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An $R L C$ circuit has a capacitance of 0.29$\mu \mathrm{F}$ . (a) What inductance will produce a resonance frequency of 95 $\mathrm{MHz}$ ? (b) It isdesired that the impedance at resonance be one-fifth the impedance at 11 $\mathrm{kHz}$ . What value of $R$ should be used to obtain thisresult?

a) 9.7 $\mathrm{pH}$b) 10 ohms

Physics 102 Electricity and Magnetism

Chapter 24

Alternating-Current Circuits

Current, Resistance, and Electromotive Force

Direct-Current Circuits

Electromagnetic Induction

Alternating Current

University of Washington

Hope College

University of Sheffield

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the first tight and we have to calculate what value often reluctance will result in a resonant frequency off 95 megahertz. Then we begin by remembering that the resonant frequency because to one divided by two pi times the square root off l time see And we can solve this expression for l. And by doing that we get the following expression for the doctors. So two pi times the resonant frequency squared Is he close to one divided by your time, See, and then l e Z close to two pipe times the resonant frequency squared times the capacitance by plugging device was given by the problem. We get the following for the doctrines to pi times 95 times 10 to the sixth squared because there is a mega here times 0.29 times 10 to the minus six because there is a micro here they're one on. This is approximately 9.7 times 10 to the minus 12 Harry's which we can write us 9.7 PICO Henry's in the next item we wanted impedance resonance to be equals two, 1/5 off the impedance at 11 kilo hertz and then we want to know what value offer resistance can satisfy this condition. We begin by remembering that at resonance the impudence is too close to the resistance. Then we get our course to 1/5 off the impedance at 11 kilohertz. Now we do the following. Remember that impedance is equal to the square it off R squared, plus the difference between the reactor sis squared. We now square both sides off this equation to get r squared. Is it close to one divided by 25 times R squared plus eggs l mines like see squared No, We can move 25 to the other side To get 25 r squared equals two wars Where plus excel mine is sexy squared Now we move these term to the other side To get 24 r squared is equals to excel Mine is sexy squared Finally divide by 24 We get our squared equals two exile miners XY squared divided by 24 Then our is equals to excel Miners XY Absolutely I you because there is no negative resistance is divided by the square root off 24. Then we remember that excel is it close to two pi times the frequency times in look tense, which is in close to two pi times. 11. Time stand to deterred because there is a key clue here, times in doctors on Docked Us is. He goes to 9.7 time Stand to my news 12 on Maxie. Is he close to one divided by two pi times f time seat because he goes to one divided by 25 times 11 time stamped with toward times the capacitance, which is you're a 0.29 Niekro for its no plugging these values in these equations would get the following for the resistance. Are these equals to that one? Divided by the square root off 24 times The absolute value Off Excel We cheese to pie Times 11 times, 10 to the third times 9.7 Time standard in minus 12 miles. Accede, which is equals to two pi times 11 times 10 to the third times 0.29 times 10 to the minus states under one, and this is approximately 10 owns

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