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Unformatted text preview: Physics 21 Fall, 2004 Solution, Hour Exam #2 The graders for the problems were: 1 Shaffer, 2 Hickman, 3 Pagan, 4 Wentzel, 5 Lowe For questions about the grading, see the grader by Nov. 12. Problem 1. For the following circuit: R L C V rms ( frequency ω ) φ ω L I rms 1/ ω C I rms R I rms V rms (a) Write the loop equation for this circuit. V − IR − L dI dt − Q C = 0 (b) Draw a phasor diagram for this circuit, and use it to find an expression for the phase angle φ in terms of R , L , and C , and ω . From the phasor diagram, we have tan φ = ωL − 1 ωC R , or sin φ = ωL − 1 ωC q ( ωL − 1 ωC ) 2 + R 2 . This expression is valid for either ωL > 1 /ωC or ωL < 1 /ωC ; in the latter case the angle φ will be negative. (c) If R = 3 . 00 Ω, L = 5 . 00 mH, and C = 1 . 25 mF, what angular frequency ω gives the maximum rms current for a fixed V rms ? What is the value of φ at this frequency? The length of the vector V rms in the diagram is V rms = p ( ωL − 1 /ωC ) 2 + R 2 I rms , so that I rms = V rms / p ( ωL − 1 /ωC ) 2 + R 2 will be largest when ωL = 1 /ωC . Then φ = 0 and ω = 1 √ LC → 1 p (0 . 005 H)(0 . 00125 Ω) = 400 rad / s (d) If the ac power source is turned off, how long does it take the oscillating current to diminish to 25% of its steady state value? Use the values of R , L , and C given in part (c). The exponential decay obeys exp( − Rt/ 2 L ), so . 25 = exp( − Rt/ 2 L ) ⇒ t = 2 L ln 4 R = 4 . 6 ms Problem 2. An electron with velocity v passes unde ﬂected through a region where there are uniform electric and magnetic fields. The components of v are v...
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 Fall '07
 Hickman
 Physics, Force, Magnetic Field, m/s, I0

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