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An $R L C$ series circuit has a 1.00 $\mathrm{k} \Omega$ resistor, a 150$\mu \mathrm{H}$ inductor, and a 25.0 $\mathrm{nF}$ capacitor. (a) Find the circuit's impedance at 500 $\mathrm{Hz}$ . (b) Find the circuit's impedance at 7.50 $\mathrm{kHz}$ . (c) lif the voltage source has $V_{\mathrm{rms}}=408 \mathrm{V},$ what is $I_{\mathrm{rms}}$ at each frequency? (d) What is the resonant frequency of the circuit? (e) What is $I_{\mathrm{rms}}$ at resonance?

$x_{c}=117085$

Physics 102 Electricity and Magnetism

Chapter 23

Electromagnetic Induction, AC Circuits, and Electrical Technologies

Electromagnetic Induction

Rutgers, The State University of New Jersey

University of Sheffield

McMaster University

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for party The impotence we can first solve the inductive reactant. This would be to pi times the frequency we know to be 500 hertz and then the induct in CE we know to be 150 micro Henry. So 150 times 10 to the negative six Henry's and the, uh inductive react Ince's equaling toe point 471 comes. We know that then the capacitive reactant would be equaling one over to pie times the frequency of 500 hertz multiplied by the capacitance of 25 nano fares, or 25 times 10 to the native fit ninth hair Fareed's And so the capacitive reactant ce would be equaling thio. 12,738 0.85 comes and we can then say that the impotent Z would be equaling the resistance squared. So one kill Liam or 1000 mums quantity squared, plus the difference between the inductive reactant ce points 471 comes minus the capacitive reactor. Since 1 12,738 10.85 arms, this difference will be squared and then this sum of the squares will be raised to the 1/2 power and we find that the impotence is equaling essentially 12,771 0.13 ums. Or we can say that this would be equaling two 12.7 kill it pumps. So this would be our answer for part A around it to three significant figures. And then we're doing the exact same thing for part B. Except the frequency is now instead of 500 hertz 75 rather 7.5 kilohertz. So this would be equaling to pi times. The frequency of 7500 hurts. So 7.5 kilohertz multiplied by again 150 times 10 to the negative. Six. Henry's this and so that we find the inductive react. It's to be 7.65 arms. We know that the capacitive reactor it's would be equaling one over to pi multiplied by 7500 hertz, and this would be multiplied by 25 Dan Affair. It's 25 times 10 to the negative. Ninth Fareed's and the Capacitive React. Ince's 849.6 17 thumbs, so the impotence then would be equaling again. 1000 arms squared, the resistance squared plus seven point 065 arms minus 849.617 thumbs quantity squared, all raised to the 1/2 power. The impotence is then found to be equaling 1307.59 bums or, we can say approximately one 0.307 times 10 to the third bums or 1.3 one homes. This would be the impotence, of course. Rounded, toothy, rounded to three significant figures. Four parts see the root means squared. Current would be equaling. The root means squared voltage divided by the influence. And so this would be equaling. 408 votes, divided by 12 point by the recon say, 12.731 times 10 to the third arms, and we find that the root means squared Current would be point 032 amps. Again, this would be 4 500 hertz, and then the root means squared. Current would be again 408 faults, divided by 1.307 times 10 to the third hums, and this is equaling 0.312 amps. This is at 7500 hertz. So these would be our two answers for part C four part D. We want to find the resonant frequency resident frequency would be equaling. Ah, one over two pi times the square root between the product of the induct ins and the capacitance. So the resident frequency is equally one over two pi times the square root of the induct in CE we know to be 150 micro Henry's or 150 times 10 to the negative. Six Henry's times the capacitance of 25 nano fares 25 times tens of negative Ninth Fareed's and we find that the resident frequency is 82 point to hurts. This would be our final answer for the resident frequency. And then we can save her part e at the residence where the inductive reactant equally equals x apart the, um, reactive The, uh, ex of our we can say that Then the value of Z is equal to the resistance. That is our my apologies. It would be except see, So when except when the inductive react Insys equaling the capacitive reactant ce. Therefore, the impotence is simply equal to the resistance that they give us. And so here, for part e, the root means squared. Current is equaling. The root means squared, bolted, JJ divided by are the resistance. This would be equaling. 408 votes divided by one killer home or 1000 hums, and this is equaling 0.408 amps. So the R M s current at resonance would be 0.408 temps. That would be our answer for part E. That is the end of the solution. Thank you for

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