329sp08hw6 - ECE 329 1 a Prove the identity Homework 6 Due...

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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 di ff erentiable. 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 di ff erentiable — 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( ω t ) flowing 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
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