# Fundamentals Of Applied Electromagnetics - University Of Michigan

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Fundamentals of Applied Electromagnetics 8ebyFawwaz T. Ulaby and Umberto RavaioliExercise SolutionsFawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

ChaptersChapter 1 Introduction: Waves and PhasorsChapter 2 Transmission LinesChapter 3 Vector AnalysisChapter 4 ElectrostaticsChapter 5 MagnetostaticsChapter 6 Maxwell’s Equations for Time-Varying FieldsChapter 7 Plane-Wave PropagationChapter 8 Wave Reflection and TransmissionChapter 9 Radiation and AntennasChapter 10 Satellite Communication Systems and Radar SensorsFawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Chapter 1 Exercise SolutionsExercise 1.1Exercise 1.2Exercise 1.3Exercise 1.4Exercise 1.5Exercise 1.6Exercise 1.7Exercise 1.8Exercise 1.9Exercise 1.10Exercise 1.11Exercise 1.12Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 1.1Given charges q1 10 mC, q2 10 mC, and q3 5 mC, all in free space, what is the direction of theforce acting on charge q3 ?yq32m2mq1q2x2mFigure E1.1Solution:yFe31q3Fe32q1q2xFe3 Fe31 Fe32Forces Fe31 and Fe32 are equal in magnitude, with Fe31 pointing along 45 above the x axis and Fe32 pointing along 45 belowthe x axis. The ŷ components cancel.Hence, Fe3 is along x̂.Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 1.2Two parallel, very long, wires carry currents I1 and I2 . The magnetic field due to current I1 alone is B1 .What is the magnetic field due to both currents at a point midway between the wires if:(a) I1 I2 and both currents flow along the ŷ direction?(b) I1 I2 , but I2 flows along the ŷ direction?I1I2B ?Figure E1.2Solution:yxzI1I2B 0(a) B1 is into the page ( ẑ) and B2 is out of the page ( ẑ). Since I1 I2 in magnitude, B 0.yzxI1I2B 2B1(b) Both magnetic fields are into the page. Hence,B 2B1 2 B1 ( ẑ).Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 1.3Consider the red wave shown in Fig. E1.3. What is the wave’s (a) amplitude, (b) wavelength, and (c)frequency, given that its phase velocity is 6 m/s?υ (volts)6420-2-4-6x (cm)123456789 10Figure E1.3Solution:(a) A 6 V.(b) λ 4 cm.(c) f up6 150 Hz.λ4 10 2Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 1.4The wave shown in red in Fig. E1.4 is given by υ 5 cos 2πt/8. Of the following four equations:(1) υ 5 cos(2πt/8 π/4),(2) υ 5 cos(2πt/8 π/4),(3) υ 5 cos(2πt/8 π/4),(4) υ 5 sin 2πt/8,(a) which equation applies to the green wave? (b) which equation applies to the blue wave?υ (volts)5t (s)01 2 3 4 5 6 7 8 9 10 11 12 13 14-5Figure E1.4Solution:(a) The green wave has an amplitude of 5 V and a period T 8 s. Its peak occurs earlier than that of the red wave; hence,its constant phase angle is positive relative to that of the red wave. A full cycle of 8 s corresponds to 2π in phase. The greenwave crosses the time axis 1 s sooner than the red wave. Hence, its phase angle isφ0 1π 2π .84Consequently,υ 5 cos(2πt/T φ0 ) 5 cos(2πt/7 π/4),which is given by #2.(b) The blue wave’s period T 8 s. Its phase angle is delayed relative to the red wave by 2 s. Hence, the phase angle isnegative and given by2πφ0 2π ,82and 2πt πυ 5 cos 82 5 sin 2πt/8,which is given by #4.Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 1.5The electric field of a traveling electromagnetic wave is given byE(z,t) 10 cos(π 107t πz/15 π/6) (V/m).Determine (a) the direction of wave propagation, (b) the wave frequency f , (c) its wavelength λ , and (d) its phase velocity up .Solution:(a) z-direction because the signs of the coefficients of t and z are both positive.