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Chapter 31: Problem 32

In an oscillating \(L C\) circuit in which \(C=6.00 \mu \mathrm{F}\), the maximum potential difference across the capacitor during the oscillations is \(1.50 \mathrm{~V}\) and the maximum current through the inductor is \(50.0 \mathrm{~mA}\). What are (a) the inductance \(L\) and (b) the frequency of the oscillations? (c) How much time is required for the charge on the capacitor to rise from zero to its maximum value?

Short Answer

Expert verified
(a) \(L = 5.4 \, mH\); (b) \(f \approx 280 \, Hz\); (c) \(0.89 \, ms\).

Step by step solution

01

Given quantities

First, identify the given quantities in the problem: - Capacitance: \(C = 6.00 \, \mu F = 6.00 \times 10^{-6} \, F\) - Maximum potential difference across the capacitor: \(V_{max} = 1.50 \, V\) - Maximum current through the inductor: \(I_{max} = 50.0 \, mA = 50.0 \times 10^{-3} \, A\)
02

Use the energy conservation equation for LC circuits

In the LC circuit, the total energy is conserved and can be given by:\[ \frac{1}{2} C V^2_{max} = \frac{1}{2} L I^2_{max} \] We will use this equation to calculate the inductance \(L\).
03

Solve for Inductance L

Rearrange the energy conservation equation to solve for \(L\):\[ L = \frac{C V^2_{max}}{I^2_{max}} \]Substitute the given values:\[ L = \frac{(6.00 \times 10^{-6} \, F)(1.50 \, V)^2}{(50.0 \times 10^{-3} \, A)^2} \]Calculate:\[ L = \frac{(6.00 \times 10^{-6})(2.25)}{(0.05)^2} = \frac{13.5 \times 10^{-6}}{2.5 \times 10^{-3}} = 5.4 \times 10^{-3} \, H \]So, \(L = 5.4 \, mH\).
04

Determine the angular frequency \(\omega\)

The angular frequency \(\omega\) is related to both capacitance \(C\) and inductance \(L\) by the formula:\[ \omega = \frac{1}{\sqrt{LC}} \]
05

Calculate frequency of oscillations

First, calculate \(\omega\) using Step 4:\[ \omega = \frac{1}{\sqrt{(5.4 \times 10^{-3} \, H)(6.00 \times 10^{-6} \, F)}} \]Calculate:\[ \omega = \frac{1}{\sqrt{3.24 \times 10^{-8}}} = \frac{1}{5.69 \times 10^{-4}} \approx 1.76 \times 10^{3} \, rad/s \]The frequency \(f\) is given by \(f = \frac{\omega}{2\pi}\):\[ f = \frac{1.76 \times 10^{3}}{2 \pi} \approx 280 \, Hz \]
06

Find the period T

The period \(T\) is the reciprocal of the frequency:\[ T = \frac{1}{f} = \frac{1}{280} \approx 3.57 \times 10^{-3} \, seconds \]
07

Compute time for charge to reach maximum from zero

To find how much time it takes for the charge to go from zero to the maximum value, note that it takes a quarter of the period:\[ t = \frac{T}{4} = \frac{3.57 \times 10^{-3}}{4} \approx 0.89 \times 10^{-3} \, seconds \]
08

Conclusion

Inductance \(L = 5.4 \, mH\), frequency of oscillations \(f \approx 280 \, Hz\), and time to reach maximum charge \( \approx 0.89 \, ms \).

