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ELEC3909 - Electromagnetic Waves

创建日期：2024-04-29 17:27:31

所属分类：电气工程electric engineering

标签： ELEC3909

Electromagnetic Waves

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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, ϵ = 4ϵ0) 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 (ϵr = 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 ϵ = 3ϵ0 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 ϵr = 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 ϵr and µr.
Answers: ϵr = 8, µr = 2
Problem 6: A plane-wave in free space has a propagation vector
k = 124 ax + 124 ay + 263 az
1
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 ϵ = 4ϵ0 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 (ϵ = 9ϵ0, 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 ϵ = 1.6ϵ0 and ϵ = 4ϵ0
at f = 25 GHz.
Answers: ϵr,coating = 2.53, Lcoating = 1.9 mm.

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