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Chapter 30: Problem 28

A 20.0 -\muF capacitor is charged by a \(150.0-\mathrm{V}\) power supply, then disconnected from the power and connected in series with a \(0.280-\mathrm{mH}\) inductor. Calculate: (a) the oscillation frequency of the circuit; (b) the energy stored in the capacitor at time \(t=0 \mathrm{ms}\) (the moment of connection with the inductor); (c) the energy stored in the inductor at \(t=1.30 \mathrm{ms}\) .

Short Answer

Expert verified
(a) 4240 Hz (b) 0.225 J (c) 0.222 J

Step by step solution

01

Calculate Oscillation Frequency

The oscillation frequency of an LC circuit is given by the formula:\[f = \frac{1}{2\pi\sqrt{LC}}\]Where:- \(L\) is the inductance (0.280 mH = 0.280 \times 10^{-3}\ H)- \(C\) is the capacitance (20.0 \mu\text{F} = 20.0 \times 10^{-6}\ F)Substitute the values into the equation:\[f = \frac{1}{2\pi\sqrt{0.280 \times 10^{-3}\ \times 20.0 \times 10^{-6}}} \approx 4240\ \text{Hz}\]
02

Calculate the Initial Energy in the Capacitor

The energy stored in a capacitor is given by:\[E = \frac{1}{2}CV^2\]Where:- \(C\) is the capacitance (20.0 \mu\text{F} = 20.0 \times 10^{-6}\ F)- \(V\) is the voltage (150.0 V)Substitute the values:\[E = \frac{1}{2} \times 20.0 \times 10^{-6} \times (150.0)^2 = 0.225\ \text{J}\]
03

Calculate the Energy in the Inductor at 1.30ms

In a LC circuit, the total energy oscillates between the capacitor and inductor. At time \(t=1.30\ \text{ms}\), we use the equation:\[E_L = E_{total} \sin^2(\omega t)\]where:- \(E_{total}\) is the total energy (from Step 2, 0.225 J)- \(\omega = 2\pi f\) is the angular frequency, \(\omega = 2\pi \times 4240\ \approx 26636.2\ \text{rad/s}\)Substitute to find \(t\):- \(t = 1.30\ \text{ms} = 1.30 \times 10^{-3}\ \text{s}\)Substitute into the equation:\[E_L = 0.225 \times \sin^2(26636.2 \times 1.30 \times 10^{-3})\]After calculating the sine value and squaring it, we find:\[E_L \approx 0.225 \times 0.987 = 0.222\ \text{J}\]

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Key Concepts

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

Oscillation Frequency
When we study an LC circuit, a key feature is its oscillation frequency. This frequency is vital because it tells us how quickly energy oscillates between the capacitor and the inductor. Think of it like a seesaw, with energy going back and forth. The formula to find the oscillation frequency \( f \) is:
  • \( f = \frac{1}{2\pi\sqrt{LC}} \)
Here:
  • \( L \) represents inductance, measured in henrys (H),
  • \( C \) is capacitance, measured in farads (F).
For our exercise, the inductor has 0.280 mH and the capacitor 20.0 \( \mu \)F. To calculate, simply substitute these values into the formula:
  • \( L = 0.280 \times 10^{-3} \) H
  • \( C = 20.0 \times 10^{-6} \) F
Putting these into the frequency formula gives approximately 4240 Hz. Such calculations help us understand how fast the oscillations occur, setting the tempo for energy exchange in the circuit.
Capacitor Energy Calculation
In an LC circuit, the initial energy is primarily stored in the capacitor when it's fully charged. This energy depends on the voltage across the capacitor and its capacitance.To find out how much energy is stored, we use:
  • \( E = \frac{1}{2} CV^2 \)
Here:
  • \( C \) is the capacitance (20.0 \( \mu \)F in this case),
  • \( V \) is the voltage across the capacitor (150.0 V).
Substituting our values provides:
  • \( E = \frac{1}{2} \times 20.0 \times 10^{-6} \times (150.0)^2 \).
Calculating this gives us 0.225 Joules initially stored in the capacitor. This stored energy will begin the oscillation, transitioning between the capacitor and inductor.
Inductor Energy Calculation
When energy oscillates in the LC circuit, it alternates between the capacitor and the inductor. At certain times, the inductor will hold a portion of the circuit's total energy.To determine the energy in the inductor, particularly at \( t = 1.30 \) ms in this exercise, we use:
  • \( E_L = E_{total} \sin^2(\omega t) \)
Where:
  • \( E_{total} \) is the total initial energy calculated as 0.225 J,
  • \( \omega = 2\pi f \) is the angular frequency, previously found as approximately 26636.2 rad/s,
  • \( t \) is the time in seconds (1.30 ms = 1.30 \times 10^{-3} s).
Substituting values into the equation involves:
  • Calculating \( \sin^2(26636.2 \times 1.30 \times 10^{-3}) \) results in a value close to 0.987.
  • Multiplying this by the total energy \( 0.225 J \).
This computation results in an inductor energy of approximately 0.222 J at 1.30 ms, illustrating the energy oscillation at that instant. This explains how the energy dynamically shifts within the circuit over time.

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Most popular questions from this chapter

A \(7.50-\mathrm{nF}\) capacitor is charged up to \(12.0 \mathrm{V},\) then disconnected from the power supply and connected in series through a coil. The period of oscillation of the circuit is then measured to be \(8.60 \times 10^{-5} \mathrm{s}\) . Calculate: (a) the inductance of the coil; \((\mathrm{b})\) the maximum charge on the capacitor; (c) the total energy of the circuit; (d) the maximum current in the circuit.

Show that \(\sqrt{L C}\) has units of time.

A toroidal solenoid has 500 turns, cross-sectional area \(6.25 \mathrm{cm}^{2},\) and mean radius \(4.00 \mathrm{cm} .\) (a) Calcualte the coil's self-inductance. (b) If the current decreases uniformly from 5.00 \(\mathrm{A}\) to 2.00 \(\mathrm{A}\) in 3.00 \(\mathrm{ms}\) , calculate the self- induced emf in the coil. (c) The current is directed from terminal \(a\) of the coil to terminal \(b\) . Is the direction of the induced emf from \(a\) to \(b\) or from \(b\) to \(a ?\)

In an \(L-\) circuit, \(L=85.0 \mathrm{mH}\) and \(C=3.20 \mu \mathrm{F}\) . During the oseillations the maximum current in the inductor is 0.850 \(\mathrm{mA}\) . (a) What is the maximum charge on the capacitor? (b) What is the magnitude of the charge on the capacitor at an instant when the current in the inductor has magnitude 0.500 \(\mathrm{mA} ?\)

A \(5.00-\mu F\) capacitor is initially charged to a potential of 16.0 \(\mathrm{V}\) . It is then connected in series with a \(3.75-\mathrm{mH}\) inductor. (a) What is the total energy stored in this circuit? (b) What is the maximum current in the inductor? What is the charge on the capacitor plates at the instant the current in the inductor is maximal?
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