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Unformatted text preview: ECE 3030: Electromagnetic Fields and Waves Fall 2009 DEMO 13 INSTRUCTOR NOTES: Diffraction and Aperture Antennas Reminder: Thanksgiving on Thursday...give thanks! (No workshop.) Date: Tuesday 11/24 1. Problem 18.2: Consider a per- fect metal sphere with radius a ( a << ). A plane wave is inci- dent on the sphere, as shown in the figure. The ~ E field of the inci- dent plane wave at the location of the sphere is given as ~ E i ( ~ r = 0) = E i z a Incident plane wave E Figure 1: A sphere scattering an E field. a. Find the dipole moment phasor ~ p (small p) that is induced in the sphere due to the incident plane wave. b. Find the far-field expression ~ E s- ff ( ~ r) for the scattered electric field. c. Find the total scattered power P s . d. Find the scattering cross section s of the metal sphere and compare it to its physical cross-sectional area, which equals a 2 . Solution: Scattering from a metal sphere. a. From Lectures 6 and 7, the induced dipole moment is ~ p = 4 a 3 E i z. b. The E-field in the far field produced by a dipole is ~ E s- ff ( ~ r ) = j k ( jp ) 4 r sin e- jkr =- k 2 a 3 r E i sin e- jkr c. P s = R 2 R | ~ E s- ff | 2 2 r 2 sin d d = 4 3 k 4 a 6 | E i | 2 d. s = P s 1 2 | E i | 2 = 8 3 ( ka ) 4 ( a 2 ). Since ka << 1, s << a 2 . For Problem 18.2, we used the fact that the far-field electric field for a Hertzian Dipole is ~ E ff ( ~ r ) = j k ( Id ) 4 r sin e- jkr and the fact that the dipole moment of a charge is p = qk yielding a current of I = jq so that Id = jqd = jp . CORNELL UNIVERSITY c WES SWARTZ (09/11/6) 131 ECE 3030: Electromagnetic Fields and Waves Fall 2009 DEMO 13 INSTRUCTOR NOTES: Diffraction and Aperture Antennas 2. Problem 18.3 In this problem you will look at the operation of a diffraction-grating spectrometer. A diffraction grating consists ofspectrometer....
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