The Hertzian potentials. With a little manipulation and the introduction of a new notation, we can maintain the wave nature of the potential functions and still provide a decomposition into purely lamellar and solenoidal components. In this analysis we shall assume lossless media only. When we chose the lorentz gauge to remove the arbitrariness of the divergence of the vector potential, we established a relationship between Ae and e. Thus we should be able to write both the electric and magnetic fields in terms of a single potential function. From the lorentz gauge we can write e 中(r,t) V·A2(r,t)dt By(5.17)and (.18)we can thus write the EM fields as E V·Ad B=V (5.34) The integro-differential representation of E in(5.33) is somewhat clumsy in appear ance. We can make it easier to manipulate by defining the Hertzian potential In differential form 4=i (5.35) With this, (5. 33)and(5.34)become E=V(V.le)-u (5.36) die (5.37) An equation for Ile in terms of the source current can be found by substituting (5. 35) nto(5.31) ∈(v2Ⅱ-k∈%I Let us define For general impressed current sources(5. 38)is just a convenient notation. However,we can conceive of an impressed polarization current that is independent of e and defined through the relation D= EoE +P+P. Then(.38)has a physical interpretation as described in(2. 119). We now have which is a wave equation for Ile. Thus the Hertzian potential has the same wave behavior as the vector potential under the lorentz gauge ②2001
The Hertzian potentials. With a little manipulation and the introduction of a new notation, we can maintain the wave nature of the potential functions and still provide a decomposition into purely lamellar and solenoidal components. In this analysis we shall assume lossless media only. When we chose the Lorentz gauge to remove the arbitrariness of the divergence of the vector potential, we established a relationship between Ae and φe. Thus we should be able to write both the electric and magnetic fields in terms of a single potential function. From the Lorentz gauge we can write φe as φe(r, t) = − 1 µ t −∞ ∇ · Ae(r, t) dt. By (5.17) and (5.18) we can thus write the EM fields as E = 1 µ ∇ t −∞ ∇ · Aedt − ∂Ae ∂t , (5.33) B =∇× Ae. (5.34) The integro-differential representation of E in (5.33) is somewhat clumsy in appearance. We can make it easier to manipulate by defining the Hertzian potential Πe = 1 µ t −∞ Ae dt. In differential form Ae = µ ∂Πe dt . (5.35) With this, (5.33) and (5.34) become E = ∇(∇ · Πe) − µ ∂2 ∂t 2 Πe, (5.36) B = µ∇ × ∂Πe ∂t . (5.37) An equation for Πe in terms of the source current can be found by substituting (5.35) into (5.31): µ ∂ ∂t ∇2 Πe − µ ∂2 ∂t 2 Πe = −µJi . Let us define Ji = ∂Pi ∂t . (5.38) For general impressed current sources (5.38) is just a convenient notation. However, we can conceive of an impressed polarization current that is independent of E and defined through the relation D = 0E + P + Pi . Then (5.38) has a physical interpretation as described in (2.119). We nowhave ∇2 Πe − µ ∂2 ∂t 2 Πe = −1 Pi , (5.39) which is a wave equation for Πe. Thus the Hertzian potential has the same wave behavior as the vector potential under the Lorentz gauge.
We can use(5. 39) to perform one final simplification of the EM field representation By the vector identity V(V.f)=Vx(VxI)+V-Ii we get V×(v× ∈ Substituting this into(5.36)we obtain P v×(x正) B=∈V (5.41) Let us examine these closely. We know that b is solenoidal since it is written as the curl of another vector(this is also clear from the auxiliary Maxwell equation VB=0). The first term in the expression for E is also solenoidal. So the lamellar part of E must be contained within the source term p if we write pl in terms of its lamellar and solenoidal omponents by using J then(5.40)becomes E=V× (V xILa PS1 Pi (5.42) So we have again succeeded in dividing E into lamellar and solenoidal components Potential functions for magnetic current. We can proceed as above to derive the field-potential relationships when J=0 but Jn #0. We assume a homogeneous, loss- less, isotropic medium with permeability u and permittivity E, and begin with Faradays ⅴ×H=Dat We write H and D in terms of two potential functions Ah and h as H D A and the differential equation for the potentials is found by substitution into (5.43) vx(×A)=Jm-∈812-a1Y Taking the divergence of this equation and substituting from the magnetic continuit quation we obtain V·Ah+∈-Vφh ②2001
