ELE345_Quiz_2_Solutions.pdf

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ELE 345 ELECTROMAGNETICS II MIDTERM 2 FALL 2013 STUDENT NAME: ID # : SECTION # : CLOSED BOOK, CLOSED NOTES, GOOD LUCK NOTATION VIOLATIONS : -10 Q1 Q2 NOTAT. TOTAL 1. (25 pt) (5 pt) a) Derive the vector wave equation for a source-free, free space starting with four Maxwell’s Equations in phasor domain. Define k = ω εµ . This equation is called Helmholtz Equation. (5 pt) b) Rewrite the Helmholtz equation for a wave E = ˆ yE y ( x, z ) in cartesian coordinates. (5 pt) c) Suppose E y ( x, z ) = u ( x ) exp( - jβz ) . Derive the wave equation for u ( x ) using the equation in part b) in terms of parameter k c , where k c = k 2 - β 2 . (5 pt) d) Suppose E y ( x, z ) = u ( x ) exp( - jβz ) travels in a medium that extends to infinity in y - and z - directions. In x - direction, the wave exists between two infinite conducting sheets located at x = 0 and x = a . Find all possible solutions for u ( x ) and β . (5 pt) e) Write down the final expression for
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