The time derivative in(4.69)is the total derivative as defined in(A58) do(r, t) a(r, t) +(v·V)s(r,t) (4.70) The second term on the right accounts for the time-rate of change of momentum per ceived as the observer moves through regions of spatially-changing momentum. Since the electron velocity is induced by the electromagnetic field, we anticipate that for a sinusoidal wave the spatial variation will be on the order of the wavelength of the field A=2c/o. Thus, while the first term in(4.70)is proportional to o, the second term is proportional to ou/c and can be neglected for non-relativistic particle velocities. Then, writing E(r, t)and v(r, t)as inverse Fourier transforms, we see that(4.69) yields -gee= jome v+ mevv (4.71) nd thu 业E (4.72) The secondary current associated with the moving electrons is(since ge is unsigned) (V- JOE (4.73) q is called the plasma frequency The frequency-domain Ampere's law for primary and secondary currents in free space s merely V×H=J+J+ja∈oE. Substitution from(4. 73) gives V×H=J+ E+joo l1 E We can determine the material properties of the plasma by realizing that the above expression can be written as V×H=J+J+joD with the constitutive relations J=oE Here we identify the conductivity of the plasma as 2 and the permittivity as E(a)=601-a2+p2 2001 by CRC Press LLC
The time derivative in (4.69) is the total derivative as defined in (A.58): d℘(r, t) dt = ∂℘(r, t) ∂t + (v · ∇)℘(r, t). (4.70) The second term on the right accounts for the time-rate of change of momentum perceived as the observer moves through regions of spatially-changing momentum. Since the electron velocity is induced by the electromagnetic field, we anticipate that for a sinusoidal wave the spatial variation will be on the order of the wavelength of the field: λ = 2πc/ω. Thus, while the first term in (4.70) is proportional to ω, the second term is proportional to ωv/c and can be neglected for non-relativistic particle velocities. Then, writing E(r, t) and v(r, t) as inverse Fourier transforms, we see that (4.69) yields − qeE˜ = jωmev˜ + meνv˜ (4.71) and thus v˜ = − qe me E˜ ν + jω. (4.72) The secondary current associated with the moving electrons is (since qe is unsigned) J˜s = −Nqev˜ = 0ω2 p ω2 + ν2 (ν − jω)E˜ (4.73) where ω2 p = Nq2 e 0me (4.74) is called the plasma frequency. The frequency-domain Ampere’s law for primary and secondary currents in free space is merely ∇ × H˜ = J˜i + J˜s + jω0E˜ . Substitution from (4.73) gives ∇ × H˜ = J˜i + 0ω2 pν ω2 + ν2 E˜ + jω0 1 − ω2 p ω2 + ν2 E˜ . We can determine the material properties of the plasma by realizing that the above expression can be written as ∇ × H˜ = J˜i + J˜s + jωD˜ with the constitutive relations J˜s = σ˜E˜ , D˜ = ˜E˜ . Here we identify the conductivity of the plasma as σ(ω) ˜ = 0ω2 pν ω2 + ν2 (4.75) and the permittivity as (ω) ˜ = 0 1 − ω2 p ω2 + ν2
We can also write Ampere' s law as V×H=J+joe"E where E is the complex permittivity (4.76) If we wish to describe the plasma in terms of a polarization vector, we merely use d Eoe+P=2E to obtain the polarization vector P=(E-EoE= EoXeE, where Xe is the electric susceptibility We note that P is directed opposite the applied field E, resulting in 2<Eo The plasma is dispersive since both its permittivity and conductivity depend on o Aso→0 we have 2c→∈ FoEr where e=1-a2/n2, and also a"~1/o, as remarked in(4. 28). As o- oo we have Ec-E0 1/02 and Ec 1/0, as mentioned in(4.29) When a transient plane wave propagates through a dispersive medium, the frequency dependence of the constitutive parameters tends to cause spreading of the waveshape We see that the plasma conductivity(4. 