329sum09hw7 - ECE 329 Homework 7 1 Verify that vector...

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ECE 329 Homework 7 Due: July 13, 2009, 5PM 1. Verify that vector identity H · ∇ × E - E · ∇ × H = ∇· ( E × H ) holds for E = 2ˆ xe - αz and H = 4ˆ ye - αz by expanding both sides of the identity. Treat α as a real constant. You should download the table of vector identities from ECE 329 web site and examine the list to familiarize yourself with the listed identities — they are widely employed in electromagnetics as well as in other branches of engineering such as Fuid dynamics. 2. a) ±or current density J = (4 z 2 ˆ x +3 x 3 y ˆ y +2 z ( y - y o ) 2 ˆ z ) A/m 2 , which is time independent, ²nd the charge density ρ (0 ,t ) at the origin (0,0,0) as a function of time t , if ρ =0 there at time t =0 , y o =1 m, and coordinates x , y , and z are speci²ed in meter units. Hint: use the continuity equation ∂ρ ∂t + ∇· J =0 . b) In part (a), deduce the physical units of the coe³cients 4 , 3 , and 2 used in J x , J y , and J z speci²cations, respectively, by applying dimensional analysis. 3. Copper is a highly conducting metal with a
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329sum09hw7 - ECE 329 Homework 7 1 Verify that vector...

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