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Ch 32: AC Circuits
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 32: AC CircuitsProblem 60a
Chapter 32, Problem 60a

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?

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1
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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Related Practice
Textbook Question

A television channel is assigned the frequency range from 54 MHz to 60 MHz. A series RLC tuning circuit in a TV receiver resonates in the middle of this frequency range. The circuit uses a 16 pF capacitor. What is the value of the inductor?

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Textbook Question

The tuning circuit in an FM radio receiver is a series RLC circuit with a 0.200 μH inductor. FM radio stations are assigned frequencies every 0.2 MHz, but two nearby stations cannot use adjacent frequencies. What is the maximum resistance the tuning circuit can have if the peak current at a frequency of 103.9 MHz, the closest frequency that can be used by a nearby station, is to be no more than 0.10% of the peak current at 104.3 MHz? The radio is still tuned to 104.3 MHz, and you can assume the two stations have equal strength.

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Textbook Question

Commercial electricity is generated and transmitted as three-phase electricity. Instead of a single emf, three separate wires carry currents for the emfs ε1 = ε0 cos ωt, ε2 = ε0 cos(ωt +120°), and ε3 = ε0 cos(ωt−120°) over three parallel wires, each of which supplies one-third of the power. This is why the long-distance transmission lines you see in the countryside have three wires. Suppose the transmission lines into a city supply a total of 450 MW of electric power, a realistic value. What would be the rms current in each wire if the transmission voltage were ε0 = 120 V rms?

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Textbook Question

In FIGURE P32.54, what is the current supplied by the emf when (a) the frequency is very small and (b) the frequency is very large?

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Textbook Question

A series RLC circuit consists of a 50 Ω resistor, a 3.3 mH inductor, and a 480 nF capacitor. It is connected to a 5.0 kHz oscillator with a peak voltage of 5.0 V. What is the instantaneous current i when ε = ε0?

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Textbook Question

Show that the power factor of a series RLC circuit is cos ϕ=R/Z.

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