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You plan to change the frequency of the generator in this circuit to produce aphase angle of smaller magnitude. The resistor is still 175$\Omega$ , theinductor is 90.0 $\mathrm{mH}$ , the capacitor is 15.0$\mu \mathrm{F}$ , and the rms volt-age is 120.0 $\mathrm{V}$ (a) Should you increase or decrease the frequency?Explain. (b) Find the frequency that gives a phase angle of $-22.5^{\circ} .$ (c) What is the rms current in the circuit at the frequency found in part (b)?

a) see explanationb) 75.59 $\mathrm{Hz}$c) $0.634A$

Physics 102 Electricity and Magnetism

Chapter 24

Alternating-Current Circuits

Current, Resistance, and Electromotive Force

Direct-Current Circuits

Electromagnetic Induction

Alternating Current

Cornell University

Rutgers, The State University of New Jersey

University of Michigan - Ann Arbor

Simon Fraser University

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in the first item. We want to increase the face angle for minors 39.3 degrees to minus 22.5 degrees. Remember that the tangent off the face and go is he goes to excel. Mine is sexy, divided by the resistance We want that angle to increase on by increasing in this particular situation the tangent off the face and low we'll be closer to zero So we want to decrease this quantity as a world by only mass Mr Frequency Not that you circulate We have XY bigger than excel So this is a capacity of circuit. The capacitance dominates the circuit. If we want to decrease the difference between excel on Maxie, we must decrease exceed while increasing excel Remembering that excel is it goes to two pi times The frequency times they look tense on backseat is equals to one divided by two pi times the frequency times like a past tense We can see that by increasing the frequency way Exactly what we want to go by increasing f we increase excel Why you decreasing XY So what happens is that excel miners XY will decrease has wanted then we should increase the frequency in the second item, we have to find the value off the new frequency. Note that we can do that by using this relation between the ancient off the phase on the property is off the circuit. Two. We got the changing off phase You close to to pie F l f prime here minus on. Also fight crime here one divided by two pi. If prime time See And this is all divided by the resistance are then we have to solve these equations for the frequency and we do it as follows. What you playing both sides by r to get our tangent off. Five Prime is he goes to two pi f prime l minus one divided by two pie. That's Brian. See, now we have to cross multiply this fractions to make the eliminator the same to get our times that kind agent off five Prime. Is it close to to pi squared f Brian squared the l C minus one divided by two pi f Brian. See? No, we can do the following. We can send this term to the other side as a multiplication to get our times to pie times see Time's left prime Time's the Tangent five. Prime is he goes to to pi squared f Prime squared times the all time See my nose one. Then we can put our terms on the same side of the equation to get a second order uh, equation. And by doing so, we get the following to pie squared times l time See times f prime squared minds to pi times our time See times the tangent off five prime left prime miners one Is it cost zero. Now all we have to do so this quadratic equation and we do so by using basket rest formula which goes as follows it is screaming on for these equation Easy clothes, too. B squared. So this term here squared to pie times our time See Time's detain Geant fight crime squared miles four times to buy square times l time. You see time's men with one Then Delta is equals two to pi squared times r squared times He's queer times Detain Geant Square off the new face Uncle plus four times two pi squared I'm zero time to see Finally the frequency is down equals two minus beat So miners, this term here. So two pi times our time see Thank the tangent off five prime plus or minus the square it off the screaming and to pi squared r squared times C squared times the tangent square lo fi prime It was four to pi squared times l Time see divided by two times to pi squared times l c Then we can plug in the values that are giving the problem to get F Brian is equal stupid to pie Times 175 times 15 times 10 to the minus six times the tangent off minus 22.5 degrees plus or minus the square root off to pi squared times 175 squared times 15 times 10 to the minus six squared Plus uh oh, not yet forgot to change it Times the tangent squared off minus 22.5 degrees plus four times to pie squared times ill which is equals 2 19 time Stan to the minus three time. See which is equal to 15 times. Stand to the minus six and this is all under a square root and finally, this is all divided by two times two pi square times L, which is 90 times 10 to the minus tree times the capacitance, which is 15 times 10 to the minus six. And these gives us two possible frequencies. We tried approximately minus 24 eight point true hurts or 75 point 59 hertz. I know that there are no negative frequencies, so we do not consider the solution. And finally, the new frequency should be 75.59 hertz finally on the last tight. And we have to calculate what is the R. M s current in the circle? It In that case, we can use homes law. The our romance is it goes to I are a mess times the impudence. But not that the following the court sign off the fees is equals. Two. The resistance divided right impedance uniform impudence. Physicals to the resistance divided by the co sign off the face Dan, the current is equals True, the r a mass divided Bye, zee because it goes to Vieira mass times the co sign off the phase divided by our we can then plugging their values that are given in the problem to get 120 times the cool sign off minus 22.5 degrees, divided by the resistance, which is 175. And these results in on Aramis current, off approximately zero point 634 and peers.

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