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30. Induction and Inductance

LC Circuits

Physics

30. Induction and Inductance

LC Circuits

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6:14 minutes

Problem 29

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.

Verified step by step guidance

Step 1: Recognize that the system described is an LC circuit, which undergoes oscillations. The frequency of oscillation is determined by the formula:

f = \frac{1}{\sqrt}

, where

L

is the inductance and

C

is the capacitance. Convert the given values of capacitance (475 pF = 475 × 10⁻¹² F) and inductance (175 mH = 175 × 10⁻³ H) into SI units before substituting into the formula.

Step 2: To find the peak current, use the relationship between the maximum charge on the capacitor and the peak current in the inductor. The maximum charge is given by

Q = C V

, where

V

is the voltage across the capacitor. The peak current is then calculated using

I = \frac{Q}{L}

. Substitute the values of

C

V

, and

L

into the equations.

Step 3: To determine the maximum energy stored in the magnetic field of the inductor, use the formula for energy in an inductor:

U = \frac{1}{2} L I^{2}

. Here,

I

is the peak current calculated in Step 2. Substitute the values of

L

and

I

into the formula.

Step 4: Verify the energy conservation in the LC circuit. The maximum energy stored in the capacitor (electrical energy) is given by

U = \frac{1}{2} C V^{2}

. This energy is transferred back and forth between the capacitor and the inductor. Confirm that the maximum energy in the inductor matches the maximum energy in the capacitor.

Step 5: Summarize the results by substituting the numerical values into the derived formulas for frequency, peak current, and maximum energy. Ensure all units are consistent and verify the calculations for accuracy.

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Key Concepts

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

Capacitance and Energy Storage

Capacitance is the ability of a capacitor to store electrical energy in an electric field, measured in farads (F). The energy (U) stored in a capacitor can be calculated using the formula U = 1/2 C V^2, where C is the capacitance and V is the voltage. In this scenario, the capacitor's capacitance of 475 pF and voltage of 135 V are crucial for determining the energy stored before connecting to the inductor.

Inductance and Magnetic Energy

Inductance is the property of an inductor that quantifies its ability to store energy in a magnetic field when an electric current flows through it. The energy (U) stored in an inductor is given by the formula U = 1/2 L I^2, where L is the inductance and I is the current. Understanding this concept is essential for calculating the maximum energy stored in the magnetic field of the inductor after the capacitor is connected.

Oscillation Frequency in LC Circuits

In an LC circuit, which consists of an inductor (L) and a capacitor (C), the frequency of oscillation (f) is determined by the formula f = 1/(2π√(LC)). This frequency represents how quickly the energy oscillates between the electric field of the capacitor and the magnetic field of the inductor. For this problem, calculating the frequency of oscillation is key to understanding the dynamics of the circuit after the capacitor is connected.

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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?

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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?

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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?

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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

2.0 mH

inductor paired with a

300 nF

A 2.0 mH inductor is connected in parallel with a variable capacitor. The capacitor can be varied from 100 pF to 200 pF. What is the range of oscillation frequencies for this circuit?

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