Question

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

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

Step-by-step solution

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.

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

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: \(f \approx 15.92 \text{ Hz}\)

Conclusion

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