Multiple Choice
An inductor, a capacitor, and a resistor are in series with a AC source. If the capacitor is , the inductor is and the resistor is . what is the impedance?
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31. Alternating Current
Series LRC Circuits
Problem 60a
Textbook Question
The tuning circuit in an FM radio receiver is a series RLC circuit with a 0.200 μH inductor. The receiver is tuned to a station at 104.3 MHz. What is the value of the capacitor in the tuning circuit?
Verified step by step guidance1
Identify the formula for the resonant frequency of an RLC circuit: \( f = \frac{1}{2\pi\sqrt{LC}} \), where \( f \) is the resonant frequency, \( L \) is the inductance, and \( C \) is the capacitance.
Rearrange the formula to solve for the capacitance \( C \): \( C = \frac{1}{(2\pi f)^2 L} \).
Substitute the given values into the formula: \( f = 104.3 \text{ MHz} = 104.3 \times 10^6 \text{ Hz} \) and \( L = 0.200 \mu\text{H} = 0.200 \times 10^{-6} \text{ H} \).
Calculate the term \( (2\pi f)^2 \) by first finding \( 2\pi f \) and then squaring the result.
Divide 1 by the product of \( (2\pi f)^2 \) and \( L \) to find the capacitance \( C \). Ensure the units are consistent throughout the calculation.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
RLC Circuit
An RLC circuit is an electrical circuit consisting of a resistor (R), inductor (L), and capacitor (C) connected in series or parallel. It is used to filter signals and can resonate at a specific frequency, determined by the values of R, L, and C. The resonance occurs when the inductive and capacitive reactances are equal, allowing maximum current to flow at the resonant frequency.
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LRC Circuits
Resonant Frequency
The resonant frequency of an RLC circuit is the frequency at which the circuit naturally oscillates due to the balance between inductive and capacitive reactance. It is given by the formula f₀ = 1 / (2π√(LC)), where f₀ is the resonant frequency, L is the inductance, and C is the capacitance. At this frequency, the circuit can efficiently select or amplify signals, making it crucial for tuning applications like FM radio.
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Capacitance Calculation
To find the capacitance required for tuning an RLC circuit to a specific frequency, we can rearrange the resonant frequency formula to solve for C: C = 1 / (4π²f₀²L). This calculation allows us to determine the necessary capacitance value to achieve resonance at the desired frequency, ensuring the circuit can effectively receive the intended radio signal.
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