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Chapter 30: Problem 25

An LC circuit consists of a 1.00 -mH inductor and a fully charged capacitor. After \(2.10 \mathrm{~ms}\), the energy stored in the capacitor is half of its original value. What is the capacitance?

Short Answer

Expert verified
Answer: The capacitance of the capacitor is approximately \(1.59 \mu F\).

Step by step solution

01

Energy stored in a charged capacitor

The energy stored in a charged capacitor is given by the formula: \(E(t) = \frac{1}{2}CV^2(t)\) Where \(E(t)\) is the energy at time \(t\), \(C\) is the capacitance, and \(V(t)\) is the voltage across the capacitor at time \(t\).
02

Energy as half of original value

Given that after 2.10 ms, the energy stored in the capacitor is half of its original value, we can write the equation: \(\frac{1}{2}CV^2(2.10\mathrm{ms}) = \frac{1}{2} \cdot \frac{1}{2}CV^2(0)\) We can simplify this equation as: \(V^2(2.10\mathrm{ms}) = \frac{1}{2}V^2(0)\)
03

Period of oscillation in an LC circuit

In an LC circuit, the period of oscillation \(T\) is given by: \(T = 2\pi\sqrt{LC}\) Since we are given the time 2.10 ms as half of the oscillation, it means that \(T = 4.20\mathrm{ms}\). We can substitute and solve for \(C\): \(4.20\mathrm{ms} = 2\pi\sqrt{1.00\mathrm{mH} \cdot C}\)
04

Solve for capacitance

Now, we can solve for the capacitance \(C\): \(C = \frac{T^2}{4\pi^2 L} = \frac{(4.20\mathrm{ms})^2}{4\pi^2 \cdot 1.00\mathrm{mH}}\) Converting values into SI units: \(C = \frac{(4.20 \times 10^{-3}\mathrm{s})^2}{4\pi^2 \cdot 1.00 \times 10^{-3}\mathrm{H}}\) Calculating the value gives: \(C \approx 1.59 \times 10^{-6}\mathrm{F}\) So, the capacitance of this capacitor is approximately \(1.59\mu F\).

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Most popular questions from this chapter

Why can't a transformer be used to step up or step down the voltage in a DC circuit?

a) A loop of wire \(5.00 \mathrm{~cm}\) in diameter is carrying a current of \(2.00 \mathrm{~A}\). What is the energy density of the magnetic field at its center? b) What current has to flow in a straight wire to produce the same energy density at a point \(4.00 \mathrm{~cm}\) from the wire?

A vacuum cleaner motor can be viewed as an inductor with an inductance of \(100 . \mathrm{mH} .\) For a \(60.0-\mathrm{Hz} \mathrm{AC}\) voltage, \(V_{\mathrm{rms}}=115 \mathrm{~V}\), what capacitance must be in series with the motor to maximize the power output of the vacuum cleaner?

When you turn the dial on a radio to tune it, you are adjusting a variable capacitor in an LC circuit. Suppose you tune to an AM station broadcasting at a frequency of \(1000 . \mathrm{kHz},\) and there is a \(10.0-\mathrm{mH}\) inductor in the tuning circuit. When you have tuned in the station, what is the capacitance of the capacitor?

A 2.00 - \(\mu\) F capacitor was fully charged by being connected to a 12.0 - \(V\) battery. The fully charged capacitor is then connected in series with a resistor and an inductor: \(R=50.0 \Omega\) and \(L=0.200 \mathrm{H}\). Calculate the damped frequency of the resulting circuit.
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