An LC circuit consists of a capacitor, C=2.50 μF, and an inductor, L=4.00 mH . The capacitor is

An LC circuit consists of a capacitor, C=2.50 μF, and an...

An LC circuit consists of a capacitor, $C=2.50 \mu \mathrm{F},$ and an inductor, $L=4.00 \mathrm{mH} .$ The capacitor is fully charged using a battery and then connected to the inductor. An oscilloscope is used to measure the frequency of the oscillations in the circuit. Next, the circuit is opened, and a resistor, $R$, is inserted in series with the inductor and the capacitor. The capacitor is again fully charged using the same battery and then connected to the circuit. The angular frequency of the damped oscillations in the RLC circuit is found to be $20.0 \%$ less than the angular frequency of the oscillations in the LC circuit. a) Determine the resistance of the resistor. b) How long after the capacitor is reconnected in the circuit will the amplitude of the damped current through the circuit be $50.0 \%$ of the initial amplitude? c) How many complete damped oscillations will have occurred in that time?

An LC circuit consists of a capacitor, $C=2.50 \mu \mathrm{F},$ and an inductor, $L=4.00 \mathrm{mH} .$ The capacitor is fully charged using a battery and then connected to the inductor. An oscilloscope is used to measure the frequency of the oscillations in the circuit. Next, the circuit is opened, and a resistor, $R$, is inserted in series with the inductor and the capacitor. The capacitor is again fully charged using the same battery and then connected to the circuit. The angular frequency of the damped oscillations in the RLC circuit is found to be $20.0 \%$ less than the angular frequency of the oscillations in the LC circuit.
a) Determine the resistance of the resistor.
b) How long after the capacitor is reconnected in the circuit will the amplitude of the damped current through the circuit be $50.0 \%$ of the initial amplitude?
c) How many complete damped oscillations will have occurred in that time?

Key Concepts

LC Circuit

An LC circuit is an electrical circuit consisting of an inductor (L) and a capacitor (C) that can oscillate with a natural frequency determined by the energy exchange between the magnetic field in the inductor and the electric field in the capacitor. Its undamped angular frequency is given by ?? = 1/?(LC), representing the rate of oscillations when no resistive elements are present.

Damped RLC Circuit

An RLC circuit includes a resistor (R) along with an inductor (L) and a capacitor (C), leading to damped oscillations. The resistor introduces energy dissipation, causing the amplitude of the oscillations to decrease over time. This damping effect changes the angular frequency of the circuit relative to the ideal LC oscillations and introduces exponential decay in the amplitude.

Angular Frequency

Angular frequency (?) is a measure of how fast an oscillatory system moves through its cycle, expressed in radians per second. In an undamped LC circuit, the angular frequency is determined solely by the LC components. For a damped RLC circuit, the effective angular frequency (?_d) is lower than ??, and it is calculated by ?_d = ?(??² - (R/(2L))²), reflecting the effect of the damping introduced by the resistor.

Damping Coefficient

The damping coefficient (often designated as ?) quantifies the rate at which the oscillations decay due to the resistor in an RLC circuit. It is defined as ? = R/(2L) and appears in both the expression for the damped angular frequency and the exponential decay term governing the amplitude of the oscillations.

Exponential Decay

Exponential decay in an RLC circuit describes how the amplitude of the oscillations decreases over time due to resistive losses. The amplitude at any time t can be expressed as A(t) = A? exp(-? t), where A? is the initial amplitude. This concept is crucial when analyzing how quickly the oscillations diminish and determining time intervals for specific amplitude reductions.

Oscillation Period

The oscillation period in a damped RLC circuit is the duration of one complete cycle of the oscillation and is given by T = 2?/?_d. This period, modified by the presence of damping, is an important parameter when calculating the number of oscillations over a certain time interval and understanding the temporal characteristics of the circuit's response.

Transcript

00:01 In this problem on the topic of alternating current circuits, we are given an lc circuit which consists of a capacitor, c, which has a rating of 2 .5 microferds, and an inductor l, which is a rating of 4 millie henries.

00:14 The capacitor is fully charged using a battery and then connected to the inductor.

00:19 The circuit is opened and a resistance r is inserted in series with the inductor and capacitor, and the capacitor is again fully charged using the same battery and connected to the circuit.

00:30 We find that the angular frequency of the dammed oscillations in the rlc circuit is 20 % less than the angular frequency of the oscillations in the lc circuit.

00:40 And using this information, we want to find the resistance of the resistor.

00:43 The time after the capacitor is reconnected in the circuit that the amplitude will, that the amplitude of the dam current will be 50 % of the initial amplitude.

00:55 And we want to find how many complete damp oscillations will have occurred in that time.

01:00 Now for the oscillation frequency, we have the frequency of the rlc circuit to be 0 .8th of the frequency for the lc circuit.

01:15 And we know the angular frequency of the rlc circuit, omega rlc, is equal to the square root of omega -0 squared minus r over 2ll.

01:32 Squared and this is equal to 0 .8 omega not which means that omega not squared minus r over 2 l squared is equal to 0 .64 omega not squared and so rearranging and solving for the resistance we get r to be 1 .2 .2 times l omega -0, which is 1 .2 times the square root of l over 1 .2, rather, times l over the square root of l c, and this is 1 .2 times the square root of l over c.

02:38 And so putting our values in, we can find this resistance.

02:42 This is 1 .2 times the square root of the inductance l given to be 4 times 10 to the minus 3 henrys divided by the capacitance c which is 2 .5 times 10 to the minus 6 ferrets...

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