75)is proportional to the collision frequency v and that, since Ec<0 by the arguments of $4.5, the plasma must be lossy. Loss arises from the transfer of electromagnetic energy into heat through electron collisions. If there the conductivity of a lossless(or "collisionless")plasma reduces to zero as expected nd are no collisions(v=0), there is no mechanism for the transfer of energy into heat, In a lowloss plasma(v-0)we may determine the time-average stored electromagnetic nergy for sinusoidal excitation at frequency We must be careful to use(4.59), which holds for materials with dispersion. If we apply the simpler formula(4.64), we find that forv→0 E2-∈0E For those excitation frequencies obeying d op we have(we)<0, implying that the material is active. Since there is no mechanism for the plasma to produce energy, this is obviously not valid. But an application of (4.59) gives e)=-E (4.77) which is always positive. In this expression the first term represents the time-average energy stored in the vacuum, while the second term represents the energy stored in the kinetic energy of the electrons. For harmonic excitation, the time-average electron kinetic energy density is (Wa)== v.v. Substituting V from(4.72)with v=0 we see that I N Nm,W=4m,6=面=E which matches the second term of (4.77) 2001 by CRC Press LLC
We can also write Ampere’s law as ∇ × H˜ = J˜i + jω˜ c E˜ where ˜ c is the complex permittivity ˜ c (ω) = (ω) ˜ + σ(ω) ˜ jω = 0 1 − ω2 p ω2 + ν2 − j 0ω2 pν ω(ω2 + ν2) . (4.76) If we wish to describe the plasma in terms of a polarization vector, we merely use D˜ = 0E˜ + P˜ = ˜E˜ to obtain the polarization vector P˜ = (˜ − 0)E˜ = 0χ˜eE˜ , where χ˜e is the electric susceptibility χ˜e(ω) = − ω2 p ω2 + ν2 . We note that P˜ is directed opposite the applied field E˜ , resulting in < ˜ 0. The plasma is dispersive since both its permittivity and conductivity depend on ω. As ω → 0 we have ˜ c → 0r where r = 1 − ω2 p/ν2, and also ˜ c ∼ 1/ω, as remarked in (4.28). As ω → ∞ we have ˜ c − 0 ∼ 1/ω2 and ˜ c ∼ 1/ω3, as mentioned in (4.29). When a transient plane wave propagates through a dispersive medium, the frequency dependence of the constitutive parameters tends to cause spreading of the waveshape. We see that the plasma conductivity (4.75) is proportional to the collision frequency ν, and that, since ˜ c < 0 by the arguments of § 4.5, the plasma must be lossy. Loss arises from the transfer of electromagnetic energy into heat through electron collisions. If there are no collisions (ν = 0), there is no mechanism for the transfer of energy into heat, and the conductivity of a lossless (or “collisionless”) plasma reduces to zero as expected. In a lowloss plasma (ν → 0) we may determine the time-average stored electromagnetic energy for sinusoidal excitation at frequency ωˇ . We must be careful to use (4.59), which holds for materials with dispersion. If we apply the simpler formula (4.64), we find that for ν → 0 we = 1 4 0|Eˇ | 2 − 1 4 0|Eˇ | 2 ω2 p ωˇ 2 . For those excitation frequencies obeying ω<ω ˇ p we have we < 0, implying that the material is active. Since there is no mechanism for the plasma to produce energy, this is obviously not valid. But an application of (4.59) gives we = 1 4 |Eˇ | 2 ∂ ∂ω 0ω 1 − ω2 p ω2 ω=ωˇ = 1 4 0|Eˇ | 2 + 1 4 0|Eˇ | 2 ω2 p ωˇ 2 , (4.77) which is always positive. In this expression the first term represents the time-average energy stored in the vacuum, while the second term represents the energy stored in the kinetic energy of the electrons. For harmonic excitation, the time-average electron kinetic energy density is wq = 1 4 Nmevˇ · vˇ∗. Substituting vˇ from (4.72) with ν = 0 we see that 1 4 Nmevˇ · vˇ∗ = 1 4 Nq2 e meωˇ 2 |Eˇ | 2 = 1 4 0|Eˇ | 2 ω2 p ωˇ 2 , which matches the second term of (4.77)
