Ch 32: AC Circuits

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경

Knight Calc 5th Edition

Ch 32: AC Circuits

문제 60a

32장, 문제 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?

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.

관련 실천

교과서 질문

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?

1860

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교과서 질문

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.

1948

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교과서 질문

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?

1691

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교과서 질문

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?

137

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교과서 질문

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?

231

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교과서 질문

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

196

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