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hw11sols

hw11sols - COLORADO STATE UNIVERSITY DEPARTMENT OF PHYSICS...

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Unformatted text preview: COLORADO STATE UNIVERSITY DEPARTMENT OF PHYSICS Spring Semester 2008 Ph142 - Physics for Scientists and Engineers PROBLEM SET 11 - CHAPTER 29 - AC CIRCUITS - PART I DUE AT THE START OF RECITATION CLASS ON 17 APRIL 2008 Some guidelines for problem solving (whether handed in and graded or not): Focus is or understanding and implementation, not “plug and chug” problems. Almost every problem will involve some level of conceptual understanding and some challenge in the mathematical set-up and parameterization. Required format: (1) Do you work on grid style engineering paper. (2) Use one side of the page only. (3) Start each problem on a new page and state the problem number clearly. (4) Put your name, section number, problem set number, and due date on every page. (5) Include proper and reasonably professional looking diagrams or sketches for each problem. (6) Assemble in problem order (#1, #2, #3, etc.) and staple your pages in the upper left corner. Note #1: For assigned problems, your work (here and on future problem sets) should be accompanied by: (a) A clear problem statement in words, diagrams, and equations (not a regurgitation of the problem text in Tipler and Mosoa, necessarily, but adequate for some one to know what the problem is without going back to the book). (b) Good clear well labeled diagrams. (c) Well developed equations with all parameters deﬁned. (d) Graphs with clearly defmed and labeled axes, etc. Note #2: You Should parameterize all problems. That is, assign algebraic parameters to all relevant variables and constants, set the problem up algebraically, and solve algebraically. Only then (if numerical evaluations are required), should you begin to plug in numbers. When you do plug in numbers, if it is a simple exercise, do as much of the evaluation as you can by hand (not by calculator). Multiply and cancel simple factors, add and subtract powers often, etc, to obtain a simpliﬁed numerical expression for evaluation. Often, you will fmd that you do not even need to do a calculator evaluation! If it is a complicated evaluation, you may wish to use some appropriate software, such as MathCad, etc. Note #4: A Problem Set will generally have from ﬁve to ten problems. If your recitation instructor decides to grade problems in some fashion, he or she may give partial credit, when deserved. However, there will be no credit for nonsense. Everything you say must make sense. Take care to do some intelligent work on ALL the problems. One completely missing problem can have a serious effect on your understanding and (if graded) on your Problem Set grade. Note #5. D Jn’t skimp on pages. Start each problem on a new page. Note #6. Start on a given problem set AS SOON AS IT IS MADE AVAILABLE. Work AT your solutions every day. Develop your solutions. Only when you are done, should you “write up” your solution set. When you do write it up, do so with understanding. Do not simply copy from your scratch notes. If you simply copy, without understanding, your graphs will not make sense, your equations will not make sense, and probably nothing will make sense. The problems (a) Tipler and Mosca, Problem 29-01. Why is this question somewhat ambiguous? (b) Tipler and Mosca, Problem 29-02. (c) Tipler and Mosca, Problem 29-04. How can you explain this intuitively. What happens in the zero and inﬁnite frequency limits? (d) Tipler and Mosca, Problem 29-05. How can you explain this intuitively. What happens in the zero and inﬁnite frequency limits? (e) Show that inductive and capacitive reactance has units of ohms. Variation on Tipler and Mosca, Problem 29-33. This is similar to the LC circuit analysis developed in class, this time with the capacitor initially charged to some iQo and an open switch in the circuit. Take the convention for positive current indicated by the down arrow on the leﬁ leg of the circuit in Fig. 29-30. Close the switch at time t: 0 and analyze the response. Write the Kirchoff loop equation. Discem the proper connection between the time dependent charge on the capacitor [taken as iq(r) such that q(0) = Q0] and the current i(t) . Use your loop equation to obtain a differential equation for q(t), apply initial conditions, and solve for q(t) and the frequency of the response. Sketch the corresponding q(t) and 1'0) responses. Sketch the capacitive and inductive energies as a function of time as well. Do tie analysis for a ac voltage source with a frequency to (angular, of course) driving an inductance L . Show that the voltage leads the current by 90°. Show that the ratio of the peak voltage to the peak current is equal to the inductive reactance X L = (:31. . Justify this result in the low and high frequency limits. Do the analysis for a ac voltage source with a frequency to (angular, of course) driving a capacitance C . Show that the voltage lags the current by 90°. Show that the ratio of the peak voltage to the peak current is equal to the capacitive reactance X c = 1/ roC . Justify this result in the low and high ﬁ'equency limits. Harmonic function equivalence. Show that the forms yC (t) = X cos(a:t+6x), ys(t) = X sin(wt+ ax), and yABU) = Acos(art)+ B sin(a)t) are mathematically equivalent. That is, one can take yC (t) = y .4 3 (t) and obtain X and 9x in terms of A and B , and likewise, one can take ys (t): yAB (t) and obtain X and ¢x in terms of A and B. Variation on Tipler and Mosca, Problem 29—40. Work (a) and (b) as stated. For (0), the equations are correct but the question is poorly stated. Work out expressions for the phase 6 and the impedance Z . Add part (d): Check that the response makes sense in the limits at —-) 0 and a) —> no. Please pay attention to this problem. In Part II, we will apply the method of phasors to solve this same problem in a much easier way. Hint: Note that the voltage, £(t) = 80 cos a): , is common to both elements. The current through the inductor lags this voltage by 90° and is given by [L (t: = (60 / X L ) sin cot . The current through the resistor is in phase with the drive voltage. Just write these down, add them up, and apply the trig identity cos(a — ﬂ) = cos a cos [3 + siua sin ,6 . Variation on the high pass ﬁlter problem, Tipler and Mosca, Problems 29-44 and 29-45. This situation is ill posed as stated. From the known fact that the ac current through a capacitor leads the voltage by 90°, the conunon current for an R and C in series will lead the overall applied input voltage by some angle between zero and 90°. The output voltage, which is across the resistor in this problem, will be in phase with the current and will lead the input voltage by this same amount. The Tipler and Mosca notation implies that the output voltage will lag the input voltage. The rigorous solution, of course, would yield a negative 5 . In order to emphasize the physics, work this problem in a slightly different way. Consider the circuit in Fig. 29-35 and start with a common current of the form 1(1) = 10 cos cat for the series R and C elements. Then work backwards to obtain the input voltage in the form, VIN = Vm pEAK cos(rnt—6). Determine VH / VIN pEAK and c3 . Do this by adding the voltage across the resistor, VR = [OH cos(a)t) = VH (in Tipler, this "H" is for "high pass"), and the voltage across the capacitor, Vc = 10X c sin(r:ot) (noting that the sin function lags the cos function by 90 °), and applying once again the trig. identity cos(a — ﬂ) = cosacos ,B+sinasinﬂ. This will give you the phase factor (5' , which will now be a positive value since we have already taken the signs into account physically, and the ratio VH / VIN FEAK = 11’1l1+l/(coRC)2 as in the text. 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