Electrodynamics, chap09

Electrodynamics, chap09 - Chapter 9: Radiating Systems,...

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Chapter 9: Radiating Systems, Multipole Fields and Radiation (covered in this course) source term in wave equation boundary Ch. 7 none An Overview of Chapters on EM Waves: plane wave in space or in two semi- spaces separated by t - he - plane Ch. 8 none conducting walls Ch. 9 , it xy e ω ρ J - outgoing wave to prescribed, as in an antenna Ch. 10 , e J outgoing wave to induced by incident EM waves, as in the case of scattering of a plane wave by a dielectric object. Ch. 14 moving charges, outgoing wave to such as electrons in a synchrotron
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9.6 Spherical Wave Solutions of the Scalar Wave Equation Although this chapter deals with radiating systems, here we first solve the scalar source-free wave equation in the spherical coordinate syatem. Th Spherical Bessel Functions and Hankel functions : e purpose is to obtain a complete set of spherical Bessel funtions and Hankel functions, with which we will expand the fields produced by the sources. The scalar source-free wave equation is [see 2 22 2 1 (6.32)] ( , ) ( , ) 0 (9.77) Let ( , ) ( , ) (9.78) Each Fourier component satisfies the Helmholtz w ω ψψ −∞ ∇− = = ct it tt te d xx 2 2 2 ave eq. ( , ) 0 with (9.79) () ψω ∇+ = = kk c x
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() 2 22 2 2 2 2 2 2 2 2 2 11 1 1 sin sin sin sin In the spherical coordinate system, ( ) 0 is written sin 0 Let ( ) ( ) ( ), we obtain sin ψψ ψ θθ ϕ θψ ψθ θ ∂∂ ∇+ = ++ + = = dQ dd U d d P dr d d d rr r dr r k rk UrP Q PQ r UQ UP N 2 2 2 2 2 1 (1 ) sin sin 0 Multiply by sin ( ) (sin ) 0 Thus, as in Sec. 3.1 of lecture notes, (cos ), (cos ); , [] ϕϕ ν =+ =− + = + = == = ±²²²³²²²´ U d d P Q d ll m mm i m i m P U dr dr UPQ kUPQ r r PP Q Qe e P Q Y (, ) θϕ lm The -dependence is isolated within this term, so this term must be a constant. Let it be - m 2 . 2 Dividing all terms by sin , we see that the -dependence is isolated within this term. So this term must be a constant. Let it be ( 1). r + 9.6 Spherical Wave Solutions… ( continued ) 21 ( ) ! (cos ) 4( ) ! mi m l m P e lm π +− + rejected because of divergence at = ±
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9.6 Spherical Wave Solutions… ( continued ) 2 22 2 2 1/2 2 2 (1 ) 2 2 / 2 ) 2 11 is a function of . Rewrite it as ( ). Then, ( ) 0 (9.81) Let ( ) ( ) l ll dd l rdr dr r l rd r r U(r) l f r kf r fr ur k + + ⎡⎤ ++ = ⎣⎦ =⇒ + + () 1 2 1 2 1 2 1 2 (1) 2 2 ( ) (9.83) ( ) ( ), ( ) [Bessel functions of fractional order] ( ) , Define and l l l rr l l l l kr l l kr J k r N k r J k r N k r jk r J k r hk r j nk r N k r π + + = ⇒= = = = (2) (1) (1) (2) (2) ( , ) ( ) ( ) ( , ) (9.92) l l lm lm l lm l lm kr in kr r j k r i n k r Ah k r A h k r Y ψω θ φ + =− + x spherical Bessel functions Hankel functions
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0 () nx 1 0 jx 1 2 2 0 (0 ) 1 == ⎡⎤ ⎣⎦ x x 9.6 Spherical Wave Solutions… ( continued ) (1) (2) See Jackson pp. 426-427 for further properties of , , , and . ll j nh h 2 2 1 (2 1)!! 2(2 3) 2(1 2 ) 1, 1 1 l l xx l l x l l x xl + ++ ⎯⎯⎯ →− + ⎯⎯⎯→− + ± ± " " ( ) 1 2 1 2 1 s i n c o s ( ) [ s p a t i a l dependence of spherical waves.] ix l l x l l x l e l x x x hx i π + ⎯⎯→ ⎯⎯→− ⎯⎯→ − ² ² ² From G. Afken, "Mathematical Methods for Physicists"
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9.6 Spherical Wave Solutions… ( continued ) 22 Solution of the Green equation ( ) ( , ) 4 ( ) (6.36) is given by (derived in Sec. 6.4.)
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Electrodynamics, chap09 - Chapter 9: Radiating Systems,...

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