329sp08hw6 - ECE 329 Homework 6 Due: Feb 25, 2008, 5PM 1....

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Unformatted text preview: ECE 329 Homework 6 Due: Feb 25, 2008, 5PM 1. a) Prove the identity H · ∇ × E- E · ∇ × H = ∇· ( E × H ) for arbitrary vector fields E and H by expanding both sides of the identity under the assumption that Cartesian components of E and H are differentiable. b) Prove the identity ∇×∇× F = ∇ ( ∇· F )-∇ 2 F for an arbitrary vector F by expanding and re-arranging the left hand side of the identity into the form given on the right, assuming that Cartesian components of F are differentiable — ∇ 2 F on right is known as Laplacian of F and it is defined in Cartesian coordinates via ∇ 2 F ≡ ( ∂ 2 ∂x 2 + ∂ 2 ∂y 2 + ∂ 2 ∂z 2 )(ˆ xF x + ˆ yF y + ˆ zF z ) . Note that the Laplacian can also be written as ∇ 2 F = ˆ x ∇ 2 F x + ˆ y ∇ 2 F y + ˆ z ∇ 2 F z in terms of operator “del dot del” ∇·∇ = ∂ 2 ∂x 2 + ∂ 2 ∂y 2 + ∂ 2 ∂z 2 ≡ ∇ 2 , also known as (loosely speaking) the “del square” operator. You should try to remember the identities proven above for future use. 2. Consider an infinite surface current J s =- ˆ xJ so cos( ωt ) flowing on z = 0 surface, where J so > is the real-valued amplitude of the surface current measured in A/m units. It is found thatreal-valued amplitude of the surface current measured in A/m units....
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This note was uploaded on 04/14/2009 for the course ECE 329 taught by Professor Franke during the Spring '08 term at University of Illinois at Urbana–Champaign.

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329sp08hw6 - ECE 329 Homework 6 Due: Feb 25, 2008, 5PM 1....

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