(b) From the given expression,ω π 107(rad/s).Hence,f ωπ 107 5 106 Hz 5 MHz.2π2π(c) From the given expression,π2π .λ15Hence λ 30 m.(d) up f λ 5 106 30 1.5 108 m/s.Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 1.6 Consider the red wave shown in Fig. E1.6. What is the wave’s (a) amplitude (at x 0), (b) wavelength, and(c) attenuation constant?υ (volts)5(2.8, 4.23)(8.4, 3.02)x (cm)01 2 3 4 5 6 7 8 9 10 11 12 13 14-5Figure E1.6Solution: The wave shown in the figure exhibits a sinusoidal variation in x and its amplitude decreases as a function of x.Hence, it can be described by the general expression 2πx αx φ0 .υ AecosλFrom the given coordinates of the first two peaks, we deduce thatλ 8.4 2.8 5.6 cm.At x 0, υ 5 V and it occurs exactly λ /2 before the first peak. Hence, the wave amplitude is 5 V, and from 5 5 cos(0 φ0 ),it follows thatφ0 π.Consequently,υ 5e αx 2πxcos π .5.6In view of the relation cos x cos(x π), υ can be expressed asυ 5e αx cos2πx5.6(V).We can describe the amplitude as 5 V for a wave with a constant phase angle of π, or as 5 V with a phase angle of zero.At x 2.8 cm, 2π 2.8 2.8αυ(x 2.8) 4.23 5ecos5.6 5e 2.8α .Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Hence,e 2.8α and4.23,5 14.23α ln 0.06 Np/cm.2.85Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 1.7The red wave shown in Fig. E1.7 is given by υ 5 cos 4πx (V). What expression is applicable to (a) theblue wave and (b) the green wave?υ (volts)5V3.52 V1.01 V5x (m)00.250.50.751.01.25-5Figure E1.7Solution: At x 0, all three waves start at their peak value of 5 V. Also, λ 0.5 m for all three waves. Hence, they sharethe general form2πxλ 5e αx cos 4πx (V).υ Ae αx cosFor the red wave, α 0.For the blue wave,3.52 5e 0.5αα 0.7 Np/m.1.01 5e 0.5αα 3.2 Np/m.For the green wave,Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 1.8An electromagnetic wave is propagating in the z-direction in a lossy medium with attenuation constantα 0.5 Np/m. If the wave’s electric-field amplitude is 100 V/m at z 0, how far can the wave travel before its amplitudewill have been reduced to (a) 10 V/m, (b) 1 V/m, (c) 1 µV/m?Solution:(a)100e αz 10100e 0.5z 10e 0.5z 0.1 0.5z ln 0.1 2.3z 4.6 m.(b)100e 0.5z 1ln 0.01z 9.2 m. 0.5(c)100e 0.5z 10 6z ln 10 8 37 m. 0.5Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 1.9Express the following complex functions in polar form:z1 (4 j3)2 ,z2 (4 j3)1/2 .Solution:z1 (4 j3)2hi2 (42 32 )1/2 tan 1 3/4 [5 36.87 ]2 25 73.7 .z2 (4 j3)1/2hi1/2 (42 32 )1/2 j tan 1 3/4 [5 36.87 ]1/2 5 18.4 .Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 1.10Show that 2 j (1 j).Solution:e jπ/2 0 j sin(π/2) jp 2 j [2e jπ/2 ]1/2 2 e jπ/4 2 [cos(π/4) j sin(π/4)] 11 2 j22 (1 j).Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 1.11A series RL circuit is connected to a voltage source given by υs (t) 150 cos ωt (V). Find (a) the phasorcurrent I and (b) the instantaneous current i(t) for R 400 Ω, L 3 mH, and ω 105 rad/s.Solution:(a) From Example 1–4,I VesR jωL150400 j105 3 10 3150 0.3 36.9 400 j300 (A).(b)e jωt ]i(t) Re[Ie 5 Re[0.3e j36.9 e j10 t ] 0.3 cos(105t 36.9 ) (A).Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 1.12A phasor voltage is given by Ve j5 V. Find υ(t).Solution:Ve j5 5e jπ/2υ(t) Re[Ve e jωt ] Re[5e jπ/2 e jωt ] π 5 sin ωt 5 cos ωt 2Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagnetics(V).c 2019 Prentice Hall

