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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 2.00 - \(\mu\) F capacitor was fully charged by being connected to a 12.0 - \(V\) battery. The fully charged capacitor is then connected in series with a resistor and an inductor: \(R=50.0 \Omega\) and \(L=0.200 \mathrm{H}\). Calculate the damped frequency of the resulting circuit.

A \(200-\Omega\) resistor, a \(40.0-\mathrm{mH}\) inductor and a \(3.0-\mu \mathrm{F}\) capacitor are connected in series with a time-varying source of emf that provides \(10.0 \mathrm{~V}\) at a frequency of \(1000 \mathrm{~Hz}\). What is the impedance of the circuit? a) \(200 \Omega\) b) \(228 \Omega\) c) \(342 \Omega\) d) \(282 \Omega\)

An LC circuit consists of a capacitor, \(C=2.50 \mu \mathrm{F},\) and an inductor, \(L=4.0 \mathrm{mH}\). The capacitor is fully charged using a battery and then connected to the inductor. An oscilloscope is used to measure the frequency of the oscillations in the circuit. Next, the circuit is opened, and a resistor, \(R\), is inserted in series with the inductor and the capacitor. The capacitor is again fully charged using the same battery and then connected to the circuit. The angular frequency of the damped oscillations in the RLC circuit is found to be \(20 \%\) less than the angular frequency of the oscillations in the LC circuit. a) Determine the resistance of the resistor. b) How long after the capacitor is reconnected in the circuit will the amplitude of the damped current through the circuit be \(50 \%\) of the initial amplitude? c) How many complete damped oscillations will have occurred in that time?

A transformer has 800 turns in the primary coil and 40 turns in the secondary coil. a) What happens if an AC voltage of \(100 . \mathrm{V}\) is across the primary coil? b) If the initial \(A C\) current is \(5.00 \mathrm{~A}\), what is the output current? c) What happens if a DC current at \(100 .\) V flows into the primary coil? d) If the initial DC current is \(5.00 \mathrm{~A}\), what is the output current?

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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