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Demo13Diffraction

# Demo13Diffraction - ECE 3030 Electromagnetic Fields and...

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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 << λ ). 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 ˆ 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 σ s of the metal sphere and compare it to its physical cross-sectional area, which equals πa 2 . Solution: Scattering from a metal sphere. a. From Lectures 6 and 7, the induced dipole moment is ~ p = 4 πε a 3 E i ˆ z. b. The E-field in the far field produced by a dipole is ~ E s- ff ( ~ r ) = j kη ( jωp ) 4 πr sin θ e- jkr ˆ θ =- k 2 a 3 r E i sin θ e- jkr ˆ θ c. P s = R 2 π R π | ~ E s- ff | 2 2 η r 2 sin θ d θ d φ = 4 π 3 η k 4 a 6 | E i | 2 d. σ s = P s 1 2 η | E i | 2 = 8 3 ( ka ) 4 ( πa 2 ). Since ka << 1, σ s << πa 2 . For Problem 18.2, we used the fact that the far-field electric field for a Hertzian Dipole is ~ E ff ( ~ r ) = j η k ( Id ) 4 πr sin θ e- jkr ˆ θ and the fact that the dipole moment of a charge is p = qk yielding a current of I = jωq so that Id = jωqd = jωp . CORNELL UNIVERSITY c WES SWARTZ (09/11/6) 13–1 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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Demo13Diffraction - ECE 3030 Electromagnetic Fields and...

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