MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.013 – Electromagnetics and Applications Problem Set 2 (five problems) Suggested Reading: Course notes (Staelin) Sections 2.1-2.3.4, 2.4, 2.7.1-2.7.3, 3.2.1-3.2.2, Appdx. C. Problem 2.1 Assume the “Whatever” vector⎯W(x,y,z) = ˆ ˆ .xsiny + yy (a) If an electric displacement vector⎯D =⎯W, what is the charge density ρ(x,y,z) [C/m3]? (b) If the magnetic field⎯H =⎯W, what is the current density⎯J(x,y,z) [A/m2], assuming⎯H is physically possible? (c) Does the magnetic field⎯B =⎯W satisfy all of Maxwell’s equations? If not, which one is violated? Problem 2.2 If the electric field E(t) = Re{E ejωt} where E is a phasor, then what is E(t) if: (a) E = 1 - j (b) E = ejπ/4 - 1 (c) ⎯E = jxˆ + (1 - j)yˆ(d) What is the complex vector⎯E if ⎯E(t) = xˆcos ωt + yˆ sin(ωt + π/4)? [Hint: E(t) = cos ωt for the case E = 1] Continued on next page - 1 - 2/9/09Problem 2.3 (a) What is the frequency f (Hz) of the wave having the magnetic field: ⎯H = xˆsin(107πt - 0.2z) + yˆ cos(107πt - 0.2z – 2.5π)? (b) What is its wavelength λ (meters)? (c) What is the velocity of light c in this medium? (d) Find the corresponding⎯E(x,y,z,t) assuming μ = μo. (e) What is the polarization of this wave? (e.g., “left circular”; polarization is usually characterized by the behavior of the electric field vector) (f) What is the shortest non-zero time delay τ(sec) that could be added to the x component of the wave in order to achieve linear polarization? In this case, what is the direction θ of the linear polarization relative to the x axis? A sketch may help. (g) Polarization can also be characterized by complex notation. What is the polarization of the z-propagating wave⎯E = (j -1)xˆ + (1 - j)y ˆ ? (e.g., “left circular”.) Problem 2.4 A 1-GHz uniform plane wave propagating in the +z direction in a medium μ, ε is characterized by⎯E = x3ˆ. (a) What is the time average intensity [W/m2] of this wave? (b) What is the magnetic energy density W3m(t) [J/m ] at x = y = z = 0? What is the electric energy density We(t) there? Problem 2.5 Using the general expression for inductance L, find L for a σ = ∞ coaxial inductor of length D and short circuited at one end, rμwhere the inner and outer radii of the two concentric i oconductors are ri and ro, respectively, as illustrated. D ro ΛμN∫∫H i da L = = Ai i - 2 - 2/9/09MIT 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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