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Unformatted text preview: ECE 222B Midterm Exam February 11th, 2010 Instructions. This is an 80 minute, closed book, examination. Calculators are not permitted. 1. (20 points) Assume that the electric field of a time-harmonic wave in a vacuum takes the form ~ E = ~ E exp h- j ~ β · ~ r i where ~ β = ~ β r- j ~ β i , and where ~ E , ~ β r , and ~ β i are all real vector constants. Prove that if both ~ β r and ~ β i are non-zero, then ~ β i · ~ β r = 0. 2. (20 points) Assume that a TEM z wave propagates in a vacuum with frequency ω . Defining ∇ 2 ⊥ = ∂ 2 ∂x 2 + ∂ 2 ∂y 2 prove that ∇ 2 ⊥ H = 0 and that ∂ 2 H ∂z 2 = ²μ ∂ 2 H ∂t 2 Hint: Do the proof using a cartesian coordinate system and the Maxwell equations. 3. Assume that in the region z > 0 you are given a homogeneous, isotropic medium with constant permittivity ² . Assume that this medium is bounded by a vacuum at z = 0 (see Figure). A time-harmonic, parallel polarized wave ( H field has y-component only) of frequency ω is incident on the surface from the vacuum side of the boundary ( z < 0), and propagates at an angle θ i with respect to the normal of the surface....
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