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Carnegie Mellon University

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Problem 100 Hard Difficulty

The lowest frequency in the FM radio band is 88.0 MHz. (a) What inductance is needed to produce this resonant frequency if it is connected to a 2.50 pF capacitor? (b) The capacitor is variable, to allow the resonant frequency to be adjusted to as high as 108 MHz. What must the capacitance be at this frequency?

Answer

$8 \cdot 6 \times 10^{-7} \mathrm{H}$

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Video Transcript

So here we know that the resident frequency is equaling one over to pi times the square root of the product between the inducting ts and the capacitance. So for part a solving for the induct in CE this would be equaling one over four pi squared times the resonant frequency square times, the capacitance. So for party, the induct in CE would be equal in one over four pi squared. Here we have a frequency on equaling 2.5. Rather, my apologies 80 eight megahertz and so this would be equaling 80 eight times. 10 to the sixth hurts quantity squared, multiplied by the capacitance of 2.5 Pico Fareed's so 2.5 times 10 to the negative 12th Fareed's And we find that the induct in CE for part A is a 1.31 times 10 to the negative Sixth Henry's This would be our answer. Four part eh For part B, we know that the capacitance solving for the capacitance this would be equaling one over four pi squared times the resonant frequency squared times the induct in ce l and so we can solve for part B. The capacitance would then be equaling here it will be one over and then again to pie. Now the frequent. Another one over four pi squared times the resident frequency. Now this resident frequency is 108 megahertz, 108 times 10 to the sixth hurts quantity squared times again 1.31 times 10 to the negative Sixth Henry's. And we find that then the capacitance see is equaling 1.67 times 10 to the rather. We can say one point 1.667 You say 1.66 times 10 to the negative 12th spirits. This would be our answer for part B. Um, and that's surrounding. We could also convert to Pico. Fared so we can say six. The capacitance is 1.66 Um, Pico ferrets. It doesn't really matter. Um, either currents, either answer is correct. That is the end of the solution. Thank you. For what

Carnegie Mellon University
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