Analysis of Electromagnetic Waves: Multipole Moments & Scattering of a Conducting Sphere, Assignments of Physics

Download Analysis of Electromagnetic Waves: Multipole Moments & Scattering of a Conducting Sphere and more Assignments Physics in PDF only on Docsity! Physics 506 Winter 2006 Homework Assignment #5 — Solutions Textbook problems: Ch. 9: 9.19, 9.22 a) and b), 9.23 Ch. 10: 10.1 9.19 Consider the excitation of a waveguide in Problem 8.19 from the point of view of multipole moments of the source. a) For the linear probe antenna calculate the multipole moment components of ~p, ~m, Qαβ , QMαβ that enter (9.69). Note that the multipole moments depend on the choice of origin of the coordinates used to describe the source. Perhaps the most natural situation is to place the origin at the bottom of the probe. In this case, the source current density can be expressed as ~J = ŷI0 sin[ωc (h− y)]δ(x)δ(z)Θ(h− y) The electric dipole moment can be computed directly from the current density ~p = i ω ∫ ~J d3x = iI0 ω ŷ ∫ h 0 sin[ωc (h− y)] dy = icI0 ω2 ( 1− cos ωh c ) ŷ = 2icI0 ω2 sin2 ( ωh 2c ) ŷ For the magnetic dipole moment, we first compute the magnetization ~M = 12~x× ~J = 1 2 (x, y, z)× ŷI0 sin[ ω c (h− y)]δ(x)δ(z)Θ(h− y) = 12I0(−z, 0, x) sin[ ω c (h− y)]δ(x)δ(z)Θ(h− y) = 0 since x = z = 0 is enforced by the delta functions. This indicates that the mag- netic dipole moment ~m = ∫ ~M d3x vanishes for the probe antenna. Furthermore, since the magnetic quadrupole moment QMαβ may be computed using the effective magnetic charge density ρM = −~∇ · ~M, it also vanishes. So we are left with the electric quadrupole moment Qαβ = ∫ (3xαxβ − δαβr2)ρ d3x where ρ = − i ω ~∇ · ~J = iI0 c cos[ωc (h− y)]δ(x)δ(z)Θ(h− y) Because of the delta functions, the only non-vanishing quadrupole moments are Qxx = Qzz = − 12Qyy = − ∫ y2ρ d3x = − iI0 c ∫ h 0 y2 cos[ωc (h− y)] dy = −2iI0c ω2 ( h− c ω sin ωh c ) The resulting multipole moments are ~p = 2icI0 ω2 sin2 ( ωh 2c ) ŷ ~m = 0 Qxx = Qzz = − 12Qyy = − 2iI0c ω2 ( h− c ω sin ωh c ) QMαβ = 0 (1) b) Calculate the amplitudes for excitation of the TE1,0 mode and evaluate the power flow. Compare the multipole expansion result with the answer given in Problem 8.19b). Discuss the reasons for agreement or disagreement. What about the comparison for excitation of other modes? We consider the normalized TEmn mode specified by the field Hz = H0 cos mπx a cos nπy b where H0 = 2iγmn µ0ω √ ab , (a → 2a if m = 0 and b → 2b if n = 0) and γ2mn = (mπ a )2 + (nπ b )2 Note that this choice of coordinates places x = y = 0 at the lower left of the rectangular waveguide. Since the magnetic moments vanish, we only need the explicit expressions for the electric field. Using ~Et = − iµ0ω γ2mn ẑ × ~∇tHz we obtain Ex = − iµ0ω γ2mn nπ b H0 cos mπx a sin nπy b Ey = iµ0ω γ2mn mπ a H0 sin mπx a cos nπy b Ez = 0 (2) we see that the modes parametrized by aM (l,m) are TE modes, while those parametrized by aE(l, m) are TM modes. In particular, the TE modes may be given by ~H = − i k ~∇× jl(kr) ~Xlm, ~E = Z0jl(kr) ~Xlm (3) We now impose the boundary conditions H⊥ = 0 and E‖ = 0, or more precisely r̂ · ~H ∣∣∣ r=a = 0, r̂ × ~E ∣∣∣ r=a = 0 These are equivalent to the condition jl(ka) = 0, and leads to the quantization klmn = xln/a where xln is the n-th zero of the spherical Bessel function jl. The TElmn frequencies are thus ωlmn = xlnc a , jl(xln) = 0, l ≥ 1, |m| ≤ l Each frequency specified by l and n is (2l + 1)-fold degenerate, with azimuthal quantum number labeled by m. The