329sp08hw4 - ECE 329 Homework 4 Due 5PM 1 Consider a...

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ECE 329 Homework 4 Due: Feb 11, 2008, 5PM 1. Consider a time-varying electric field specified as E = ˆ yE o cos( ω t - β z ) V/m where E o (amplitude), ω (radian frequency), and β (wavenumber) are constant real scalars. a) Find the corresponding magnetic fields B and H = μ - 1 o B using Faraday’s law ∇ × E = - B t (in di ff erential form) and an additional constraint that B = 0 if E o = 0 . b) Show that E specified above satisfies Gauss’s law · D = ρ with D = o E and ρ = 0 . c) Show that D and H from above satisfy the Ampere’s law ∇ × H = J + D t with J = 0 and ω 2 β 2 = 1 μ o o c 2 . d) What is the numerical value of c (speed of light) in SI? e) Suppose that β = + ω c , in consistency with the constraint in (c), and ω = 2 π × 10 6 rad/s. Plot (by hand, not with computer) E y = E o cos( ω t - β z ) in that case as function of z at time instants t = 0 and t = π / 4 ω , and from the displacement of one of the crests of E y waveform infer the velocity of propagation , v p , of the field E = ˆ yE y in z direction.
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