Fall 2011 Homework 20 Solutions

Fall 2011 Homework 20 Solutions - Physics 21 Fall 2011...

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Physics 21 Fall, 2011 Solution to HW-20 31-21 You have a 207Ω resistor, a 0 . 408H inductor, a 4 . 99 μ F capacitor, and a variable-frequency ac source with an amplitude of 2 . 93V. You connect all four elements to- gether to form a series circuit. (a) At what frequency will the current in the circuit be greatest? (b) What will be the current amplitude at this frequency? (c) What will be the current amplitude at an angular frequency of 410 rad/s? (d) At this frequency, will the source voltage lead or lag the current? (a) We need to look at the relationship V = IZ . Since V is fixed, in order to maximize I we need to minimize Z . Z depends on ω via X L and X C : Z = radicalBig R 2 + ( X L - X C ) 2 Clearly, Z is minimized when X L - X C = 0. Using the definitions of X L and X C , 1 ωC = ωL ω 2 = 1 LC ω = 1 LC = 1 radicalbig (0 . 408H)(4 . 99 μ F) = 701rad / s (b) When X L - X C = 0, Z = R , so I = V Z = 2 . 93V 207Ω = 14 . 2 × 10 3 A (c) Now, X L - X C = (410rad / s)(0 . 408H) - 1 (410rad / s)(4 . 99 μ F) = - 322Ω I = V Z = 2 . 93V radicalBig (207Ω) 2 + ( - 322Ω) 2 = 7 . 66 × 10 3 A (d) Since our calculation of X L - X C in part (c) came out negative, the circuit is mostly capacitive. This means that
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