HW3 - UNIVERSITY OF CALIFORNIA Santa Barbara Department of...

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UNIVERSITY OF CALIFORNIA Santa Barbara Department of Electrical and Computer Engineering Problem Set No. 3 Issued: October 17, 2007 Fall 2007 Due: October 24, 2007 ECE 201A 1. A general plane wave in an isotropic, homogenous, uniform and a source free medium can be described as 0 jk r EE e = G G i G G (a) Using Gauss' law show that 0 kE = G G i . (b) Substitute the given solution 0 jk r e = G G i G G into the homogenous wave equation 22 0 ωμε ∇+ = G G and show that general dispersion relationship for a plane wave can be written as 2 kk = G G i . (c) If the medium is lossy μ or ε or both can be complex hence k G is complex. Therefore '" kkj k =− G GG where ' k G and " k G are real vectors that describe the real and imaginary parts of the k G vector. Assuming that k G is in the xz plane write ' k G and " k G in terms of their components. Write an expression for the general plane wave having this complex k G vector. Comment on the directions of ' k G and " k G . (d) Show that the dispersion relationship found in part b can be written as two real equations given below. () 2 R e −= G G and 2 1 I m 2 G G i (e) Now assume that a plane wave is incident on a planar interface between two lossy media. Let's define the plane of incidence as the xz plane x being along the interface and z normal to the interface. Then the k G vector of the incident wave is known and can be expressed as '"' " ' " ˆˆ i i i ix ix x iz iz z k k j k a k j k a =− = − + − G Write similar equations for the k G vector of the reflected and transmitted waves.
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HW3 - UNIVERSITY OF CALIFORNIA Santa Barbara Department of...

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