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ps2 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science Problem Set No. 2 6.630 Electromagnetic Theory Issued: Week 2 Fall Term 2006 Due: Week 3 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Reading assignment : Section 1.3-1.4; J. A. Kong, “ Electromagnetic Wave Theory ,” EMW, 2005. Problem P2.1 Prove the following identities: ∇ × ∇ × E = · E - ∇ 2 E · E × H = H · ∇ × E - E · ∇ × H · ∇ × A = 0 ∇ × ( φ ) = 0 Problem P2.2 Consider an electromagnetic wave propagating in the ˆ z -direction with E = ˆ xe x cos( kz - ω t + ψ x ) + ˆ ye y cos( kz - ω t + ψ y ) where e x , e y , ψ x , and ψ y are all real numbers. (a) Let e x = 2 , e y = 1 , ψ x = π / 2 , ψ y = π / 4 . What is the polarization? (b) Let e x = 1 , e y = ψ x = 0 . This is a linearly polarized wave. Prove that it can be expressed as the superposition of a right-hand circularly polarized wave and a left- hand circularly polarized wave. (c) Let e x = 1 , ψ x = π / 4 , ψ y = - π / 4 , e y = 1 . This is a circularly polarized wave. Prove that it can be decomposed into two linearly polarized waves.
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