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Chapter 31: Problem 2

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

Short Answer

Expert verified
(a) 5 microseconds, (b) 2.5 microseconds, (c) 1.25 microseconds.

Step by step solution

01

Understanding frequency and period of oscillation

The frequency of an oscillating circuit refers to how many complete cycles occur in one second, typically measured in Hertz (Hz). The given frequency is 200 kHz, which is 200,000 Hz. The period of oscillation is the reciprocal of frequency, given by \( T = \frac{1}{f} \).
02

Calculating period of oscillation

Using the formula for the period, we find \( T = \frac{1}{200,000} \) seconds. Converting from seconds to microseconds (1 second = 1,000,000 microseconds), we have \( T = 5 \) microseconds. This is the time for one complete cycle.
03

Finding earliest time for maximum positive charge on same plate (a)

Since one full cycle returns the charge to its original state, the earliest time for plate A to again have maximum positive charge is after one period, i.e., \( t = 5 \) microseconds.
04

Determining earliest time for maximum positive charge on opposite plate (b)

The other plate will have maximum positive charge halfway through the cycle, which is at \( \frac{T}{2} \). Since \( T = 5 \) microseconds, this time is \( \frac{5}{2} = 2.5 \) microseconds.
05

Identifying earliest time for maximum magnetic field in the inductor (c)

At maximum magnetic field, the current in the circuit is at its peak, which occurs at a quarter of a cycle (or a quarter period) after the initial condition. Thus, the earliest time is \( \frac{T}{4} \). So, \( \frac{T}{4} = \frac{5}{4} = 1.25 \) microseconds.

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Key Concepts

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

frequency of oscillation
In an LC circuit, the frequency of oscillation describes how often the circuit completes a full cycle of energy transfer between the capacitor and the inductor per second. The unit of frequency is Hertz (Hz), which signifies one cycle per second. For the problem at hand, the oscillator is set at a frequency of 200 kHz. This means that the circuit goes through 200,000 complete cycles every second. To put it simply, the frequency tells us just how quickly the circuit is oscillating. This high frequency is common in electronic communications where rapid oscillations are required.
capacitor charge
The capacitor in an LC circuit alternates between storing electrical charge and being completely discharged. At time zero, the capacitor's plate A is maximally charged positively, meaning it has accumulated the most charge possible. As the circuit oscillates, this charge transfers back and forth. The maximum positive charge will appear on the other plate only halfway through a cycle, which means that every half-period or 2.5 microseconds for this circuit, the plate will achieve this condition. Understanding how the charge flows in the capacitor is essential for comprehending energy storage in circuits.
magnetic field in inductor
The inductor in an LC circuit temporarily stores energy in the form of a magnetic field. This occurs when the current flowing through the inductor is at its maximum. In this context, the magnetic field reaches its peak a quarter-way through the oscillating cycle, which is 1.25 microseconds in our specific problem. At this point, all energy initially stored in the capacitor has transferred to the inductor. This oscillation between electric charge in the capacitor and magnetic energy in the inductor is what keeps the LC circuit oscillating over time.
period of oscillation
The period of oscillation is the amount of time it takes for an LC circuit to complete one full cycle of current and charge transfer. It's the inverse of the frequency, meaning it shows how long each oscillation lasts. Given the frequency of 200 kHz, the period here is 5 microseconds. This time is crucial because it dictates the timing and intervals at which the circuit repeats specific states such as maximum charge on a capacitor plate or maximum magnetic field in the inductor. Understanding the period helps predict the circuit's behavior over time, ensuring components are timed correctly.

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Most popular questions from this chapter

A series \(R L C\) circuit is driven by a generator at a frequency of \(2000 \mathrm{~Hz}\) and an emf amplitude of \(170 \mathrm{~V}\). The inductance is \(60.0 \mathrm{mH},\) the capacitance is \(0.400 \mu \mathrm{F},\) and the resistance is \(200 \Omega .\) (a) What is the phase constant in radians? (b) What is the current amplitude?

An ac voltmeter with large impedance is connected in turn across the inductor, the capacitor, and the resistor in a series circuit having an alternating emf of \(100 \mathrm{~V}\) (rms); the meter gives the same reading in volts in each case. What is this reading?

A series \(R L C\) circuit is driven in such a way that the maximum voltage across the inductor is 1.50 times the maximum voltage across the capacitor and 2.00 times the maximum voltage across the resistor. (a) What is \(\phi\) for the circuit? (b) Is the circuit inductive, capacitive, or in resonance? The resistance is \(49.9 \Omega\), and the current amplitude is \(200 \mathrm{~mA}\). (c) What is the amplitude of the driving emf?

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?

An alternating emf source with a variable frequency \(f_{d}\) is connected in series with an \(80.0 \Omega\) resistor and a \(40.0 \mathrm{mH}\) inductor. The emf amplitude is \(6.00 \mathrm{~V}\). (a) Draw a phasor diagram for phasor \(V_{R}\) (the potential across the resistor) and phasor \(V_{L}\) (the potential across the inductor). (b) At what driving frequency \(f_{d}\) do the two phasors have the same length? At that driving frequency, what are (c) the phase angle in degrees, (d) the angular speed at which the phasors rotate, and (e) the current amplitude?
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