We can use (5.39) to perform one final simplification of the EM field representation. By the vector identity ∇(∇ · Π) =∇× (∇ × Π) + ∇2Π we get ∇ (∇ · Πe) =∇× (∇ × Πe) − 1 Pi + µ ∂2 ∂t 2 Πe. Substituting this into (5.36) we obtain E =∇× (∇ × Πe) − Pi , (5.40) B = µ∇ × ∂Πe ∂t . (5.41) Let us examine these closely. We knowthat B is solenoidal since it is written as the curl of another vector (this is also clear from the auxiliary Maxwell equation ∇ · B = 0). The first term in the expression for E is also solenoidal. So the lamellar part of E must be contained within the source term Pi . If we write Pi in terms of its lamellar and solenoidal components by using Ji s = ∂Pi s ∂t , Ji l = ∂Pi l ∂t , then (5.40) becomes E = ∇ × (∇ × Πe) − Pi s − Pi l . (5.42) So we have again succeeded in dividing E into lamellar and solenoidal components. Potential functions for magnetic current. We can proceed as above to derive the field–potential relationships when Ji = 0 but Ji m = 0. We assume a homogeneous, lossless, isotropic medium with permeability µ and permittivity , and begin with Faraday’s and Ampere’s laws ∇ × E = −Ji m − ∂B ∂t , (5.43) ∇ × H = ∂D ∂t . (5.44) We write H and D in terms of two potential functions Ah and φh as H = −∂Ah ∂t − ∇φh, D = −∇ × Ah, and the differential equation for the potentials is found by substitution into (5.43): ∇ × (∇ × Ah) = Ji m − µ ∂2Ah ∂t 2 − µ ∂ ∂t ∇φh. (5.45) Taking the divergence of this equation and substituting from the magnetic continuity equation we obtain µ ∂2 ∂t 2 ∇ · Ah + µ ∂ ∂t ∇2 φh = − ∂ρi m ∂t .
Under the lorentz gauge condition this reduces to -h p xpanding the curl-curl operation in(5.45)we have v(V·Ah)-V2Ah=∈Jm-H∈ which, upon substitution of the lorentz gauge condition gives vA-∈4 J (5.46) We can also derive a Hertzian potential for the case of magnetic current. Letting all 47 and employing the lorentz condition we have aIn H=V(v·In)-μ∈ The wave equation for IIh is found by substituting(5.47)into(5.46) to give 72I Defining M through J M re write the wave equation as V-lh-u aln_-M We can think of M as a convenient way of representing Jm, or we can conceive of an impressed magnetization current that is independent of H and defined through B o(H+M+M). With the help of(5. 48) we can also write the fields as H=×(×I)-M dIh ②2001
Under the Lorentz gauge condition ∇ · Ah = −µ ∂φh ∂t this reduces to ∇2 φh − µ ∂2φh ∂t 2 = −ρi m µ . Expanding the curl-curl operation in (5.45) we have ∇(∇ · Ah) − ∇2 Ah = Ji m − µ ∂2Ah ∂t 2 − µ ∂ ∂t ∇φh, which, upon substitution of the Lorentz gauge condition gives ∇2 Ah − µ ∂2Ah ∂t 2 = −Ji m. (5.46) We can also derive a Hertzian potential for the case of magnetic current. Letting Ah = µ ∂Πh ∂t (5.47) and employing the Lorentz condition we have D = −µ∇ × ∂Πh ∂t , H = ∇(∇ · Πh) − µ ∂2Πh ∂t 2 . The wave equation for Πh is found by substituting (5.47) into (5.46) to give ∂ ∂t ∇2 Πh − µ ∂2Πh ∂t 2 = − 1 µ Ji m. (5.48) Defining Mi through Ji m = µ ∂Mi ∂t , we write the wave equation as ∇2 Πh − µ ∂2Πh ∂t 2 = −Mi . We can think of Mi as a convenient way of representing Ji m, or we can conceive of an impressed magnetization current that is independent of H and defined through B = µ0(H + M + Mi ). With the help of (5.48) we can also write the fields as H =∇× (∇ × Πh) − Mi, D = −µ∇ × ∂Πh ∂t .
