Find the transmissivity

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

 E and

magnetic field

 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

50Ω transmission line. The line has length

0.6λ , and is terminated in a load

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

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

−Q0 . (b) [15] Using Ampère’s law, compute the magnetic field produced by an infinite straight wire carrying current

I0 .

(c) [15] For a given vector potential (in cylindrical coordinates)

 A = −ˆ z µ0J0

4 r2, find the magnetic

field

 H .

4. [60 points] Plane wave propagation Consider a plane wave described by

˜ E z( ) = ˆ x ax + ˆ y aye jδ( )e+αze+ jβz . We assume this wave is propagating in seawater, with

f =103 Hz,

εr = 80 ,

µr =1, and

σ = 4 S/m. (a) [10] Assuming

ax is a given constant, for what values of

ay and

δ will this wave be right- hand circularly polarized?

(b) [15] If

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

β , attenuation constant

α , and complex impedance

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

ω propagating at an angle

θi with respect to the surface normal of the boundary between two semi-infinite materials. You may assume that the boundary is at

z = 0, material 1 is found in the region

z < 0, and that

ε1 = ε0,

µ1 = µ0,

ε2 = εrε0 , and

µ2 = µ0 . The incident electric field phasor is given by

˜ E i = ˆ x cosθi − ˆ z sinθi( )E0i exp − jk1 x sinθi + zcosθi( )[ ], where

k1 =ω µ1ε1 =ω µ0ε0 = ω c

.

(a) [10] Find the corresponding magnetic field phasor for the incident wave

˜ H i . (b) [15] Find the time-dependent magnetic field for the reflected wave

 H r x,z,t( ) .

(c) [15] Find the electric field phasor of the transmitted wave

˜ E t . (d) [10] Find the transmissivity

T .