Homework07MatBC - instruments. Now we better check that the...

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ECE 3030: Electromagnetic Fields and Waves Fall 2009 Homework 7 SOLUTIONS for GRADER: Reminder: Date: Tuesday 10/20 Problem 10.12 Given a wave frequency of 30 GHz in a collisionless plasma where its wavelength is 1.5 cm=0.015 m. a. The wavenumber is k = 2 π λ = 418 . 9m - 1 and we use this to determine the plasma frequency via k 2 = ω 2 μ 0 ε 0 " 1 - ω 2 p ω 2 # or ω p = s ω 2 - k 2 μ 0 ε 0 = r (2 π · 30 × 10 9 ) 2 - 418 . 9 2 4 π × 10 - 7 · 8 . 85 × 10 - 12 = 1 . 405 × 10 11 s - 1 giving f p = 22 . 37 GHz. The density is then N = f 2 p / 8 . 9 2 = 6 . 32 × 10 18 m - 3 . b. The pulse travels at the group velocity of v g = c " 1 - f 2 p f 2 # 1 / 2 = 0 . 666 c = 2 × 10 8 m / s Problem 10.17 Assume the good conductor approximation holds, and check this assumption later. The power in the wave attenuates following h S ( z ) i ∝ E 2 e - 2 z/δ , where the skin depth is δ = 1 / πfμσ . So h S (100 m) i h S (0 m) i = e - 2 z/δ = 10 - 2 W / m 2 100 W / m 2 = 10 - 4 = - 2 z δ = ln10 - 4 = - 9 . 24 = Then, δ = 21 . 7 m, and f = 1 πμ 0 σδ 2 = 1 π [4 π × 10 - 7 ]4(21 . 7) 2 = 134 Hz! This low frequency requirement has application to communications with submarines and other under-the-sea
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Unformatted text preview: instruments. Now we better check that the “good conductor” approximation really holds, i.e., is σ >> ωε ? ωε = (2 π )(134)(81)(8 . 85 × 10-12 ) = 6 . 04 × 10-7 σ = 4 is indeed much bigger than this. Problem 12.11 Phase propagation in the x direction must match on either side of the boundary at z = 0. Hence, k ix = k t x = ⇒ k tx = ω c sin θ i , which is independent of the material. k tz = s ω 2 c 2 ± 1-j 30 4 ¶-k 2 tx = ω c s ± 1-j 30 4 ¶-1 4 = ω c √ 3 2 p 1-j 10 Now we find the angle of transmission from θ t = tan-1 • k tx <{ k tz } ‚ = tan-1 " 1 / 2 √ 3 2 <{ √ 1-j 10 } # = tan-1 • 1 √ 3 <{ √ 1-j 10 } ‚ = 13 . 8 ◦ CORNELL UNIVERSITY c ± WES SWARTZ (09/9/15) 7–1...
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This note was uploaded on 01/16/2010 for the course ECE 3030 at Cornell University (Engineering School).

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