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Ch. 31 - Maxwell's Equations and Electromagnetic Waves
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Giancoli Douglas 5th editionCh. 31 - Maxwell's Equations and Electromagnetic Waves문제 47
30장, 문제 47

An amateur radio operator wishes to build a receiver that can tune a range from 14.0 MHz to 15.0 MHz. A variable capacitor has a minimum capacitance of 95 pF.
(a) What is the required value of the inductance?
(b) What is the maximum capacitance used on the variable capacitor?

검증된 단계별 안내
1
Step 1: Understand the problem. The receiver uses an LC circuit (inductor-capacitor circuit) to tune frequencies. The resonant frequency of an LC circuit is given by the formula: f = rac{1}{2 extpi extsqrt{LC}}, where f is the frequency, L is the inductance, and C is the capacitance.
Step 2: Rearrange the formula to solve for inductance L. Using the minimum capacitance C = 95 \(\text{ pF}\) and the maximum frequency f = 15.0 \(\text{ MHz}\), substitute these values into the formula: L = rac{1}{(2 extpi extsqrt{f^2C})}. Ensure units are consistent (convert MHz to Hz and pF to F).
Step 3: To find the maximum capacitance, use the same formula for resonant frequency but substitute the minimum frequency f = 14.0 \(\text{ MHz}\) and the calculated inductance L. Rearrange the formula to solve for C: C = rac{1}{(4 extpi^2f^2L)}. Substitute the values and ensure unit consistency.
Step 4: Perform dimensional analysis to ensure all units are consistent. Convert MHz to Hz (1 \(\text{ MHz}\) = 10^6 \(\text{ Hz}\)) and pF to F (1 \(\text{ pF}\) = 10^{-12} \(\text{ F}\)). This ensures the calculations are accurate.
Step 5: After substituting the values and performing the calculations, you will have the required inductance for part (a) and the maximum capacitance for part (b). These values will allow the LC circuit to tune the desired frequency range.

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주요 개념

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

Resonant Frequency

The resonant frequency is the frequency at which a circuit naturally oscillates when not subjected to a continuous external force. In an LC circuit, this frequency is determined by the inductance (L) and capacitance (C) values, following the formula f = 1/(2π√(LC)). Understanding this concept is crucial for determining the appropriate inductance needed to tune the radio receiver to the desired frequency range.
추천 영상:
가이드 코스
05:23
Resonance in Series LRC Circuits

Inductance

Inductance is a property of an electrical component, typically a coil or inductor, that quantifies its ability to store energy in a magnetic field when an electric current flows through it. The value of inductance is essential for tuning circuits, as it directly affects the resonant frequency. In this context, calculating the required inductance allows the radio operator to achieve the desired frequency range for the receiver.
추천 영상:
가이드 코스
12:59
Mutual Induction

Capacitance

Capacitance is the ability of a component, usually a capacitor, to store electrical energy in an electric field. It is measured in farads and is critical in tuning circuits, as varying the capacitance alters the resonant frequency. In this scenario, knowing the minimum capacitance and calculating the maximum capacitance helps the operator design a variable capacitor that can effectively tune the receiver across the specified frequency range.
추천 영상:
가이드 코스
08:02
Capacitors & Capacitance (Intro)
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