892_Physics ProblemsTechnical Physics

892_Physics ProblemsTechnical Physics - 232 *P31.54...

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232 Faraday’s Law * P31.54 (a) Consider an annulus of radius r , width dr , height b , and resistivity ρ . Around its circumference, a voltage is induced according to εω π ω =− ⋅ =− =+ N d dt d dt Bt r B r t BA 1 22 max max cos sin bg . The resistance around the loop is ρπ A A r bdr x = 2 . The eddy current in the ring is dI Br t b d r r b d r t == = ε πω ω ωω resistance max max sin sin 2 2 2 . The instantaneous power is dd I b d r t i P πω max sin 23 2 2 2 . The time average of the function sin cos 2 1 2 1 2 2 tt is 1 2 0 1 2 −= so the time-averaged power delivered to the annulus is d b d r = max 2 4 . The power delivered to the disk is PP zz d Bb rd r R max 3 0 4 F H G I K J = R BR b max max 4 24 2 44 0 16 . (b) When B max gets two times larger, B max 2 and get 4 times larger. (c) When f and ωπ = 2 f double, 2 and get (d) When R doubles, R 4 and become 21 6 4 = times larger.
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