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329sum11hw7 - J = σ E Gauss’s law ∇ E = ρ ± o and...

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ECE 329 Homework 7 Due: Fri, July 8, 2011, 5PM 1. Verify that vector identity H · ∇ × E - E · ∇ × H = · ( E × H ) holds for E = 2ˆ xe - α z and H = 4ˆ ye - α z 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 fluid dynamics. 2. a) For 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, find 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 specified in meter units. Hint: use the continuity equation ∂ρ t + · J = 0 . b) In part (a), deduce the physical units of the coe ffi cients 4 , 3 , and 2 used in J x , J y , and J z specifications, respectively, by applying dimensional analysis. 3. a) Show that in a homogeneous conductor where
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Unformatted text preview: J = σ E , Gauss’s law ∇· E = ρ ± o and the continuity equation ∂ρ ∂t + ∇· J = 0 can be used together to derive a di´erential equation ∂ρ ∂t + σ ± o ρ = 0 for the charge density ρ . b) ±ind the solution of the di´erential equation above for t > if at t = 0 the charge density is ρ ( x, y, z, 0) = cos(100 x ) C/m 3 over all space. c) According to the solution found in part (b), how long would it take for ρ to reduce to . 01 cos(100 x ) C/m 3 ? Assume that σ = 5 . 8 × 10 7 S/m. 4. If E = cos( ωt-βz ) ˆ x V m , ω β = c , and μ = μ o , ²nd the corresponding H by using ±araday’s law ∇× E =-∂ B ∂t . 5. If H = cos( ωt + βz ) ˆ y A m , σ = 0 , ω β = 2 3 c and ± = 2 . 25 ± o , ²nd the corresponding E by using Ampere’s law ∇× H = J + ∂ D ∂t in which J = σ E and D = ± E . 1...
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