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EELE 334 ELECTROMAGNETIC THEORY I Fall 2017—Practice Final Exam 1. [40 points] Short answer (a) [8] State the boundary conditions for the normal components of the electric field
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E and
magnetic field
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H at the interface between two dielectric materials. You may assume no surface
currents or charges at the boundary. (b) [8] Explain the concept of a displacement current. Why is it important in electromagnetics? (c) [8] Explain hysteresis in ferromagnetic materials. (d) [8] State the current-charge continuity relation. (e) [8] What is the Brewster angle? Under what conditions does it exist? 2. [55 points] Transmission Lines Consider an ideal lossless
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50Ω transmission line. The line has length
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0.6λ , and is terminated in a load
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ZL = 50 + 25 j( )Ω. (a) [10] Find the voltage reflection coefficient and the standing wave ratio. (b) [15] Find the input impedance of the line plus load. (c) [30] Now assume that we attach a shunt resistance of
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30Ω at the midpoint of the transmission line. Find the input impedance to the line. 3. [45 points] Electrostatics & Magnetostatics Please answer the following questions, clearly stating any assumptions you make! (a) [15] Using the integral form of Gauss’s law, compute the electric field produced by a point charge with charge
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−Q0 . (b) [15] Using Ampère’s law, compute the magnetic field produced by an infinite straight wire carrying current
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I0 .
(c) [15] For a given vector potential (in cylindrical coordinates)
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A = −ˆ z µ0J0
4 r2, find the magnetic
field
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H .
4. [60 points] Plane wave propagation Consider a plane wave described by
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˜ E z( ) = ˆ x ax + ˆ y aye jδ( )e+αze+ jβz . We assume this wave is propagating in seawater, with
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f =103 Hz,
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εr = 80 ,
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µr =1, and
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σ = 4 S/m. (a) [10] Assuming
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ax is a given constant, for what values of
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ay and
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δ will this wave be right- hand circularly polarized?
(b) [15] If
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ay = 2ax , is the wave linearly, circularly, or elliptically polarized, or indeterminate? Please explain your answer. (c) [10] For this wave, can the material be approximated as a good conductor, a lossless dielectric, or neither? Please explain your answer. (d) [10] Find the propagation constant
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β , attenuation constant
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α , and complex impedance
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ηc . (e) [15] Assuming the wave has an initial average power density of 1 W/m2, what is the maximum depth at which the signal can be received if the minimum detectable signal is 100 µW/m2? 5. [50 points] Plane waves & Oblique incidence Consider a plane wave of angular frequency
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ω propagating at an angle
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θi with respect to the surface normal of the boundary between two semi-infinite materials. You may assume that the boundary is at
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z = 0, material 1 is found in the region
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z < 0, and that
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ε1 = ε0,
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µ1 = µ0,
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ε2 = εrε0 , and
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