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450sp11hw3 - ECE 450 Homework 3 Due Tue Feb 8 2011 5 PM 1 A...

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ECE 450 Homework 3 Due: Tue, Feb 8, 2011, 5 PM 1. A Hertzian radiation source with the current density J ( r , t ) = ˆ xI o Δ x δ ( x - 100 λ ) δ ( y ) δ ( z - 100 λ ) cos( ω t ) A m 2 , is embedded in free space and its oscillation frequency f = ω 2 π = 300 MHz. Assuming that I o Δ x = 1 A.m, determine the numerical values (in appropriate units) of a) wavelength λ of the radiation field, b) the corresponding wavenumber k , c) retarded vector potential phasor ˜ A at r = 200 λ ˆ z , d) the radiation electric field phasor ˜ E at r = 200 λ ˆ z , e) the radiation magnetic field intensity phasor ˜ H at r = 200 λ ˆ z , and f) the time-average Poynting vector E × H at the same location. Note: the vector quantities in parts (c-f) should be specified in terms of unit vectors ˆ x , ˆ y , ˆ z . 2. Three identical Hertzian dipoles I Δ are co-located at the origin aligned with the positive x -, y -, and z -axis, respectively. a) Determine the total retarded vector potential phasor ˜ A of the dipoles acting in unison. b) Express the corresponding radiation field phasor ˜ E of the dipoles at (i) ( x, y, z ) = ( d, 0 , 0) , and (ii) ( x, y, z ) = ( d, d, d ) in terms of dipole moment I Δ , wavenumber
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