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1 MIT OpenCourseWare /ESD.13J Electromagnetcs and Applcatons, Fall 5 Please use the followng ctaton format: Markus Zahn, Erch Ippen, and Davd Staeln, 6.13/ESD.13J Electromagnetcs and Applcatons, Fall 5. (Massachusetts Insttute of Technology: MIT OpenCourseWare). (accessed MM DD, YYYY). Lcense: Creatve Commons Attrbuton- Noncommercal-Share Alke. Note: Please use the actual date you accessed ths materal n your ctaton. For more nformaton about ctng these materals or our Terms of Use, vst:

2 Massachusetts Insttute of Technology Department of Electrcal Engneerng and Computer Scence 6.13 Electromagnetcs and Applcatons Problem Set #6 Issued: 1/1/5 Fall Term 5 Due: 1/6/5 Readng Assgnment: Quz 1: Thursday, Oct., 5 n lecture, 1-11AM. Covers materal up to and ncludng P.S. #5. Problem 6.1 An electrc feld s present wthn a plasma of delectrc permttvtyε wth conducton consttuent relaton J f = ω pεe, where ω p t = qn mε wth qn, and m beng the charge, number densty (number per unt volume) and mass of each charge carrer. (a) Poyntng s theorem s S + w EM = E J f t For the plasma medum, EJ f, can be wrtten as w EJ k f =. t What s w k (b) What s the velocty v of the charge carrers n terms of the current densty J f parameters qnand, m defned above and (c) Wrte w k of part (a) n terms of v, q, n, and m. What knd of energy densty s w k (d) Assumng that all felds vary snusodally wth tme as: E( r, ) = Re ˆ j t t E( r)e ω wrte Mawell s equatons n comple ampltude form wth the plasma consttutve law. (e) Reduce the comple Poyntng theorem from the usual form 1 ˆ ˆ * 1 ˆ ˆ * E( r ) H () r + jω < w >= E J EM f 1

3 to 1 ˆ r ˆ * r E() H () + j ω (< w EM >+< w k > )= What are < w EM > and < w k > (f) Show that 1 < w EM >+< w k >= 1 μ H ε( ω ) E 4 4 What s ε ( ω ) and compare to the results from Problem 5.3b Problem 6. A TEM wave ( E, H ) propagates n a medum whose delectrc permttvty and magnetc y permeablty are functons of z, ε ( z ) and μ ( z). (a) Wrte down Mawell s equatons and obtan a sngle partal dfferental equaton n H y. (b) Consder the dealzed case where ε ( z ) = ε a e +α z and μ( z ) = μ a e α z. Show that the equaton of (a) for H y reduces to a lnear partal dfferental equaton wth constant coeffcents of the form H H H y y y β γ = z z t What are β and γ (c) Infnte magnetc permeablty regons wth zero magnetc feld etend for z< and z>d. A j t current sheet Re Ke ω s placed at z =. Take the magnetc feld of the form ˆ ω κ H = Re[ y H y e ( j t z) ] and fnd values of κ that satsfy the governng equaton n (b) for <z<d. (d) What are the boundary condtons on H (e) Superpose the solutons found n (c) and fnd H that satsfes the boundary condtons of (d). (f) What s the electrc feld for <z<d

4 Problem 6.3 j t k y A sheet of surface charge wth charge densty σ f = Re[ σˆe (ω y ) ] s placed n free space ( ε, μ ) at z =. j t k y σ f = Re[ σˆe (ω y ) ] ε, μ y ε, μ z The comple magnetc feld n each regon s of the form ˆ ( y + z ) He j k y k z Ĥ = 1 ˆ ( ) H e j k y y k z z z > z < (a) What s k z (b) What s the comple electrc feld for z < and z > n terms of Hˆ, Hˆ, k, k and ω 1 y z (c) Usng the boundary condtons at z =, what are Ĥ 1 and Ĥ (d) For what range of the frequency wll the waves for z < and z > be evanescent (e) What surface current flows on the charge sheet at z = 3

5 Problem 6.4 A TM wave s ncdent onto a medum wth a delectrc permttvty ε from a medum wth delectrc permttvty ε 1 at the Brewster s angle of no reflecton, θ B. Both meda have the same magnetc permeablty μ = μ μ. The reflecton coeffcent for a TM wave s 1 Ê r η1 cos θ η cos θt = R = Ê η cosθ + η cos θ 1 t (a) What s the transmtted angle θ t when θ = θ B How are θ B and θ t related (b) What s the Brewster angle of no reflecton ε μ ε, μ 1, θ = θ B θ t E H (c) What s the crtcal angle of transmsson θc when μ 1 = μ μ For the crtcal angle to est, what must be the relatonshp between ε 1 and ε 4

6 (d) A Brewster prsm wll pass TM polarzed lght wthout any loss from reflectons. θ θ = θ B ε, μ ε, μ n = ε ε For the lght path through the prsm shown above what s the ape angle θ Evaluate for glass wth n = (e) In the Brewster prsm of part (d), determne the output power n terms of the ncdent power for TE polarzed lght wth n =1.45. The reflecton coeffcent for a TE wave s Ê r η cos θ η cos θ = R = 1 t Ê η cos θ + η cos θ 1 t 5