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Chapter 21: Problem 54

In an RLC series circuit, these three elements are connected in scries: a resistor of \(60.0 \Omega,\) a 40.0 -mH inductor, and a \(0.0500-\mathrm{F}\) capacitor. The series elements are connected across the terminals of an ac oscillator with an rms voltage of \(10.0 \mathrm{V}\). Find the resonant frequency for the circuit.

Short Answer

Expert verified
Answer: The resonant frequency of the given RLC series circuit is approximately 15.92 Hz.

Step by step solution

01

Formula for resonant frequency of an RLC circuit

The resonant frequency (f) for an RLC circuit is given by the formula: \(f = \dfrac{1}{2\pi\sqrt{LC}}\) Where: - L is the inductance of the inductor (in henry) - C is the capacitance of the capacitor (in farad) Now, we can plug the values of L and C into the formula to obtain the resonant frequency.
02

Convert the inductance value L to henry

The given inductance value is 40.0 mH (millihenry). To convert it to henry, we use the following conversion: 1 henry (H) = 1000 millihenry (mH) So, L = 40.0 mH = \(\dfrac{40.0}{1000}\) H = 0.0400 H
03

Calculate the resonant frequency

Now we have all the values needed. Let's plug them into the resonant frequency formula: \(f = \dfrac{1}{2\pi\sqrt{LC}} = \dfrac{1}{2\pi\sqrt{(0.0400 \text{ H})(0.0500 \text{ F})}}\) Calculating the result, we get: \(f \approx 15.92 \text{ Hz}\)
04

Conclusion

The resonant frequency of the given RLC series circuit is approximately 15.92 Hz.

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Most popular questions from this chapter

An RLC series circuit is driven by a sinusoidal emf at the circuit's resonant frequency. (a) What is the phase difference between the voltages across the capacitor and inductor? [Hint: since they are in series, the same current \(i(t) \text { flows through them. }]\) (b) At resonance, the rms current in the circuit is \(120 \mathrm{mA}\). The resistance in the circuit is \(20 \Omega .\) What is the rms value of the applied emf? (c) If the frequency of the emf is changed without changing its rms value, what happens to the rms current? (W) tutorial: resonance)
In the crossover network of Problem \(61,\) the inductance \(L\) is $1.20 \mathrm{mH}$. The capacitor is variable; its capacitance can be adjusted to set the crossover point according to the frequency response of the woofer and tweeter. What should the capacitance be set to for a crossover point of $180 \mathrm{Hz} ?[\text {Hint}:$ At the crossover point, the currents are equal in amplitude. \(]\)
An RLC series circuit is built with a variable capacitor. How does the resonant frequency of the circuit change when the area of the capacitor is increased by a factor of \(2 ?\)
Three capacitors $(2.0 \mu \mathrm{F}, 3.0 \mu \mathrm{F}, 6.0 \mu \mathrm{F})$ are connected in series to an ac voltage source with amplitude \(12.0 \mathrm{V}\) and frequency \(6.3 \mathrm{kHz}\). (a) What are the peak voltages across each capacitor? (b) What is the peak current that flows in the circuit?
For a particular \(R L C\) series circuit, the capacitive reactance is $12.0 \Omega,\( the inductive reactance is \)23.0 \Omega,$ and the maximum voltage across the \(25.0-\Omega\) resistor is \(8.00 \mathrm{V}\) (a) What is the impedance of the circuit? (b) What is the maximum voltage across this circuit?
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