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329fall08hw3sol

# 329fall08hw3sol - ECE-329 Fall 2008 Homework 3(Solution 1...

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ECE-329 Fall 2008 Homework 3 (Solution) September 15, 2008 1. In this problem, we will study Faraday’s law, ˛ C ~ E · d ~ l = - d dt ˆ S ~ B · d ~ S, which states that the electromotive force E = ¸ C ~ E · d ~ l around any closed loop C equals the time rate of change of the magnetic flux Ψ B = ´ S ~ B · d ~ S through the surface S bounded by the loop. (a) If ~ B = 0 at all times, the magnetic flux is also zero ( Ψ B = 0 ), and therefore, according to Faraday’s law, the electromotive force is zero ( E = 0 ) over any closed loop C. (b) If ~ B 6 = 0 then it is possible for E 6 = 0 if the path C is disturbed or being displaced in such a way that the magnetic flux Ψ B = ´ S ~ B · d ~ S varies in time. (c) Let us define a closed loop C passing through the fixed points P 1 and P 2 (see the next figure). P 1 P 2 dl dl path A path B C Since ~ B is time-independent, the corresponding magnetic flux Ψ B is also time independent, and therefore, ˛ C ~ E · d ~ l = - d Ψ B dt = 0 . Breaking the closed path integral into two parts, we have ˛ C ~ E · d ~ l = ˆ P 1 P 2 path A ~ E · d ~ l + ˆ P 2 P 1 path B ~ E · d ~ l = 0 . Reversing the direction of integration of the second integral, it can be shown that ˆ P 1 P 2 path A ~ E · d ~ l = ˆ P 1 P 2 path B ~ E · d ~ l. In consequence, the line integral ´ 12 ~ E · d ~ l does not depend on the path taken between P 1 and P 2 . 1

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ECE-329 Fall 2008 2. Given ~ B = B 0 ( t cos( ωt ) ˆ x + sin( ωt ) ˆ z ) Wb / m 2 , we can apply Faraday’s law to compute the emf E around the following closed paths. Since the surfaces bounded by the closed paths are not varying in time, and also since ~
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329fall08hw3sol - ECE-329 Fall 2008 Homework 3(Solution 1...

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