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Figure 4.26: Geometryof a perfectlyconducting wedge illuminated bya line source.


                        This in turn gives us the surface current induced on the cylinder. From the boundary
                                                                       ˜
                                                      ˜
                                         ˜
                                 ˜
                                                           ˆ ˜
                        condition J s = ˆ n × H| ρ=a = ˆρ × [ˆρH ρ + φH φ ]| ρ=a = ˆ zH φ | ρ=a we have
                                              ∞
                                        j    #      n j −n        (2)
                                           ˜

                            J s (φ, ω) =−  ˆ zE 0  (2)    J (k 0 a)H n  (k 0 a) − J n (k 0 a)H n (2)  (k 0 a) cos nφ,
                                                           n
                                        η 0      H n (k 0 a)
                                             n=0
                        and an application of (E.93) gives us
                                                          ˜   ∞       −n
                                                         2E 0  #     n j
                                             J s (φ, ω) = ˆ z             cos nφ.             (4.365)
                                                                   (2)
                                                       η 0 k 0 πa  H n (k 0 a)
                                                              n=0
                          Computation of the scattered field for a magnetically-polarized impressed field pro-
                        ceeds in the same manner. The impressed electric and magnetic fields are assumed to be
                                                                    ˆ
                                                                           ˜
                                    ˜ i
                                               ˜
                                    E (r,ω) = ˆ yE 0 (ω)e − jk 0 x  = (ˆρ sin φ + φ cos φ)E 0 (ω)e − jk 0 ρ cos φ ,
                                               ˜
                                                             ˜
                                               E 0 (ω)  − jk 0 x  E 0 (ω)  − jk 0 ρ cos φ
                                    ˜ i
                                    H (r,ω) = ˆ z   e    = ˆ z    e       .
                                                η 0           η 0
                        For a perfectlyconducting cylinder, the total magnetic field is
                                      ∞     n j −n
                                   ˜ #
                              ˜    E 0                    (2)              (2)
                             H z =                 J n (k 0 ρ)H n  (k 0 a) − J (k 0 a)H n  (k 0 ρ) cos nφ.  (4.366)
                                                                    n
                                           (2)
                                   η 0   H n (k 0 a)
                                     n=0
                        The details are left as an exercise.
                        Boundary value problems in cylindrical coordinates: scattering by a perfectly
                        conducting wedge.   As a second example, consider a perfectlyconducting wedge im-
                        mersed in free space and illuminated by a line source (Figure 4.26) carrying current
                        ˜ I(ω) and located at (ρ 0 ,φ 0 ). The current, which is assumed to be z-invariant, induces
                        a secondarycurrent on the surface of the wedge which in turn produces a secondary

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