Figure 4.3: Integration contour used in Kronig-Kramers relations to find 2 from Efor a non-magnetized plasma The complex permittivity of a plasma(4.76) obviously obeys the required frequency- symmetry conditions(4.27). It also obeys the Kronig-Kramers relations required for a causal material. From(4.76) we see that the imaginary part of the complex plasma permittivity is Substituting this into(4.37)we have E(a)-0- P V s2(32+v2)g We can evaluate the principal value integral and thus verify that it produces Ecby Ising the contour method of$ A 1. Because the integrand is even we can extend the domain of integration to(-oo, oo) and divide the result by two. Th ∈0o2v ds P V (g-ju)(g+j)(2-c)(s2 We ate around the closed contour shown in Figure 4.3. Since the integrand falls off as 1/$24 the contribution from Co is zero. The contributions from the semicircles C and C-o are given by j times the residues of the integrand at S=o and at &2=-O, respectively, which are identical but of opposite sign. Thus, the semicircle contributions cancel and leave only the contribution from the residue at the upper-half-plane pole &=jv. Evaluation of the residue gives +jv(v-odv+o) 2001 by CRC Press LLC
Figure 4.3: Integration contour used in Kronig–Kramers relations to find ˜ c from ˜ c for a non-magnetized plasma. The complex permittivity of a plasma (4.76) obviously obeys the required frequencysymmetry conditions (4.27). It also obeys the Kronig–Kramers relations required for a causal material. From (4.76) we see that the imaginary part of the complex plasma permittivity is ˜ c(ω) = − 0ω2 pν ω(ω2 + ν2) . Substituting this into (4.37) we have ˜ c (ω) − 0 = − 2 π P.V. ∞ 0 − 0ω2 pν (2 + ν2) 2 − ω2 d. We can evaluate the principal value integral and thus verify that it produces ˜ c by using the contour method of § A.1. Because the integrand is even we can extend the domain of integration to (−∞,∞) and divide the result by two. Thus ˜ c (ω) − 0 = 1 π P.V. ∞ −∞ 0ω2 pν ( − jν)( + jν) d ( − ω)( + ω). We integrate around the closed contour shown in Figure 4.3. Since the integrand falls off as 1/4 the contribution from C∞ is zero. The contributions from the semicircles Cω and C−ω are given by π j times the residues of the integrand at = ω and at = −ω, respectively, which are identical but of opposite sign. Thus, the semicircle contributions cancel and leave only the contribution from the residue at the upper-half-plane pole = jν. Evaluation of the residue gives ˜ c (ω) − 0 = 1 π 2π j 0ω2 pν jν + jν 1 (jν − ω)(jν + ω) = − 0ω2 p ν2 + ω2
nd thu e(a)=∈o1 which matches(4.76) as expecte 4.6.2 Complex dyadic permittivity of a magnetized plasma When an electron plasma is exposed to a magnetostatic field, as occurs in the earth ionosphere the behavior of the plasma is altered so that the secondary current is no longer aligned with the electric field, requiring the constitutive relationships to be written in terms of a complex dyadic permittivity. If the static field is Bo, the velocity field of the plasma is determined by adding the magnetic component of the Lorentz force to(4.71) giving qcE+ⅴ×Bo]= gOme+mv) or equivalently (o E (4.78) Writing this expression generically as (4.79) we can solve for v as follows. Dotting both sides of the equation with C we quickly establish that C. V=C. A. Crossing both sides of the equation with C, using(B7), and substituting c.a for c.v. we have v×C=AxC+v(C·C)-C(A·C) Finally, substituting v x C back into(4.79) we obtain A-A×C+(A·C)C (4.80) 1+C·C Let us first consider a lossless plasma for which v=0. We can solve(4.78)for v by etting - E Bo Here ac = ge Bol Jwl is called the electron cyclotron frequency. Substituting these (480) Since the secondary current produced by the moving electrons is just J=-Nqev,we have E (4.81) 2001 by CRC Press LLC
and thus ˜ c (ω) = 0 1 − ω2 p ν2 + ω2 , which matches (4.76) as expected. 4.6.2 Complex dyadic permittivity of a magnetized plasma When an electron plasma is exposed to a magnetostatic field, as occurs in the earth’s ionosphere, the behavior of the plasma is altered so that the secondary current is no longer aligned with the electric field, requiring the constitutive relationships to be written in terms of a complex dyadic permittivity. If the static field is B0, the velocity field of the plasma is determined by adding the magnetic component of the Lorentz force to (4.71), giving −qe[E˜ + v˜ × B0] = v˜(jωme + meν) or equivalently v˜ − j qe me(ω − jν) v˜ × B0 = j qe me(ω − jν) E˜ . (4.78) Writing this expression generically as v + v × C = A, (4.79) we can solve for