TM modes are similar, although the boundary conditions are somewhat more involved. The modes themselves are given by ~H = jl(kr) ~Xlm, ~E = Z0 i k ~∇× jl(kr) ~Xlm (4) This time, the H⊥ = 0 boundary condition is automatic, while the E‖ = 0 condition gives ~r × (~∇× jl(kr) ~Xlm) ∣∣∣ r=a = 0 This vector quantity may be simplified using ~r× (~∇× ~V ) = ~∇(~r · ~V )− ~V − (~r · ~∇)~V = ~∇(~r · ~V )− (1+ r∂r)~V = ~∇(~r · ~V )− ∂rr~V Using ~V = jl(kr) ~Xlm with ~r · ~Xlm = 0 gives ~r × (∇× jl(kr) ~Xlm) = −∂r(rjl(kr)) ~Xlm (5) Hence the E‖ = 0 boundary condition leads to the TMlmn frequencies ωlmn = ylnc a , d dx [xjl(x)] ∣∣∣∣ z=yln = 0, l ≥ 1, |m| ≤ 1 The yln correspond to zeros of [xjl(x)]′ or equivalently jl(x) + xj′l(x). b) Calculate numerical values for the wavelength λlm in units of the radius a for the four lowest modes for TE and TM waves. The numerical values for the wavelengths are obtained from the zeros xln and yln. For TE modes, the first four zeros of jl(x) are x11 = 4.4934, x21 = 5.7635, x31 = 6.9879, x12 = 7.7253 Since klmn = xln/a and λlmn = 2π/klmn, we end up with λlmn/a = 2π/xln or λ1m1 a = 1.398, λ2m1 a = 1.090, λ3m1 a = 0.899, λ1m2 a = 0.813 All these modes are (2l + 1)-fold degenerate. For TM modes, the first four zeros of [xjl(x)]′ are y11 = 2.7437, y21 = 3.8702, y31 = 4.9734, y41 = 6.0619 with corresponding wavelengths λ1m1 a = 2.290, λ2m1 a = 1.623, λ3m1 a = 1.263, λ4m1 a = 1.036 Note that the next mode, given by y12 = 6.1168 is nearly degenerate with y41. 9.23 The spherical resonant cavity of Problem 9.22 has nonpermeable walls of large, but finite, conductivity. In the approximation that the skin depth δ is small compared to the cavity radius a, show that the Q of the cavity, defined by equation (8.86), is given by Q = a δ for all TE modes Q = a δ ( 1− l(l + 1) x2lm ) for TM modes where xlm = (a/c)ωlm for TM modes. In order to calculate the Q factor, we need to obtain both the stored energy and the power loss at the walls. We start with the simpler case of TE modes, given by (3). The energy density for harmonic fields is u = 0 4 | ~E|2 + µ0 4 | ~H|2 However, the energy is equally distributed between ~E and ~H. Thus for TE modes we may immediately write down u = 0 2 | ~E|2 = µ0 2 jl(kr)2| ~Xlm|2 The stored energy is given by integrating this over the volume of the sphere U = µ0 2 ∫ jl(kr)2| ~Xlm|2 r2drdΩ = µ0 2 ∫ a 0 jl(kr)2r2dr We now use the normalization integral for spherical Bessel functions∫ a 0 jl(xlmρ/a)jl(xlnρ/a)ρ2dρ = 12a 3[j′l(xln)] 2δmn to obtain Ulmn = µ0a 3 4 j′l(xln) 2 (6) The power loss is given in terms of the tangential magnetic field at the conducting surface P = 1 2σδ ∫ |r̂ × ~H|2da Using ~h = −(i/k)~∇× jl(kr) ~Xlm as well as the vector identity (5) gives Plmn = 1 2σδ ∫ r=a ( 1 kr d dr rjl(kr) )2 | ~Xlm|2r2dΩ = 1 2σδk2 ([rjl(kr)]′)2 ∣∣∣∣ r=a = 1 2σδk2 (jl(ka) + kaj′l(ka)) 2 = a2 2σδ j′l(xln) 2 (7) where in the last line we made use of the fact that ka = xln and that jl(xln) = 0. Combining (6) and (7) then gives the Q factor for TE modes Qlmn = ω Ulmn Plmn = µ0σωδa 2 = a δ where we made use of the definition of the skin depth δ = √ 2/µ0σω. The calculation for TM modes is similar. However, the appropriate spherical Bessel function normalization integral needs to be modified for integrating to zeros of [xjl(x)]′. Here we simply state that the appropriate normalization integral may be written as∫ a 0 jl(αmρ/a)jl(αnρ/a)ρ2dρ = 12a 3 ( 1 + p(p− 1)− l(l + 1) α2n ) [jl(αn)]2δmn where αn is the n-th positive zero of [xpjl(x)]′ = 0