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Key Concepts

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

Capacitance
Capacitance is a measure of a capacitor's ability to store charge in an electric field. In an LC circuit, the capacitor stores electrical energy when charged. A higher capacitance means more charge and energy stored for the same voltage.
A capacitor in an LC circuit will release its stored energy into the inductor, causing oscillations. It is crucial for regulating the voltage and maintaining energy flow. Here, the given capacitance is 6.00 \(\mu F\), which is equal to 6.00 x 10^{-6} F.
The key factors influencing capacitance are:
  • Plate area: Larger plates can hold more charge.
  • Distance between plates: Closer plates increase capacitance.
  • Material between plates: A dielectric material can increase capacitance.
Inductance
Inductance is a property of an inductor that opposes changes in current. It plays a vital role in LC circuits by storing energy in a magnetic field. The energy moves between magnetic fields and electric fields, creating oscillations.
The inductance value is crucial, as it determines the circuit's frequency of oscillation and affects the overall dynamics of the circuit. In the provided exercise, the inductance is calculated using the energy conservation equation.The relationship between inductance, capacitance, and energy is expressed as:
  • The energy stored in a capacitor: \(\frac{1}{2} CV^2_{max}\)
  • The energy stored in an inductor: \(\frac{1}{2} LI^2_{max}\)
This property is measured in henrys (H), and for this problem, the inductance is found to be 5.4 mH.
Oscillation Frequency
Oscillation frequency is a measure of how many cycles of oscillation happen in one second in the LC circuit. This frequency depends on both the inductance and capacitance. A key point to remember is:
  • Higher capacitance or inductance results in lower frequency.
  • Lower capacitance or inductance leads to higher frequency.
The frequency is essential to determine the behavior of the circuit and how it responds to changes. It can be calculated from the angular frequency using:\[ f = \frac{\omega}{2\pi} \]For the given LC circuit, the frequency is approximately 280 Hz, indicating that the circuit completes 280 oscillations per second.
Energy Conservation
In an LC circuit, energy conservation is a core principle. It refers to how energy transitions between being stored in an electric field (capacitor) and a magnetic field (inductor), without loss.The energy conservation equation in an LC circuit is:\[ \frac{1}{2} CV^2_{max} = \frac{1}{2} LI^2_{max} \]This equation ensures that all energy is conserved between the capacitor and inductor, allowing the system to continuously oscillate.Key aspects of energy conservation in an LC circuit include:
  • No energy loss: In ideal conditions, the system does not lose energy to its surroundings.
  • High efficiency: Oscillations ideally continue indefinitely without energy loss.
  • Energy storing components: Capacitors and inductors cycle energy back and forth.
Angular Frequency
Angular frequency \(\omega\) defines the rate of change of phase of a sinusoidal waveform, like the oscillations in an LC circuit. It provides insight into how "fast" the oscillations are.The equation for angular frequency in an LC circuit is:\[ \omega = \frac{1}{\sqrt{LC}} \]From this, we see the dependency of angular frequency on both the capacitance and inductance, and how a change in either affects the oscillation speed.For the LC circuit in the exercise, the angular frequency was calculated to be about 1760 rad/s, indicating rapid oscillations. This helps to determine the precise behavior and timing of the oscillations in the circuit.

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Most popular questions from this chapter

In a certain oscillating \(L C\) circuit, the total energy is converted from clectrical cnergy in the capacitor to magnetie cnergy in the inductor in \(2.50 \mu \mathrm{s}\). What are (a) the period of oscillation and (b) the frequency of oscillation? (c) How long after the magnetic energy is a maximum will it be a maximum again? (d) In one full cycle, how many times will the electrical energy be maximum?

Figure 31-23 shows an ac generator connected to a "black box" through a pair of terminals. The box contains an \(R L C\) circuit, possibly even a multiloop circuit, whose ele- ments and connections we do not know. Measurements outside the box reveal that $$ \mathscr{E}(t)=(61.4 \mathrm{~V}) \sin \omega_{a} t $$ and \(i(t)=(0.930 \mathrm{~A}) \sin \left(\omega_{u} t+42.0^{\circ}\right) .\) (a) What is the power factor? (b) Does the current lead or lag the emf? (c) Is the circuit in the box largely inductive or largely capacitive? (d) Is the circuit in the box in resonance? (e) Must there be a capacitor in the box? (f) An inductor? (g) A resistor? (h) At what average rate is energy delivered to the box by the generator? (i) Why don't you need to know \(\omega_{d}\) to answer all these questions?

A transformer has 400 primary turns and 10 secondary turns. (a) If \(V_{p}\) is \(120 \mathrm{~V}\) (rms), what is \(V_{s}\) with an open circuit? If the secondary now has a resistive load of \(27 \Omega\), what is the current in the (b) primary and (c) secondary?

(a) At what frequency would a \(12 \mathrm{mH}\) inductor and a \(10 \mu \mathrm{F}\) capacitor have the same reactance? (b) What would the reactance be? (c) Show that this frequency would be the natural frequency of an oscillating circuit with the same \(L\) and \(C\).

A coil of inductance \(62 \mathrm{mH}\) and unknown resistance and a \(0.94 \mu \mathrm{F}\) capacitor are connected in series with an alternating emf of frequency \(930 \mathrm{~Hz}\). If the phase constant between the applied voltage and the current is \(82^{\circ}\), what is the resistance of the coil?
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