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329fall08hw6 - ECE 329 1 a Prove the identity Homework 6...

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ECE 329 Homework 6 Due: Tue, Oct 7, 2008, 5PM 1. a) Prove the identity H · ∇ × E - E · ∇ × H = ∇· ( E × H ) for arbitrary vector Felds E and H by expanding both sides of the identity under the assumption that Cartesian components of E and H are di±erentiable. b) Another important identity is ∇×∇× F = ( ∇· F ) -∇ 2 F , where 2 F on right is known as Laplacian of F and it is deFned in Cartesian coordinates via 2 F ( 2 ∂x 2 + 2 ∂y 2 + 2 ∂z 2 )(ˆ xF x yF y zF z ) . ²or the special case of F = x 2 ˆ x + z ˆ y , show that ∇×∇× F and ( ∇ · F ) -∇ 2 F are equal in consistency with the general identity quoted above. 2. Consider an inFnite surface current J s = - ˆ xJ so cos( ωt ) ³owing on z =0 surface, where J so > 0 is the real-valued amplitude of the surface current measured in A/m units. It is found that J s injects Feld energy into propagating electromagnetic waves away from z =0 plane at an average rate of 1 W/m
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329fall08hw6 - ECE 329 1 a Prove the identity Homework 6...

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