Quiz 3 in Fields and Waves I, Spring 2011 by James J.-Q. Lu - Prof. James T. Lu, Quizzes of Electrical and Electronics Engineering

Download Quiz 3 in Fields and Waves I, Spring 2011 by James J.-Q. Lu - Prof. James T. Lu and more Quizzes Electrical and Electronics Engineering in PDF only on Docsity! Fields and Waves I Quiz 3 Spring 2011 James J.-Q. Lu 2 May 2011 1 Name ________Solutions________ Section _______________________ Question # (points) 1. (20 Pts) ____________ 2. (15 Pts) ____________ 3. (20 Pts) ____________ 4. (25 Pts) ____________ 5. (20 Pts) ____________ 6. (10 Pts) ____________ Total ____________ Notes: 1. Please read over all questions before you begin your work. There may be some information in a later question that helps you with an earlier question. 2. Show your work details on problems to receive full credit. 3. For short answer questions, you may add some comments to justify your answer. 4. Try to arrange your results to the simplest expressions, so you can verify their meaning and check if they are correct. http://coraifeartaigh.wordpress.com/2010/02/10/radiation/ Fields and Waves I Quiz 3 Spring 2011 James J.-Q. Lu 2 May 2011 2 1. MULTIPLE CHOICE AND/OR SHORT ANSWER QUESTIONS Circle all the right answers. More than one correct answer is possible. A proportional positive (or negative) point will be received for a correct (or wrong) answer. 1.1. Maxwell Equations (4 pts) Which of the following is/are true? A B D a. The displacement current density t DJd ∂ ∂ = r r can generate magnetic field as what conduction current does. b. The charge continuity equation 0= ∂ ∂ +•∇ t J v ρr is a mathematical statement of the law of conservation of electric charge. c. The continuity equation for static fields is equivalent to Kirchhoff’s voltage law. d. The quasi-static approximation for electromagnetic fields can be applied when all system dimensions are much smaller than a wavelength. 1.2. Plane Wave in Charge-Free Homogeneous Medium (4 pts) A B C Which of the following is/are true? a. TEM stands for Transverse Electromagnetic wave. b. Electric and magnetic fields of a TEM wave are normal to each other and to the direction of propagation. c. The magnitude of TEM electric field and magnetic field are related as E=ηH. d. The TEM electric field in a lossy material decays more rapidly than the Poynting vector. 1.3. Wave Polarization (4 pts) Which of the following is/are true? A B a. Any plane wave can be described as a sum of an x-component and a y-component, such as ),(ˆ),(ˆ),( tzEytzExtzE yx += r for an electric field vector. b. A linear polarization of waves happens when the phase difference is zero (in-phase) or π (out of phase). c. A wave incident upon an observer is right-hand circularly (RHC) polarized if its electric field appears to the observer to rotate in a clockwise direction along a circle as a function of time in a fixed plane perpendicular to the direction of wave travel. d. A wave is circularly polarized when the phase difference is ±π/2 for any values of its x- component and y-component. 1.4. Normal Incident of Plane Wave (4 pts) Which of the following can be a practical choice for the thickness of a. a radome to make it invisible to a radar signal? λ8 λ 4 3 8 λ λ 2 5 8 λ 3 4 λ 7 8 λ λ 9 8 λ 5 4 λ 11 8 λ 3 2 λ b. an anti-reflection coating to eliminate reflections for a specific wave length (or frequency)? 0 λ8 λ 4 3 8 λ λ 2 5 8 λ 3 4 λ 7 8 λ λ 9 8 λ 5 4 λ 11 8 λ 3 2 λ Fields and Waves I Quiz 3 Spring 2011 James J.