Chapter 2 Exercise SolutionsExercise 2.1Exercise 2.2Exercise 2.3Exercise 2.4Exercise 2.5Exercise 2.6Exercise 2.7Exercise 2.8Exercise 2.9Exercise 2.10Exercise 2.11Exercise 2.12Exercise 2.13Exercise 2.14Exercise 2.15Exercise 2.16Exercise 2.17Exercise 2.18Exercise 2.19Exercise 2.20Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 2.1 Use Table 2-1 to compute the line parameters of a two-wire air line whose wires are separated by a distanceof 2 cm, and each is 1 mm in radius. The wires may be treated as perfect conductors with σc .Solution: Two-wire air line: Because medium between wires is air, ε ε0 , µ µ0 , and σ 0.d 2 cm, 1/2π f µc 0Rs σca 1 mm,σc R0 0 s 2µ0 ddL0 ln 1 π2a2a s 2 7204π 1020 ln 1 π22 4 10 7 ln[10 99] 1.2 (µH/m).G0 0C0 because σ 0 lnπε0 qd2a d 2 12a π 8.85 10 12 9.29ln[10 99]Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagnetics(pF/m).c 2019 Prentice Hall

Exercise 2.2Calculate the transmission line parameters at 1 MHz for a rigid coaxial air line with an inner conductordiameter of 0.6 cm and an outer conductor diameter of 1.2 cm. The conductors are made of copper [see Appendix B for µcand σc of copper].Solution: Coaxial air line: Because medium between wires is air, ε ε0 , µ µ0 , and σ 0.a 0.3 cm,Rs b 0.6 cm,µc µ0 ,σc 5.8 107 S/mpπ f µc /σc [π 106 4π 10 7 /(5.8 107 )]1/2 2.6 10 4 Ω.RsR 2π0 1 1 a b 2.6 10 4 2π 11 33 106 10 3 µ0b4π 10 7L ln ln 2 0.142πa2π0G0 0C0 2.07 10 2(Ω/m)(µH/m)because σ 02πε2π 8.85 10 12 80.3ln(b/a)ln 2(pF/m).Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 2.3Verify that Eq. (2.26a) is indeed a solution of the wave equation given by Eq. (2.21).Solution:Ve (z) V0 e γz V0 eγzd 2 Ve (z)? γ 2 Ve (z) 02dzd 2 γz?(V0 e V0 eγz ) γ 2 (V0 e γz V0 eγz ) 02dzγ 2V0 e γz γ 2V0 eγz γ 2V0 e γz γ 2V0 eγz 0.Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 2.4A two-wire air line has the following line parameters: R0 0.404 (mΩ/m), L0 2.0 (µH/m), G0 0, and0C 5.56 (pF/m). For operation at 5 kHz, determine (a) the attenuation constant α, (b) the phase constant β , (c) the phasevelocity up , and (d) the characteristic impedance Z0 .Solution: Given:R0 0.404 (mΩ/m),G0 0,L0 2.0 (µH/m),C0 5.56 (pF/m).(a)noα Re [(R0 jωL0 )(G0 jωC0 )]1/2n Re [(0.404 10 3 j2π 5 103 2 10 6 )o· (0 j2π 5 103 5.56 10 12 )]1/2 Re[3.37 10 7 j1.05 10 4 ]α 3.37 10 7(Np/m).(b) From part (a),noβ Im [(R0 jωL0 )(G0 jωC0 )]1/2 1.05 10 4(rad/m).(c)up ω2π 5 103 3 108 β1.05 10 4(m/s).(d)R0 j2πωL0α jβ0.404 10 3 j5 103 2 10 6 3.37 10 7 j1.05 10 4 (600 j1.9) Ω.Z0 Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 2.5A microstrip transmission line uses a strip of width w and height h 1 mm, over a substrate of relativepermittivity εr 4. What should w be so that the characteristic impedance of the line is Z0 50 Ω?Solution: Since Z0 50 (44 2εr ) (44 8) 36, we should use Eqs. 2.43(a) and (b):r εr 1 Z0εr 10.12p 0.23 2 60εr 1εrr 4 1 504 10.12 0.23 2604 14 1.474.8e ps 2p 2.05.e 2w sh 2.05 1 mm 2.05 mm.Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 2.6For a lossless transmission line, λ 20.7 cm at 1 GHz. Find εr of the insulating material.Solution:λ0λ εr 2 2 2λ0c3 108εr 2.1.λfλ1 108 20.7 10 2Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 2.7 A lossless transmission line uses a dielectric insulating material with εr 4. If its line capacitance is C 0 10(pF/m), find (a) the phase velocity up , (b) the line inductance L0 , and (c) the characteristic impedance Z0 .Solution:(a)c3 108up 1.5 108 m/s.εr4(b)1up ,L0C0L0 u2p 1.L0C011 4.45u2pC0 (1.5 108 )2 10 10 12(µH/m).(c)rZ0 L0 C0 4.45 10 610 10 12 1/2 667.1 Ω.Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 2.8A 50-Ω lossless transmission line is terminated in a load impedance ZL (30 j200) Ω. Calculate thevoltage reflection coefficient at the load.Solution:ZL Z0ZL Z030 j200 50 20 j200 0.93 27.5 .(30 j200) 5080 j200Γ Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 2.9A 150-Ω lossless line is terminated in a capacitor whose impedance is ZL j30 Ω. Calculate Γ.Solution:ZL Z0ZL Z0 j30 150 1 157.4 . j30 150Γ Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 2.10impedance zL .Given that the reflection coefficient at the load is Γ 0.6 j0.3, determine the normalized loadSolution: Solving Eq. (2.59) for zL givesΓ zL 1.zL 1Solving for zL :zL Γ Γ zL 1,which leads tozL 1 Γ 1 (0.6 j0.3) 1 Γ 1 (0.6 j0.3)1.6 j0.3 2.2 j2.4.0.4 j0.3Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 2.11Use CD Module 2.4 to generate the voltage and current standing-wave patterns for a 50-Ω line oflength 1.5λ , terminated in an inductance with ZL j140 Ω.Solution: Standing-wave patterns generated with the help of DVD Module 2.4 are shown.Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 2.12load.If Γ 0.5 60 and λ 24 cm, find the locations of the voltage maximum and minimum nearest to theSolution:Γ 0.5 60 ,λ 24 cmθr λ λ (because θr is negative)2 4π ( π/3) 24 24 cm [ 2 12] cm 10 cm. 4π2lmax λlmin lmax (because lmax λ /4)4 24 10 cm 4 cm.4Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 2.13A 140-Ω lossless line is terminated in a load impedance ZL (280 j182) Ω. If λ 72 cm, find (a) thereflection coefficient Γ, (b) the voltage standing-wave ratio S, (c) the locations of voltage maxima, and (d) the locations ofvoltage minima.Solution:Z0 140 Ω,ZL (280 j182) Ω(a)ZL Z0ZL Z0280 j182 140 140 j182 0.5 29 .280 j182 140 420 j182Γ (b)S 1 Γ 1 0.5 1.5 3.1 Γ 1 0.5 0.5(c)θr λ nλ ,n 0, 1, 2, . . .4π2(29π/180) 0.72 n 0.72 4π2 (2.9 36n) (cm),n 0, 1, 2, . . .lmax (d)λlmin lmax 4 72cm (2.9 36n) 4 (20.9 36n) cm,n 0, 1, 2, . . .Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 2.14A 50 Ω lossless transmission line uses an insulating material with εr 2.25. When terminated in an opencircuit, how long should the line be for its input impedance to be equivalent to a 10 pF capacitor at 50 MHz?Solution: For a 10 pF capacitor at 50 MHz,11000 j j Ω6 12jωC 2π 50 10 10 10π 2π εr 2π f εr2π β λλ0c 2π 5 107 2.25 1.57 (rad/m).3 108Zc For lossless lines with open-circuit termination,Zin jZ0 cot β l j50 cot 1.57lHence, j1000 j50 cot 1.57lπorl 9.92(cm).Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 2.15 A 300-Ω feedline is to be connected to a 3-m long, 150-Ω line terminated in a 150-Ω resistor. Both lines arelossless and use air as the insulating material, and the operating frequency is 50 MHz. Determine (a) the input impedance ofthe 3-m long line, (b) the voltage standing-wave ratio on the feedline, and (c) the characteristic impedance of a quarter-wavetransformer were it to be used between the two lines in order to achieve S 1 on the feedline.Solution: At 50 MHz,λ λ0 c3 108 6 m.f5 107(a)l3 0.5.λ6Hence, Zin ZL 150 Ω. (Zin ZL if Z nλ /2.)(b)Zin Z0 150 300 1501 .Zin Z0 150 30045034/31 Γ 1 31 2.S 11 Γ 1 32/3Γ (c)2Z02 Z1 Z3 300 150 45,000Z02 212.1 Ω.where Z1 is the feedline and Z3 is Zin of part (a).Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 2.16 Through multiple trials, it was determined that a load with unknown impedance ZL can be perfectly matchedto a feedline with Zin 50 Ω by using a λ /4 transformer section with characteristic impedance of 60 Ω. What is ZL ?Solution: From Eq. (2.97),ZL Z02602 72 Ω.Zin50Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 2.17For a 50-Ω lossless transmission line terminated in a load impedance ZL (100 j50) Ω, determine thefraction of the average incident power reflected by the load.Solution:ZL Z0ZL Z0100 j50 5050 j50 0.45 26.6 .100 j50 50 150 j50Γ Fraction of reflected power Γ 2 (0.45)2 20%.Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 2.18For the line of Exercise 2.17, what is the magnitude of the average reflected power if V0 1 V?Solution:rPav Γ 2 V0 2 0.2 1 22Z02 50(mW).Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 2.19Use the Smith chart to find the values of Γ corresponding to the following normalized load impedances:(a) zL 2 j0, (b) zL 1 j1, (c) zL 0.5 j2, (d) zL j3, (e) zL 0 (short circuit), (f) zL (open circuit), (g) zL 1(matched 250.260.240.270.230.250.240.260.23COEFFICIENT IN0.27REFLECTIONDEGRLE .21.00.90.80.70.60.50.40.30.2 180A500.150G ORESISTANCE COMPONENT (R/Zo), OR CONDUCTANCE COMPONENT (G/Yo)0.2Fθ1 0o200.410o)jB/YE 21.6-60641.4-700.150.351.20.10.3θ2 -63.4o0.14-800.361.070.90.130.8θ3 0.0.63.00.3144θ4 PONENT( jX/Z0.4159 030.20.4970.30.30.48 RD LOADTOWATHS-170ENGVELWA-1600.1600.240 VETICIPACA0.34310. WAVELENGTHSTOWARD0.49GENERA0.48TO170R0.47160)/Yo( jBCEANPTCESSU0120.167010Eθ5 .110.190.0-900.120.130.380.370.11-1000.40.39(a)Γ OAθ1 0.33OR(b)Γ OBθ2 0.45 63.4 ORΓ OCθ3 0.83 50.9 OR(c)(d)Γ ODθ4 1 36.9 ORFawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

