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# 329sp09hw6 - ECE 329 1 Verify that vector identity Homework...

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ECE 329 Homework 6 Due: Tue, March 3, 2009, 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. 2. Consider an infinite surface current density J s = - ˆ xJ so cos( ω t ) flowing on z = 0 surface, where J so > 0 is real-valued amplitude of the monochromatic surface current measured in A/m units. It is found that J s injects field energy into propagating transverse electromagnetic (TEM) waves away from the z = 0 plane at an average rate of 1 W/m 2 — that is, the magnitude of the average Poynting vector ¯ P is 1 2 W/m 2 for the waves excited by the surface current. a) Denoting the TEM waves excited by J s (above and below the z = 0 plane) as E = ˆ xE o cos( ω t β z ) V/m and H = ± ˆ yH o cos( ω t β z ) A/m, where wavenumber β = ω c , determine the numerical values of wave amplitudes E o and H o in V/m and A/m units (assuming wave propagation in free space). Hint: equate the expression for the magnitude of ¯ P for each wave to 1 2 W/m 2 .

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329sp09hw6 - ECE 329 1 Verify that vector identity Homework...

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