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30. Induction and Inductance

LC Circuits

Physics

30. Induction and Inductance

LC Circuits

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5:13 minutes

Problem 46

Textbook Question

A certain FM radio tuning circuit has a fixed capacitor C = 680 pF. Tuning is done by a variable inductance. What range of values must the inductance have to tune stations from 88 MHz to 108 MHz?

Verified step by step guidance

The problem involves a resonant LC circuit, where the resonant frequency \( f \) is determined by the formula \( f = \frac{1}{2 \pi \sqrt{L C}} \). Here, \( L \) is the inductance, \( C \) is the capacitance, and \( f \) is the frequency.

Rearrange the formula to solve for \( L \): \( L = \frac{1}{(2 \pi f)^2 C} \). This equation will allow us to calculate the inductance \( L \) for a given frequency \( f \).

Substitute the given capacitance \( C = 680 \; \text{pF} = 680 \times 10^{-12} \; \text{F} \) into the formula. The inductance \( L \) will now depend on the frequency \( f \).

Calculate the inductance for the lower frequency \( f = 88 \; \text{MHz} = 88 \times 10^6 \; \text{Hz} \) using the formula \( L = \frac{1}{(2 \pi f)^2 C} \). This gives the maximum inductance required.

Similarly, calculate the inductance for the higher frequency \( f = 108 \; \text{MHz} = 108 \times 10^6 \; \text{Hz} \) using the same formula. This gives the minimum inductance required. The range of inductance values is then from the minimum to the maximum calculated values.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Resonant Frequency

The resonant frequency of an LC circuit is the frequency at which the circuit naturally oscillates. It is determined by the inductance (L) and capacitance (C) of the circuit, following the formula f = 1/(2π√(LC)). This concept is crucial for understanding how varying the inductance can tune the circuit to different radio frequencies.

Inductance

Inductance is a property of an electrical component, typically a coil, that quantifies its ability to store energy in a magnetic field when an electric current flows through it. In the context of radio tuning, adjusting the inductance allows the circuit to resonate at different frequencies, enabling the selection of various radio stations.

Capacitance

Capacitance is the ability of a component, usually a capacitor, to store electrical energy in an electric field. In this question, the fixed capacitor value of 680 pF plays a critical role in determining the resonant frequency of the circuit when combined with the variable inductance. Understanding capacitance is essential for calculating the necessary inductance values to tune the desired frequency range.

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Textbook Question

In recent years it has been possible to buy a 1.0 F capacitor. This is an enormously large amount of capacitance. Suppose you want to build a 1.0 Hz oscillator with a 1.0 F capacitor. You have a spool of 0.25-mm-diameter wire and a 4.0-cm-diameter plastic cylinder. How long must your inductor be if you wrap it with 2 layers of closely spaced turns?

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Textbook Question

BIO An MRI machine needs to detect signals that oscillate at very high frequencies. It does so with an LC circuit containing a 15 mH coil. To what value should the capacitance be set to detect a 450 MHz signal?

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Textbook Question

An electric oscillator is made with a 0.10 μF capacitor and a 1.0 mH inductor. The capacitor is initially charged to 5.0 V. What is the maximum current through the inductor as the circuit oscillates?

655

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Textbook Question

How much resistance must be added to a pure LC circuit (L = 350 mH, C = 1600 pF) to change the oscillator’s frequency by 0.25%? Will it be increased or decreased?

436

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Textbook Question

(II) A 475-pF capacitor is charged to 135 V and then quickly connected to a 175-mH inductor. Determine (a) the frequency of oscillation, (b) the peak value of the current, and (c) the maximum energy stored in the magnetic field of the inductor.

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Multiple Choice

An LC circuit with an inductor of 0.05 H and a capacitor of 35 μF begins with the current of -1A. The capacitor plates have a maximum charge of 2.65mC at any time during the oscillation. What is the phase angle of this oscillation?

In an oscillating LC circuit in which the capacitance C = 4μF and the maximum voltage across the capacitor V = 1.50V, the maximum current measured across the inductor is 50mA. What is the angular frequency of this LC circuit?

What is the resonant frequency of an LC circuit consisting of a