(e)Γ OEθ5 1 180 1OR(f)Γ OFθ1 1OR(g)Γ OG 0ORFawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 2.20Use the Smith chart to find the normalized input impedance of a lossless line of length l terminated in anormalized load impedance zL for each of the following combinations: (a) l 0.25λ , zL 1 j0, (b) l 0.5λ , zL 1 j1,(c) l 0.3λ , zL 1 j1, (d) l 1.2λ , zL 0.5 j0.5, (e) l 0.1λ , zL 0 (short circuit), (f) l 0.4λ , zL j3, (g) l 0.2λ ,zL (open circuit).Solution:(a)zin 1 j0Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

(b)zin 1 j1Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

(c)zin 0.76 j0.84Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

(d)zin 0.59 j0.66Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

(e)zin 0 j0.73Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

(f)zin 0 j0.72Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

(g)zin 0 j0.32Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Chapter 3 Exercise SolutionsExercise 3.1Exercise 3.2Exercise 3.3Exercise 3.4Exercise 3.5Exercise 3.6Exercise 3.7Exercise 3.8Exercise 3.9Exercise 3.10Exercise 3.11Exercise 3.12Exercise 3.13Exercise 3.14Exercise 3.15Exercise 3.16Exercise 3.17Exercise 3.18Exercise 3.19Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 3.1Find the distance vector between P1 (1, 2, 3) and P2 ( 1, 2, 3) in Cartesian coordinates.Solution: P1 P2 x̂(x2 x1 ) ŷ(y2 y1 ) ẑ(z2 z1 ) x̂( 1 1) ŷ( 2 2) ẑ(3 3) x̂2 ŷ4.Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 3.2Find the angle θ between vectors A and B of Example 3-1 using the cross product between them.Solution: B n̂AB sin θABA B A AB ( x̂ ŷ5 ẑ) (x̂2 ŷ3 ẑ3) 22 27 ẑ10 ŷ2 ẑ3 x̂3 ŷ3 x̂15 22 27 x̂12 ŷ ẑ7 144 1 49 0.5722 2722 27sin θAB θAB sin 1 (0.57) 34.9 or 145.1 .Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 3.3Find the angle that vector B of Example 3-1 makes with the z-axis.Solution:B · ẑ B cos θ ( x̂ ŷ5 ẑ) · ẑ 27 cos θ 1cos θ 27θ 101.1 .Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 3.4 Vectors A and B lie in the y-z plane and both have the same magnitude of 2 (Fig. E3.4). Determine (a) A · B B.and (b) A zB2 30 A2yxFigure E3.4Solution:(a)A · B AB cos(90 30 ) 2 2 cos 120 2.(b)A ŷ 2B ŷ 2 cos 60 ẑ 2 cos 30 ŷ 1 ẑ 1.73 B ŷ 2 ( ŷ 1 ẑ 1.73)A x̂ 3.46.Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 3.5If A · B A · C, does it follow that B C?Solution: The answer is No, which can be demonstrated through the following example. LetA x̂ 1,B x̂ 2 ŷ 1,C x̂ 2 ŷ 2.A · B 2,A · C 2,butB 6 C.y2C1BAx12Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 3.6A circular cylinder of radius r 5 cm is concentric with the z-axis and extends between z 3 cm andz 3 cm. Use Eq. (3.44) to find the cylinder’s volume.Solution:d V r dr dφ dzZ 5 cm Z 2π Z 3 cmV r dr dφ dzr 0φ 0 z 3 cm5 cm2r3 cm φ 2π0 z 3 cm2 025 2π 6 471.2 cm3 .2Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 3.7 Point P (2 3, π/3, 2) is given in cylindrical coordinates. Express P in spherical coordinates.Solution:q p r2 z2 (2 3)2 ( 2)2 4πφ (unchanged)3 ! π2π 1 r 1 2 3 60 orθ tan tan.z 233R Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 3.8Transform vectorA x̂(x y) ŷ(y x) ẑzfrom Cartesian to cylindrical coordinates.Solution:A x̂(x y) ŷ(y x) ẑz (r̂ cos φ φ̂φ sin φ )(r cos φ r sin φ ) (r̂ sin φ φ̂φ cos φ )(r sin φ r cos φ ) ẑz r̂(cos2 φ cos φ sin φ sin2 φ cos φ sin φ )r φ̂φ ( sin φ cos φ sin2 φ sin φ cos φ cos2 φ )r ẑz r̂r φ̂φ r ẑz.Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 3.9Given V x2 y xy2 xz2 , (a) find the gradient of V , and (b) evaluate it at (1, 1, 2).Solution:V x2 y xy2 xz2(a) V V V ŷ ẑ x y z22 x̂ (2xy y z ) ŷ (x2 2xy) ẑ 2xz. V x̂(b) V (1, 1,2) x̂ ( 2 1 4) ŷ (1 2) ẑ 4 x̂ 3 ŷ ẑ 4.Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 3.10Find the directional derivative of V rz2 cos 2φ along the direction A r̂2 ẑ and evaluate it at (1, π/2, 2).Solution:V rz2 cos 2φ1 V V V φ̂φ ẑ V r̂ rr φ z2r r̂ z2 cos 2φ φ̂φ z2 sin 2φ ẑ 2rz cos 2φrdV V · âldlA V ·Ar̂ 2 ẑ (r̂ z2 cos 2φ φ̂φ 2z2 sin 2φ ẑ 2rz cos 2φ ) · 52z2 cos 2φ 2rz cos 2φ 5dVdl2 4 cos π 2 2 cos π 5(1,π/2,2) 4/ 5. Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 3.11The power density radiated by a star [Fig. E3.11(a)] decreases radially as S(R) S0 /R2 , where R is theradial distance from the star and S0 is a constant. Recalling that the gradient of a scalar function denotes the maximum rateof change of that function per unit distance and the direction of the gradient is along the direction of maximum increase,generate an arrow representation of S.S(a) S(b)Figure E3.11Solution:S(R) S0.R2 1 1 θ̂θ φ̂φ RR θR sin θ φS0 R̂ 2 3 .R S R̂Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagnetics S0R2c 2019 Prentice Hall

