329fall08hw6sol - ECE-329 Fall 2008 Homework 6 Solution...

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ECE-329 Fall 2008 Homework 6 — Solution October 5, 2008 1. Some identities. a) Given E = E x ˆ x + E y ˆ y + E z ˆ z and H = H x ˆ x + H y ˆ y + H z ˆ z , we show the following, H · ∇ × E - E · ∇ × H = H x ∂E z ∂y - ∂E y ∂z + H y ∂E x ∂z - ∂E z ∂x + H z ∂E y ∂x - ∂E x ∂y - E x ∂H z ∂y - ∂H y ∂z - E y ∂H x ∂z - ∂H z ∂x - E z ∂H y ∂x - ∂H x ∂y = E y ∂H z ∂x + H z ∂E y ∂x - E z ∂H y ∂x - H y ∂E z ∂x + E z ∂H x ∂y + H x ∂E z ∂y - E x ∂H z ∂y - H z ∂E x ∂y + E x ∂H y ∂z + H y ∂E x ∂z - E y ∂H x ∂z - H x ∂E y ∂z = ∂x ( E y H z - E z H y ) + ∂y ( E z H x - E x H z ) + ∂z ( E x H y - E y H x ) = ∇ · ( E × H ) . b) We will verify that F = x 2 ˆ x + z ˆ y satisfies the following identity ∇ × ∇ × F = ( ∇ · F ) - ∇ 2 F . The left-hand side gives ∇ × ∇ × F = ∇ × ˆ x ˆ y ˆ z ∂x ∂y ∂z x 2 z 0 = ∇ × ( - ˆ x ) = 0 , and the right hand side gives ( ∇ · F ) - ∇ 2 F = ( ∇ · ( x 2 ˆ x + z ˆ y )) - ∇ 2 ( x 2 ˆ x + z ˆ y ) = (2 x ) - x = x - x = 0 , therefore, the identity is verified. 2. A surface current J s = - J so cos( ωt ) ˆ x flowing on the x - y plane induces electromagnetic waves on both sides of the sheet. The induced fields are given by
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