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Chapter 30: Problem 60

A series RLC circuit has a source of time-varying emf providing \(12.0 \mathrm{~V}\) at a frequency \(f_{0}\), with \(L=7.00 \mathrm{mH}\), \(R=100 . \Omega,\) and \(C=0.0500 \mathrm{mF}\) a) What is the resonant frequency of this circuit? b) What is the average power dissipated in the resistor at this resonant frequency?

Short Answer

Expert verified
Question: Calculate the resonant frequency and the average power dissipated in the 100 Ω resistor of an RLC circuit with an inductor of 7.00 mH, a capacitor of 0.0500 μF, and a voltage of 12.0 V. Answer: The resonant frequency of the RLC circuit is approximately 429.16 Hz, and the average power dissipated in the 100 Ω resistor at this frequency is 1.44 W.

Step by step solution

01

Resonant Frequency Formula

To find the resonant frequency of the RLC circuit, we use the following formula: $$ f_0 = \frac{1}{2\pi \sqrt{LC}} $$ Where \(L\) is the inductance in henrys (H), \(C\) is the capacitance in farads (F), and \(f_0\) is the resonant frequency in hertz (Hz).
02

Find the Resonant Frequency

Now substitute the given values of the inductor and the capacitor into the formula to find the resonant frequency: $$ f_0 = \frac{1}{2\pi \sqrt{(7.00 \times 10^{-3})(0.0500\times 10^{-3})}} $$ Calculate \(f_0\) to get the resonant frequency.
03

Average Power Dissipated in the Resistor Formula

The average power dissipated in the resistor at resonant frequency can be found using the following formula: $$ P = \frac{V^2}{R} $$ Where \(P\) is the average power in watts (W), \(V\) is the voltage in volts (V), and \(R\) is the resistance in ohms (Ω).
04

Find the Average Power Dissipated in the Resistor at Resonant Frequency

Now substitute the given values of voltage and resistance, and the calculated resonant frequency into the formula to find the average power dissipated in the resistor: $$ P = \frac{(12.0)^2}{100} $$ Calculate \(P\) to get the average power dissipated in the resistor at the resonant frequency.

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Most popular questions from this chapter

A series circuit contains a \(100.0-\Omega\) resistor, a \(0.500-\mathrm{H}\) inductor, a 0.400 - \(\mu\) F capacitor, and a time-varying source of emf providing \(40.0 \mathrm{~V}\). a) What is the resonant angular frequency of the circuit? b) What current will flow through the circuit at the resonant frequency?

A standard North American wall socket plug is labeled \(110 \mathrm{~V}\). This label indicates the __________ value of the voltage. a) average b) maximum c) root-mean-square (rms) d) instantaneous

A 10.0 - \(\mu\) F capacitor is fully charged by a 12.0 - \(\mathrm{V}\) battery and is then disconnected from the battery and allowed to discharge through a 0.200 - \(\mathrm{H}\) inductor. Find the first three times when the charge on the capacitor is \(80.0-\mu \mathrm{C},\) taking \(t=0\) as the instant when the capacitor is connected to the inductor.

A capacitor with capacitance \(C=5.00 \cdot 10^{-6} \mathrm{~F}\) is connected to an AC power source having a peak value of \(10.0 \mathrm{~V}\) and \(f=100 . \mathrm{Hz} .\) Find the reactance of the capacitor and the maximum current in the circuit.

What are the maximum values of (a) current and (b) voltage when an incandescent 60 -W light bulb (at \(110 \mathrm{~V})\) is connected to a wall plug labeled \(110 \mathrm{~V} ?\)
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