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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 Ω, 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 computer draws an rms current of \(2.80 \mathrm{A}\) at an \(\mathrm{rms}\) voltage of \(120 \mathrm{V}\). The average power consumption is 240 W. (a) What is the power factor? (b) What is the phase difference between the voltage and current?
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?
A variable inductor can be placed in series with a lightbulb to act as a dimmer. (a) What inductance would reduce the current through a \(100-\) W lightbulb to \(75 \%\) of its maximum value? Assume a \(120-\mathrm{V}\) rms, \(60-\mathrm{Hz}\) source. (b) Could a variable resistor be used in place of the variable inductor to reduce the current? Why is the inductor a much better choice for a dimmer?
A series combination of a resistor and a capacitor are connected to a \(110-\mathrm{V}\) rms, \(60.0-\mathrm{Hz}\) ac source. If the capacitance is \(0.80 \mu \mathrm{F}\) and the rms current in the circuit is $28.4 \mathrm{mA},$ what is the resistance?
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