ELEC3909 – Electromagnetic Waves
Problem Set #4
Problem 1: A plane wave with E = 30 cos(ωt z) ax V/m in air normally hits a lossless medium
(μ = μ0, = 4 0) at z = 0.
(a) Find Γ, τ , and s.
(b) Calculate the reflected electric and magnetic fields.
Answers: (a) Γ = 0.333, τ = 0.667, s = 2. (b) Er = 10 cos(ωt + z) ax V/m, Hr =
0.0265 cos(ωt+ z) ay A/m.
Problem 2: A 150 Mrad/s uniform wave with Ei = 10 sin(ωt βz) ax V/m is propagating in a
dielectric medium ( r = 4, μr = 1). If it enters free space normally at z = 0, determine the power
densities in both media.
Answers: Pdielectric,ave = 0.236 az W/m
2, Pfree space,ave = 0.236 az W/m
2.
Problem 3: A uniform plane wave in air is normally incident on an infinite lossless dielectric
material occupying z > 0 and having = 3 0 and μ = μ0. If the incident wave is Ei = 10 cos(ωt
z) ay V/m, find
(a) λ and ω of the wave in air and the transmitted wave in the dielectric medium
(b) The incident Hi field
(c) Γ and τ
(d) The total electric field and the time-average power in both regions
Answers: (a) λair = 6.28 m, λdielectric = 3.63 m, ω = 3 × 108 rad/s. (b) Hi = 0.0265 cos(ωt
z) ax A/m. (c) Γ = 0.268, τ = 0.732. (d) Eair,total = 10 cos(ωt z) ay 2.68 cos(ωt+ z) ay V/m,
Edielectric,total = 7.32 cos(ωt z) ay V/m,Pair,ave = 0.123 az W/m2,Pdielectric,ave = 0.123 az W/m2.
Problem 4: An EM wave with an amplitude of 1 V/m is normally incident from air into a dielec-
tric material with r = 9. Find the amplitude of the transmitted and reflected magnetic intensities.
Answers: Hro = 0.0013 A/m, Hto = 0.004 A/m.
Problem 5: A uniform plane wave in air is normally incident on a certain lossless medium. If the
reflection coefficient is found to be Γ = 1/3 and the phase velocity is reduced by a factor of 4,
determine r and μr.
Answers: r = 8, μr = 2
Problem 6: A plane-wave in free space has a propagation vector
k = 124 ax + 124 ay + 263 az
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Find the wavelength, frequency, and angles k makes with the x , y , and z axes.
Answers: λ = 0.0199 m, f = 15.1 GHz, θx = 66.9
, θy = 66.9 , θz = 33.7 .
Problem 7: A parallel-polarized wave in air with
E = (8 ay 6 az) sin(ωt 4y 3z) V/m
impinges a dielectric half-space with = 4 0 and μ = μ0, as shown in Fig. 1. Find
(a) The incidence angle θi.
(b) The time-average power in air.
(c) The reflected and transmitted E fields.
Figure 1: Illustration for Problem 7.
Answers: (a) θi = 36.9
. (b) Pi,ave = 0.106 ay + 0.0796 az W/m2. (c) Er = ( 1.52 ay
2.02 az) sin(ωt+ 4y 3z) V/m, Et = (1.88 ay 5.98 az) sin(ωt 9.5y 3z) V/m.
Problem 8: In a dielectric medium ( = 9 0, μ = μ0), a plane wave with
H = 0.2 cos(109t nx n
√
8z) ay A/m
is incident on an air boundary at z = 0. Find
(a) θr and θt
(b) n
(c) The wavelength in the dielectric and in air
(d) The incident E
(e) The transmitted and reflected E
(f) The Brewster angle
2
Answers: (a) θr = 19.5
, θt = 90 . (b) n = 3.33. (c) λdielectric = 0.628 m, λair = 1.88 m. (d)
E = (23.6 ax 8.36 az) cos(wt nx n
√
8z) V/m. (e) Er = ( 23.6 ax 8.33 az) cos(wt nx +
n
√
8z) V/m, Et = 150 cos(wt 3.34x) az V/m. (f) θB,|| = 18.4 .
Problem 9: Design an anti-reflection coating between two dielectrics with = 1.6 0 and = 4 0
at f = 25 GHz.
Answers: r,coating = 2.53, Lcoating = 1.9 mm.
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