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Chapter 33: Problem 31

Oscillating \(L \boldsymbol{C}\) Circuit In an oscillating \(L C\) circuit, \(L=3.00 \mathrm{mH}\) and \(C=2.70 \mu \mathrm{F}\). At \(t=0 \mathrm{~s}\) the charge on the capacitor is zero and the current is \(2.00\) A. (a) What is the maximum charge that will appear on the capacitor? (b) In terms of the period \(T\) of oscillation, how much time will elapse after \(t=0\) until the energy stored in the capacitor will be increasing at its greatest rate? (c) What is this greatest rate at which energy is transferred to the capacitor?

Short Answer

Expert verified
Maximum charge: 1.80 脳 10鈦烩伒 C. Time: 1.41 脳 10鈦烩伒 s. Maximum power: 1.33 脳 10鲁 W.

Step by step solution

01

Identify given values

Given in the problem are: Inductance, \[L = 3.00 \text{ mH} = 3.00 \times 10^{-3} \text{ H},\] Capacitance, \[C = 2.70 \text{ 渭F} = 2.70 \times 10^{-6} \text{ F},\] and Initial current, \[I_0 = 2.00 \text{ A}.\]Also, the initial charge on the capacitor is zero, \[Q_0 = 0 \text{ C}.\]
02

Calculate the angular frequency

The angular frequency of an LC circuit is given by: \[\omega = \frac{1}{\sqrt{LC}}.\]Substituting the given values: \[\omega = \frac{1}{\sqrt{(3.00 \times 10^{-3})(2.70 \times 10^{-6})}} \approx 1.11 \times 10^{5} \text{ rad/s}.\]
03

Determine the maximum charge on the capacitor

The maximum charge on the capacitor is given by the relationship between current and angular frequency, \[I_0 = \omega Q_\text{max}.\]Rearranging to solve for \(Q_\text{max}\): \[Q_\text{max} = \frac{I_0}{\omega}.\]Substituting the given values: \[Q_\text{max} = \frac{2.00}{1.11 \times 10^{5}} \approx 1.80 \times 10^{-5} \text{ C}.\]
04

Calculate the period of oscillation

The period of oscillation, T, is given by: \[T = \frac{2\pi}{\omega}.\]Substituting the value of \(\omega\): \[T = \frac{2\pi}{1.11 \times 10^{5}} \approx 5.65 \times 10^{-5} \text{ s}.\]
05

Find the time when energy in the capacitor is increasing at the greatest rate

The energy stored in the capacitor is increasing at the maximum rate when the current is at its maximum. The current is a sine function of time, and is at its maximum at time: \[t = \frac{T}{4}.\]So, substituting the period, \[t = \frac{5.65 \times 10^{-5}}{4} \approx 1.41 \times 10^{-5} \text{ s}.\]
06

Calculate the greatest rate at which energy is transferred to the capacitor

The power transferred to the capacitor is \[P = IV.\]At time \[t = \frac{T}{4},\]the current is at its maximum, and the voltage is: \[V = L \frac{dI}{dt} = L \omega I_0.\]So, \[P_\text{max} = I_0 \cdot L \omega I_0 = L \omega I_0^2.\]Substituting the values, \[P_\text{max} = (3.00 \times 10^{-3}) (1.11 \times 10^{5}) (2.00)^2 \approx 1.33 \times 10^{3} \text{ W}.\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Oscillating Circuit
An oscillating LC circuit consists of an inductor (L) and a capacitor (C) connected together. Such circuits can store and transfer energy back and forth between the capacitor and the inductor, leading to oscillations. When a capacitor is charged and then connected to an inductor, the electric energy stored in the capacitor gets converted to magnetic energy in the inductor. This process repeats, causing the circuit to oscillate.
Capacitance and Inductance
Capacitance and inductance are two fundamental properties in an LC circuit. Capacitance (C) is the ability of a capacitor to store an electric charge. It is measured in farads (F). In this problem, the capacitance is given as 2.70 microfarads (碌F). Inductance (L) is the property of an inductor to store energy in a magnetic field when electric current flows through it. It is measured in henrys (H), and here it's given as 3.00 millihenrys (mH). The relationship between these two quantities determines the oscillatory behavior of the circuit.
Energy Transfer in Capacitors
In an LC circuit, energy oscillates between the capacitor and inductor. When the capacitor discharges, its electric energy is converted to magnetic energy in the inductor. When the capacitor is fully discharged, the energy is entirely in the inductor's magnetic field. As the inductor's magnetic field collapses, it recharges the capacitor with reversed polarity, but no net energy is lost during ideal oscillation. The maximum energy transfer rate to the capacitor happens when the current is highest, and this is analyzed mathematically by the relationships given for power and energy in the solution.

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Most popular questions from this chapter

Air Conditioner An air conditioner connected to a \(120 \mathrm{~V} \mathrm{rms}\) ac line is equivalent to a \(12.0 \Omega\) resistance and a \(1.30 \Omega\) inductive reactance in series. (a) Calculate the impedance of the air conditioner. (b) Find the average rate at which energy is supplied to the appliance.

Maximum Positive Charge The frequency of oscillation of a certain \(L C\) circuit is \(200 \mathrm{kHz}\). At time \(t=0 \mathrm{~s}\), plate \(A\) of the capacitor has maximum positive charge. At what times \(t>0 \mathrm{~s}\) will (a) plate \(A\) again have maximum positive charge, \((\mathrm{b})\) the other plate of the capacitor have maximum positive charge, and (c) the inductor have maximum magnetic field?

Toroidal Inductor A toroidal inductor with an inductance of \(90.0 \mathrm{mH}\) encloses a volume of \(0.0200 \mathrm{~m}^{3}\). If the average energy density in the toroid is \(70.0 \mathrm{~J} / \mathrm{m}^{3}\), what is the current through the inductor?

Initially a Maximum In an oscillating \(L C\) circuit with \(L=50 \mathrm{mH}\) and \(C=4.0 \mu \mathrm{F}\), the current is initially a maximum. How long will it take before the capacitor is fully charged for the first time?

Capacitive Reactance An ac generator with \(\mathscr{E}^{\max }=220 \mathrm{~V}\) and operating at \(400 \mathrm{~Hz}\) causes oscillations in a series \(R L C\) circuit having \(R=220 \Omega, L=150 \mathrm{mH}\), and \(C=24.0 \mu \mathrm{F}\). Find (a) the capacitive reactance \(X_{C},(\mathrm{~b})\) the impedance \(Z\), and \((\mathrm{c})\) the current amplitude \(I .\) A second capacitor of the same capacitance is then connected in series with the other components. Determine whether the values of (d) \(X_{C}\), (e) \(Z\), and (f) \(I\) increase, decrease, or remain the same.
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