic field accompanies the field of the wave, as is the case in the earth’s ionosphere for example, we cannot ignore the magnetic component of the Lorentz force. This case will be considered in § 4.6.2. 4. We assume that the mechanical interactions between particles can be described using a collision frequency ν, which describes the rate at which a directed plasma velocity becomes random in the absence of external forces. With these assumptions we can write the equation of motion for the plasma medium. Let v(r, t) represent the macroscopic velocity of the plasma medium. Then, by Newton’s second law, the force acting at each point on the medium is balanced by the time-rate of change in momentum at that point. Because of collisions, the total change in momentum density is described by F(r, t) = −NqeE(r, t) = d℘(r, t) dt + ν℘(r, t) (4.69) where ℘(r, t) = Nmev(r, t) is the volume density of momentum. Note that if there is no externally-applied electromagnetic force, then (4.69) becomes d℘(r, t) dt + ν℘(r, t) = 0. Hence ℘(r, t) = ℘0(r)e−νt , and we see that ν describes the rate at which the electron velocities move toward a random state, producing a macroscopic plasma velocity v of zero