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Chapter 21: Problem 75

At what frequency does the maximum current flow through a series RLC circuit containing a resistance of \(4.50 \Omega,\) an inductance of \(440 \mathrm{mH},\) and a capacitance of \(520 \mathrm{pF} ?\)

Short Answer

Expert verified
Question: Calculate the frequency at which the maximum current flows through a series RLC circuit with the following values: resistance R = 120 惟, inductance L = 440 mH, and capacitance C = 520 pF. Answer: The maximum current flows through the series RLC circuit at a frequency of approximately \(1.047 \times 10^{6} \, \text{Hz}\) or \(1.047 \,\text{MHz}\).

Step by step solution

01

Convert given values to SI units

We first need to convert the given values to their proper SI units: Inductance (L) = 440 mH = \(440 \times 10^{-3}\) H Capacitance (C) = 520 pF = \(520 \times 10^{-12}\) F.
02

Calculate the resonant frequency

Apply the formula for resonant frequency, which is \(f_r = \dfrac{1}{2\pi\sqrt{LC}}\). Substitute the given values of L and C: \( f_r = \dfrac{1}{2\pi\sqrt{(440 \times 10^{-3}) \cdot (520 \times 10^{-12})}} \).
03

Calculate the result

Simplify and solve for \(f_r\): \( f_r = \dfrac{1}{2\pi\sqrt{2.288 \times 10^{-10}}} \). \( f_r \approx 1.047 \times 10^{6} \,\text{Hz} \).
04

Present the final answer

The maximum current flows through the series RLC circuit at a frequency of approximately \(1.047 \times 10^{6} \, \text{Hz}\) or \(1.047 \,\text{MHz}\).

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Most popular questions from this chapter

A series \(R L C\) circuit has \(R=500.0 \Omega, L=35.0 \mathrm{mH},\) and $C=87.0 \mathrm{pF} .$ What is the impedance of the circuit at resonance? Explain.
Suppose that an ideal capacitor and an ideal inductor are connected in series in an ac circuit. (a) What is the phase difference between \(v_{\mathrm{C}}(t)\) and \(v_{\mathrm{L}}(t) ?\) [Hint: since they are in series, the same current \(i(t)\) flows through both. \(]\) (b) If the rms voltages across the capacitor and inductor are \(5.0 \mathrm{V}\) and \(1.0 \mathrm{V}\), respectively, what would an ac voltmeter (which reads rms voltages) connected across the series combination read?
An RLC series circuit has \(L=0.300 \mathrm{H}\) and \(C=6.00 \mu \mathrm{F}\) The source has a peak voltage of \(440 \mathrm{V}\). (a) What is the angular resonant frequency? (b) When the source is set at the resonant frequency, the peak current in the circuit is \(0.560 \mathrm{A}\). What is the resistance in the circuit? (c) What are the peak voltages across the resistor, the inductor, and the capacitor at the resonant frequency?
An ac circuit has a single resistor, capacitor, and inductor in series. The circuit uses \(100 \mathrm{W}\) of power and draws a maximum rms current of $2.0 \mathrm{A}\( when operating at \)60 \mathrm{Hz}\( and \)120 \mathrm{V}$ rms. The capacitive reactance is 0.50 times the inductive reactance. (a) Find the phase angle. (b) Find the values of the resistor, the inductor, and the capacitor.
A hair dryer has a power rating of \(1200 \mathrm{W}\) at \(120 \mathrm{V}\) rms. Assume the hair dryer circuit contains only resistance. (a) What is the resistance of the heating element? (b) What is the rms current drawn by the hair dryer? (c) What is the maximum instantaneous power that the resistance must withstand?
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