Summary of potential relations for lossless media. When both electric and mag netic sources are present, we may superpose the potential representations derived above We assume a homogeneous, lossless medium with time-invariant parameters u and E. For the scalar/vector potential representation we have E Vφ--V×A (5.49 at Here the potentials satisfy the wave equations (5.51) aA ∈Jm and are linked by the lorentz conditions We also have the Hertz potential representation a正 E=v(V·I) x(×I) P al H=∈V (×m)-M The Hertz potentials satisfy the wave equations a2)[I IIh Potential functions for the frequency-domain fields. In the frequency domain it is much easier to handle lossy media. Consider a lossy, isotropic, homogeneous medium described by the frequency-dependent parameters A, E, and o. Maxwells curl equations E=-J V×H=J+joEE Here we have separated the primary and secondary currents through J=J+oE, and used the complex permittivity E=E+o/jo. As with the time-domain equations we ②2001
Summary of potential relations for lossless media. When both electric and magnetic sources are present, we may superpose the potential representations derived above. We assume a homogeneous, lossless medium with time-invariant parameters µ and . For the scalar/vector potential representation we have E = −∂Ae ∂t − ∇φe − 1 ∇ × Ah, (5.49) H = 1 µ ∇ × Ae − ∂Ah ∂t − ∇φh. (5.50) Here the potentials satisfy the wave equations ∇2 − µ ∂2 ∂t 2 Ae φe = −µJi −ρi , (5.51) ∇2 − µ ∂2 ∂t 2 Ah φh = −Ji m −ρi m µ , and are linked by the Lorentz conditions ∇ · Ae = −µ ∂φe ∂t , ∇ · Ah = −µ ∂φh ∂t . We also have the Hertz potential representation E = ∇(∇ · Πe) − µ ∂2Πe ∂t 2 − µ∇ × ∂Πh ∂t =∇× (∇ × Πe) − Pi − µ∇ × ∂Πh ∂t , (5.52) H = ∇ × ∂Πe ∂t + ∇(∇ · Πh) − µ ∂2Πh ∂t 2 = ∇ × ∂Πe ∂t +∇× (∇ × Πh) − Mi . (5.53) The Hertz potentials satisfy the wave equations ∇2 − µ ∂2 ∂t 2 Πe Πh = −1 Pi −Mi . Potential functions for the frequency-domain fields. In the frequency domain it is much easier to handle lossy media. Consider a lossy, isotropic, homogeneous medium described by the frequency-dependent parameters µ˜ , ˜, and σ˜. Maxwell’s curl equations are ∇ × E˜ = −J˜i m − jωµ˜ H˜ , (5.54) ∇ × H˜ = J˜i + jω˜ c E˜ . (5.55) Here we have separated the primary and secondary currents through J˜ = J˜i + σ˜E˜ , and used the complex permittivity ˜ c = ˜ + σ/˜ jω. As with the time-domain equations we
introduce the potential functions using superposition. If Jm=0 and J#0 then we ay introduce the electric potentials through the relationships E=-Voe -JoA (5.57) assuming the Lorentz conditio V·A=- Joueφ we find that upon substitution of (5.56)-(5.57) into(5.54)-(5.55) the potentials must obey the Helmholtz equation If Jm#0 and J=0 then we may ce the magnetic potentials through E H=-Voh-jo assuming φh we find that upon substitution of (5.58)-(5.59) into(5.54)-(5. 55) the potentials must (V2+k2) -pm/i J When both electric and magnetic sources are present, we use superposition H=V×A2-Vdh-jaAb2 Using the Lorentz conditions we can also write the fields in terms of the vector potentials E=-V(V·A A2-V×A (5.60) 且=×A-Bv(A)-joA (5.61) We can also define Hertzian potentials for the frequency-domain fields. When Jm= 0 J≠0 A。= joue Il。 E=V(VⅡ)+k=Vx(Vx) Joe (5.62) ②2001
introduce the potential functions using superposition. If J˜i m = 0 and J˜i = 0 then we may introduce the electric potentials through the relationships E˜ = −∇φ˜ e − jωA˜ e, (5.56) H˜ = 1 µ˜ ∇ × A˜ e. (5.57) Assuming the Lorentz condition ∇ · A˜ e = − jωµ˜ ˜ c φ˜ e, we find that upon substitution of (5.56)–(5.57) into (5.54)–(5.55) the potentials must obey the Helmholtz equation ∇2 + k2 φ˜ e A˜ e = −ρ˜i /˜ c −µ˜ J˜i . If J˜i m = 0 and J˜i = 0 then we may introduce the magnetic potentials through E˜ = − 1 ˜ c ∇ × A˜ h, (5.58) H˜ = −∇φ˜ h − jωA˜ h. (5.59) Assuming ∇ · A˜ h = − jωµ˜ ˜ c φ˜ h, we find that upon substitution of (5.58)–(5.59) into (5.54)–(5.55) the potentials must obey ∇2 + k2 φ˜ h A˜ h = −ρ˜i m/µ˜ −˜ cJ˜i m . When both electric and magnetic sources are present, we use superposition: E˜ = −∇φ˜ e − jωA˜ e − 1 ˜ c ∇ × A˜ h, H˜ = 1 µ˜ ∇ × A˜ e − ∇φ˜ h − jωA˜ h. Using the Lorentz conditions we can also write the fields in terms of the vector potentials alone: E˜ = − jω k2 ∇(∇ · A˜ e) − jωA˜ e − 1 ˜ c ∇ × A˜ h, (5.60) H˜ = 1 µ˜ ∇ × A˜ e − jω k2 ∇(∇ · A˜ h) − jωA˜ h. (5.61) We can also define Hertzian potentials for the frequency-domain fields. When J˜i m = 0 and J˜i = 0 we let A˜ e = jωµ˜ ˜ c Π˜ e and find E˜ = ∇(∇ · Π˜ e) + k2 Π˜ e =∇× (∇ × Π˜ e) − J˜i jω˜ c (5.62)