MASSACHUSETTS ISTITUTE OF TECHOLOGY Department of Electrical Engineering and Computer Science 6.013 – Electromagnetics and Applications Problem Set 5 Suggested Reading: Course notes, Sections 9.5.1 - 9.5.3, 9.2.1 - 9.2.4. 9.1, 9.2.6. Problem 5.1 -6A certain pair of Polaroid sheets transmits light with minimum transmission (10 ) when their attenuating axes are at right angles, and with an ideal 0.5 transmission when these axes are aligned. We wish to design a light intensity modulator by electrically controlling the permittivity εfast along the fast axis of a perfectly transparent plate of -5thickness d inserted between these two polaroids, where εfast = εslow + 10 V , εslow = 2εo is the permittivity of the orthogonal slow axis, and V is the control voltage applied along the fast axis. (a) In terms of d and εo, what control voltage V yields a quarter-wave plate (QWP) between the polaroids, and what is the maximum fraction F of the incident light that can then pass through the modulator when the two Polaroid sheets are at right angles (i.e. blocking ~all light in the absence of the QWP)? By what angle θ is this QWP rotated relative to the polaroids when transmission through the three sheets is maximum? Would a half-wave plate perform better and, if so, how? Explain briefly. (b) This optimum modulator can then vary the fraction, F, of transmitted power between -6 10 and the maximum value found in part (a), depending on V. Find the expression for F(V,d). (c) If the transparent dielectric plate has ε = 4εo, what fraction Fr of the incident wave power is reflected by the plate’s front surface at normal incidence? A nearly comparable amount is reflected by its back surface as well. (d) What is Brewster’s angle θB for the plate of part (c)? Problem 5.2 (a) Consider a metal with roughly one free electron per atom and assume a frequency 30 -3such that it behaves like a plasma with µ = µo, ε = εo, and ne = 10 [m ]. What is the skin or penetration depth δ [m] of radio waves into this plasma in the low-frequency limit (ω → 0)? Appendix A.4 of the Course Notes contains useful physical constants. - 1 -DHS 3/18/09(b) For a metal with conductivity σ = 107 [S/m], at what frequency f [Hz] is the penetration “skin depth” δ the same as for (a)? Problem 5.3 John knows that the electron density in the ionosphere generally increases with altitude sufficiently slowly that upward bound radio waves suffer very little reflection except below the plasma frequency and at incidence angles beyond the critical angle θc. John can hear his local 1-MHz radio station WMIT out to the suburbs but then loses it as he drives further, until suddenly about 200 miles away he can again hear WMIT; from this he concludes θc = 45o. (a) What is the ionospheric plasma frequency fp o ) For those waves impacting the ionosphere at 45 thtensity, S(t) 2, is 1 mW/m . What is the complex Pointingnosphere at this position when the wave is incident exa = θc = 45°? [Hz] on this occasion? (b e incident time-averaged in vector, S , just inside the io ctly at the critical angle, θiProblem 5.4 (a) An x-polarized 1-GHz uniform plane wave of one watt per square meter is propagating in the +z direction in a very slightly lossy medium characterized by εo, µo, and σ = 10-4 [S/m]. What are the complex electric and magnetic fields E(z) and H(z) in this case? Are they in phase? (b) The same wave is now propagating in the +z direction in vacuum and is reflected from a perfect conductor at z = 0. What now are the complex electric and magnetic fields E(z) and H(z) for z < 0? AreE andH in phase in time and space? Explain briefly. (c) What are the total E(x, z) and H(x, z) for the case where a TM uniform plane wave of wavelength λ is propagating in the x-z plane for z < 0 and is reflected at 45o incidence angle from a perfect conductor at z = 0? AssumeH(0,0) = Ho, the maximum value ofH(x,z). (d) Sketch the total electric fieldE(x,y,t) for z < 0 and t = 0 for the waves of part (c). - 2 -DHS 3/18/09MIT OpenCourseWare http://ocw.mit.edu 6.013 Electromagnetics and Applications Spring 2009 For information about citing these materials or our Terms of Use, visit:
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