Exercise 3.12The graph in Fig. E3.12(a) depicts a gentle change in atmospheric temperature from T1 over the sea to T2over land. The temperature profile is described by the functionT (x) T1 (T2 T1 )/(e x 1),where x is measured in kilometers and x 0 is the sea-land boundary. (a) In which direction does T point and (b) at whatvalue of x is it a maximum?TT2T1xSeaLand(a) TxSeaLand(b)Figure E3.12Solution:T (x) T1 T2 T1e x 1 T x T2 T1 x̂T1 x xe 1 x(e 1) 1 x̂(T2 T1 ) x x̂(T2 T1 )e x (e x 1) 2 T x̂ x̂e x (T2 T1 ).(e x 1)2Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 3.13Given A e 2y (x̂ sin 2x ŷ cos 2x), find · A.Solution:A e 2y (x̂ sin 2x ŷ cos 2x) Ax Ay Az ·A x y z 2y (e sin 2x) (e 2y cos 2x) x y 2y 2y 2e cos 2x 2e cos 2x 0.Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 3.14Given A r̂ r cos φ φ̂φ r sin φ ẑ 3z, find · A at (2, 0, 3).Solution:A r̂ r cos φ φ̂φ r sin φ ẑ 3z1 Aφ Az1 (rAr ) ·A r rr φ z1 21 (r cos φ ) (r sin φ ) (3z)r rr φ z 2 cos φ cos φ 3 · A (2,0,3) 2 1 3 6.Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 3.15at the origin.If E R̂AR in spherical coordinates, calculate the flux of E through a spherical surface of radius a, centeredSolution:E R̂ARZSE · ds Z π Z 2πθ 0 φ 0R̂AR · R̂R2 sin θ dθ dφ R ahiπ 2πAR3 cos θ 0R a 4πAa3 .Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 3.16 Verify the divergence theorem by calculating the volume integral of the divergence of the field E of Exercise3.15 over the volume bounded by the surface of radius a.Solution:ZDivergence Theorem:From Exercise 3.11,ZV · E dV ZE · dsSE · ds 4πAa3 .SFor the left side of Divergence Theorem, with E R̂AR, ·E ZV1 1 (R2 ER ) 2(AR3 ) 3AR2 RR R · E dV Z a Z π Z 2π3A · R2 sin θ dR dθ dφ0 00a33AR ( cos θ π0 ) π 2π0303 4πAa .Hence, Divergence Theorem is verified.Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 3.17The arrow representation in Fig. E3.17 represents the vector field A x̂ x ŷ y. At a given point in space,A has a positive divergence · A if the net flux flowing outward through the surface of an imaginary infinitesimal volumecentered at that point is positive, · A is negative if the net flux is into the volume, and · A 0 if the same amount of fluxenters into the volume as leaves it. Determine · A everywhere in the x–y plane.Figure E3.17Solution:A x̂ x ŷ y Ax Ay Az ·A x y z x y x y 1 1 0.Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 3.18 A at (2, 0, 3) in cylindrical coordinates for the vector fieldFind A r̂10e 2r cos φ ẑ10 sin φ .Solution:A r̂10e 2r cos φ ẑ10 sin φ 1 Az Aφ Ar Az1 Ar A r̂ φ̂φ ẑrAφ r φ z z rr r φ 1 2r(10 sin φ ) φ̂φ(10e cos φ ) (10 sin φ ) r̂r φ z r1 ẑ( 10e 2r cos φ )r φ10e 2r10 cos φ ẑsin φ r̂rr A (2,0,3) r̂ 5.Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 3.19 A at (3, π/6, 0) in spherical coordinates for the vector field A θ̂θ 12 sin θ .Find Solution:A θ̂θ 12 sin θ1 (RAθ ) A φ̂φR R1 φ̂φ(12R sin θ )R R12 sin θ φ̂φ.R A (3,π/6,0) φ̂φ 4 sin 30 φ̂φ 2.Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Chapter 4 Exercise SolutionsExercise 4.1Exercise 4.2Exercise 4.3Exercise 4.4Exercise 4.5Exercise 4.6Exercise 4.7Exercise 4.8Exercise 4.9Exercise 4.10Exercise 4.11Exercise 4.12Exercise 4.13Exercise 4.14Exercise 4.15Exercise 4.16Exercise 4.17Exercise 4.18Exercise 4.19Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 4.1A square plate in the x–y plane is situated in the space defined by 3 m x 3 m and 3 m y 3 m.Find the total charge on the plate if the surface charge density is given by ρs 4y2 (µC/m2 ).Solution:ρs 4y2ZQ Sρs dsZ 3Z 3 4y2 dx dy 3 34y3 x 333 432 µC 0.432(mC). 3 3Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagneticsc 2019 Prentice Hall

Exercise 4.2 A thick spherical shell centered at the origin extends between R 2 cm and R 3 cm. If the volume chargedensity is ρv 3R 10 4 (C/m3 ), find the total charge contain

Chapter 1 Exercise Solutions Exercise 1.1 Exercise 1.2 Exercise 1.3 Exercise 1.4 Exercise 1.5 Exercise 1.6 Exercise 1.7 Exercise 1.8 Exercise 1.9 Exercise 1.10 Exercise 1.11 Exercise 1.12 Fawwaz T. Ulaby and Umberto Ravaioli, Fundamentals of Applied Electromagnetics c 2019 Prentice Hall

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