v as follows. Dotting both sides of the equation with C we quickly establish that C · v = C · A. Crossing both sides of the equation with C, using (B.7), and substituting C · A for C · v, we have v × C = A × C + v(C · C) − C(A · C). Finally, substituting v × C back into (4.79) we obtain v = A − A × C + (A · C)C 1 + C · C . (4.80) Let us first consider a lossless plasma for which ν = 0. We can solve (4.78) for v˜ by setting C = − j ωc ω , A = j 0ω2 p ωNqe E˜ , where ωc = qe me B0. Here ωc = qeB0/me = |ωc| is called the electron cyclotron frequency. Substituting these into (4.80) we have ω2 − ω2 c v˜ = j 0ωω2 p Nqe E˜ + 0ω2 p Nqe ωc × E˜ − j ωc ω 0ω2 p Nqe ωc · E˜ . Since the secondary current produced by the moving electrons is just J˜s = −Nqev˜, we have J˜s = jω − 0ω2 p ω2 − ω2 c E˜ + j 0ω2 p ω(ω2 − ω2 c ) ωc × E˜ + ωc ω2 0ω2 p ω2 − ω2 c ωc · E˜ . (4.81)
Now, by the Ampere-Maxwell law we can write for currents in free space V×H=J+J+jooE. (4.82) Considering the plasma to be a material implies that we can describe the gas in terms of a complex permittivity dyadic E such that the Ampere-Maxwell law is V×H=J+joe·E Substituting(4.81)into(4.82), and defining the dyadic ac so that acE XE.we identify the dyadic permittivity (o)=6o-6o 1+J( “cc 4.83 Note that in rectangular coordinates acz 0 (4.84) cy aer 0 To examine the properties of the dyadic permittivity it is useful to write it in matrix form. To do this we must choose a coordinate system. We shall assume that bo is aligned along the z-axis such that Bo= iBo and w= ioc. Then(4.84)become 0-a20 [aa]=a200 000 and we can write the permittivity dyadic(4.83)as e(a)=j6∈0 86 where ∈022 Note that the form of the permitt yadic is that for a lossless gyrotropic material (233) Since the plasma is lossless, equation(4.49) shows that the dyadic permittivity must be hermitian. Equation(4.86)confirms We also note that since the sign of we is determined by the sign of Bo, the dyadic permittivity obeys the symmetry relation o)=E;(-B0) (4.87) as does the permittivity matrix of any material that has anisotropic properties dependent on an externally applied magnetic field [141]. We will find later in this section that the permeability matrix of a magnetized ferrite also obeys such a symmetry condition 2001 by CRC Press LLC
Now, by the Ampere–Maxwell law we can write for currents in free space ∇ × H˜ = J˜i + J˜s + jω0E˜ . (4.82) Considering the plasma to be a material implies that we can describe the gas in terms of a complex permittivity dyadic ˜¯ c such that the Ampere–Maxwell law is ∇ × H˜ = J˜i + jω˜¯ c · E˜ . Substituting (4.81) into (4.82), and defining the dyadic ω¯ c so that ω¯ c · E˜ = ωc × E˜ , we identify the dyadic permittivity ˜¯ c (ω) = 0 − 0 ω2 p ω2 − ω2 c ¯ I + j 0ω2 p ω(ω2 − ω2 c ) ω¯ c + 0ω2 p ω2(ω2 − ω2 c ) ωcωc. (4.83) Note that in rectangular coordinates [ω¯ c] = 0 −ωcz ωcy ωcz 0 −ωcx −ωcy ωcx 0 . (4.84) To examine the properties of the dyadic permittivity it is useful to write it in matrix form. To do this we must choose a coordinate system. We shall assume that B0 is aligned along the z-axis such that B0 = zˆB0 and ωc = zˆωc. Then (4.84) becomes [ω¯ c] = 0 −ωc 0 ωc 0 0 0 00 (4.85) and we can write the permittivity dyadic (4.83) as [˜¯(ω)] = − jδ 0 jδ 0 0 0 z (4.86) where = 0 1 − ω2 p ω2 − ω2 c , z = 0 1 − ω2 p ω2 , δ = 0ωcω2 p ω(ω2 − ω2 c ) . Note that the form of the permittivity dyadic is that for a lossless gyrotropic material (2.33). Since the plasma is lossless, equation (4.49) shows that the dyadic permittivity must be hermitian. Equation (4.86) confirms this. We also note that since the sign of ωc is determined by the sign of B0, the dyadic permittivity obeys the symmetry relation ˜ c i j(B0) = ˜ c ji(−B0) (4.87) as does the permittivity matrix of any material that has anisotropic properties dependent on an externally applied magnetic field [141]. We will find later in this section that the permeability matrix of a magnetized ferrite also obeys such a symmetry condition.