-Q. Lu 2 May 2011 5 4. Wave Reflection and Transmission (25 pts) A plane wave in free space has an electric field magnitude of 100 V/m, and is normally incident of a perfect dielectric of εr = 25. Find the following: a. (6 pts) The reflection and transmission coefficients. πηη 12001 == , 5 1 0 02 η ε ηη == r 3 2 1 5 1 1 5 1 12 12 −= + − = + − =Γ ηη ηη Γ+== + = + = 1 3 1 1 5 1 5 2 2 12 2 ηη ητ b. (3 pts) The standing wave ratio in the free space region. 5 3 21 3 21 1 1 = − + = Γ− Γ+ =SWR c. (6 pts) The average power densities (Poynting vector) of the incident, reflected and transmitted wave. η2 2 0ESav = Incident = 263.13 377 100 2 1 2 = W/m2 Reflected = 895.5 3772 1 2 0 = ΓE W/m2 Transimitted = 368.7 2 1 2 2 0 = η τ E W/m2 d. (6 pts) If the incident angle is no more normal to the dielectric surface, prove that the refracted EM wave propagates almost vertically down to the dielectric. 5 1/ 020 00 22 11 21 === εεμ εμ εμω εμω r kk tkk θθ sinsin 211 = 2.05/1sinsin 2 1 =≤= it k k θθ °=≤ − 5.11)2.0(sin 1tθ Thus the refracted EM wave propagates “almost” vertically down to the dielectric e. (4 pts) Following d, find the polarizing angle of the incident wave. Polarizing angle = Brewster’s angle °=== −− 7.78 1 25tantan 1 1 21 || ε ε θB Free space Dielectric η1     εr=1 η2     εr=25 Free space Dielectric η1     εr1=1 η2     εr2=25 θt θi Fields and Waves I Quiz 3 Spring 2011 James J.-Q. Lu 2 May 2011 6 5. Wave Reflection and Transmission (20 pts) An aircraft is attempting to communicate with a submerged submarine in seawater (εr = 81 and σ = 4 S/m) at a frequency of 1 MHz. The incident wave magnitude is 1 kV/m right before the EM wave reaches the sea surface. The aircraft is far above the sea, so we can treat the EM wave as a plane wave. a. (3pts) Explain if the seawater is a good conductor, good insulator or neither. ⇒>×= ××× = − 23 96 1010 9 8 10 36 81102 4 π πωε σ good conductor b. (6pts) Find the propagation constants (k and/or α, β) and the wave impedance (η) in air and seawater. In air 021.0 103 102 8 6 001 =× × === πωεμω c k m-1 πηη 12001 == Ω In seawater, as a good conductor 222 βαγ j+= , 0μμ = 497.3104104 6722 ≈=××××=== −πππμσβα f m-1 jjj +≈+=+= 1 4 97.3)1()1(2 σ αη (Ω) c. (5pts) Find reflection and transmission coefficients for normal incident. °∠= ++ −+ = + − =Γ 7.179995.0 3771 3771 12 12 j j ηη ηη °∠×=×≈ ++ + = + = −− 8.441041.7105.7 3771 )1(22 34/3 12 2 π ηη ητ je j j d. (6pts) Write the expressions of E and H phasors in seawater. Assume ExE ˆ~ = , we know mVmkVEi /1000/1 0 == 00 it EE τ= 0 2 0 1 tt EH η = zjjzit eexeExE )(8.4430 10001041.7ˆˆ~ 2 βαγτ +−°−− ××== °+−−= 8.4497.397.341.7ˆ jzjzeex (V/m) °−−−°+−− = + == 2.097.397.38.4497.397.3 2 25.5ˆ 1 41.7ˆ1ˆ~ jzjzjzjztt eeyeej yEyH η (A/m) Fields and Waves I Quiz 3 Spring 2011 James J.-Q. Lu 2 May 2011 7 6. Displacement Current (10 pts) A co-axial capacitor of length l = 6 cm uses an insulating dielectric material with εr = 9. The radii of the cylindrical conductors are 0.5 cm and 1 cm. If the voltage applied across the capacitor is V(t) = 50sin(120π t ) (V). Find the displacement current. (Hint: the capacitance per unit length for a co-ax cable is )/ln( 2 ba C πε=′ F/m) pFl ba lCC r 32.43 )/ln( 2 0 ==⋅′= επε The displacement current: )120cos(10166.8)]120sin(50[1032.43 712 tt dt d dt dVCId ππ −− ×=××== (A) The reason we use Id = CdV/dt can be found from Chapter 6-7 or as follows, t DJd ∂ ∂ = r r ∫∫∫∫ •∂ ∂ =• ∂ ∂ =• ∂ ∂ =•= Ssss dd SdD t SdD t Sd t DSdJI rrrrr r rr Using Gauss’s law: .enclS QSdD∫ =• rr and C=Q/V t VC t CV t QSdD t I Sd ∂ ∂ = ∂ ∂ = ∂ ∂ =• ∂ ∂ = ∫ rr Note: 0== EJc rr σ because σ = 0 in the lossless dielectric. a b dJ r Gaussian surface, S ε D r