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LC Oscillations Page 1 of 5

LC Oscillations

NAME: ____________________________________

NAME: ____________________________________

NAME: ____________________________________

RECITATION SECTION: __________________________

INSTRUCTOR: __________________________

DATE: __________________________

This activity is based on the following concepts:

 LC Oscillations: If a charged capacitor is connected across an inductor, the charge on the capacitor and the current in the circuit oscillate as a function of time:

0 ( ) cos( )q t q t

with the angular frequency ω given by:

LC

1  .

 Damped LC Oscillations: If there is some resistance in an LC circuit, the LC oscillations are damped, and the frequency of the oscillations changes from that of an undamped LC

circuit. We observe also that the amplitude of the charge oscillation on the capacitor

decays exponentially:

2 0

oscillatingexponentially termdamped

amplitude of oscillation

( ) cos( ) Rt

Lq t q e t  

 

 The energy stored in an inductor is given by: 2

2

1 LiU 

 The energy stored in a capacitor is given by: QV C

Q CVU

2

1

2

1

2

1 2

2 

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In the circuit shown below, the switch S is kept in position A for a long time. It is then thrown to position B at a time defined as t = 0.

Exercise 1:

First assume that R = 0.

Q1. Calculate the angular frequency ω of the LC oscillations that result when the switch is

thrown to position B. What is the frequency f of these oscillations?

Q2. Calculate the time t when the magnitude of the current in the inductor first reaches its

maximum value. Justify your answer.

Q3. Calculate the amplitude of the current oscillations. Rather than simply using an equation

from the book, try reasoning this out using energy conservation. (Note that ‘amplitude’ refers to

the maximum magnitude of the current.)

+ −

120 V

R 100 μF S

10 mH

A

B

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Exercise 2:

Now assume that R takes on a non-zero value. Using energy conservation, describe as

quantitatively as possible, what happens to the total energy in the circuit as a function of time. In

your discussion, consider the following points:

Q4. What is the initial energy in the circuit?

Q5. Given that the amplitude of the charge on the capacitor decays exponentially as shown by

the equation on the cover page, how does the amplitude of the energy stored in the capacitor

decay with time?

Q6. If we look at the total energy stored in the circuit as a function of time (that is, the sum of the

energy stored in the capacitor and the energy stored in the inductor at any instant), we would

observe that it too decreases exponentially over time. Throughout your physics courses,

however, you have continually encountered the Principle of Conservation of Energy, which says

that the energy of a closed, isolated system must remain constant throughout time. How can we

reconcile our observation with this fundamental principle of physics? What can you say about

the energy “missing” from the circuit?

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Exercise 3:

A circuit consists of a battery with emf V, an inductor L, a capacitor C, and two resistors, each

with resistance R, as shown in the sketch. The capacitor is initially uncharged and there is no

current flowing anywhere in the circuit. The switch S has been open for a long time, and is then

closed, as shown in the diagram.

Q7. Using Faraday’s Law, what is the sum of the potential drops around the outer loop (the loop

including both the battery and the inductor) if we move clockwise around the loop? Note that

although we ask you to find this differential equation, you do not have to solve it to answer any

later questions.

Q8. Just after the switch S is closed, what are the currents i1, i2, and i3 in terms of the given

quantities? Assume that the left loop of the circuit has zero inductance.

Q9. A long, long time after switch S is closed, what are the currents i1, i2, and i3?

LC Oscillations Page 5 of 5

Q10. A long, long time after switch S is closed, what is the charge on the capacitor?

Q11. The switch S is now opened again. Just after the switch is opened again, what are the

currents i1, i2, and i3 in terms of the given quantities? Assume that the left loop of the circuit

has zero inductance.

Q12. After approximately how long will the current through the inductor first fall to zero?