## Oscillations in an LC Circuit

10.5 Oscillations in an LC Circuit

### Learning Objectives

By the end of this section, you will be able to:

- Explain why charge or current oscillates between a capacitor and inductor, respectively, when wired in series
- Describe the relationship between the charge and current oscillating between a capacitor and inductor wired in series

### LC Oscillations

**Practice!**

Practice 10.5.1 |
---|

A fully charged capacitor is connected in series with an inductor, as shown in the figure. What happens in the brief amount of time (maybe a few microseconds) just after the switch is closed to complete the circuit? |

In the brief amount of time just after the switch is closed to complete the circuit, **the capacitor begins to discharge and the current through the inductor increases at the same rate as the charge on the capacitor decreases**. The charge on the capacitor provides a voltage that will drive a current when the switch is closed. However, it takes some time for the capacitor to fully discharge. The electrons on one plate of the capacitor will be attracted to the positive plate and will discharge but, depending on the capacitance and the resistance in the circuit, this will take time. The more charge there is on the capacitor, the more time it will take to discharge.

As the capacitor discharges, the current increases in this circuit. The current is the flow of electrons from the negative plate of the capacitor to the positive plate. As the rate of discharge increases, so does the current in the circuit. This increasing current in the inductor induces a back-*emf* in the inductor, which helps to increase the time required for the capacitor to fully discharge. The induced *emf* in the inductor is not strong enough to stop the flow of current though. The back-*emf* only reduces the rate at which the current increases.

Practice 10.5.2 |
---|

A fully charged capacitor is connected in series with an inductor, as shown in the figure. When the switch is closed to complete the circuit, the capacitor can discharge through the circuit. What is true about the current in the inductor at the instant the capacitor is fully discharged (when Q = 0 on the capacitor)? |

After the switch is closed to complete the circuit, **the current in the inductor is a maximum at the instant the capacitor is fully discharged**. As the capacitor discharges, the current increases in this circuit. The current is the flow of electrons from the negative plate of the capacitor to the positive plate. As the rate of discharge increases, so does the current in the circuit. This increasing current in the inductor induces a back-*emf* in the inductor, which helps to increase the time required for the capacitor to fully discharge. The induced *emf* in the inductor is not strong enough to stop the flow of current though. The back-*emf* only reduces the rate at which the current increases. This means the current is increasing but slowing down as it does so.

At the instant that the current reaches its maximum value, all the charge has come off the capacitor such that *Q* = 0. At this instant the current has reached its maximum value and won’t get any larger because there isn’t a voltage on the capacitor to drive the current anymore.

Practice 10.5.3 |
---|

A fully charged capacitor is connected in series with an inductor, as shown in the figure. When the switch is closed to complete the circuit, the capacitor can discharge through the circuit. If T is the oscillation period of the LC circuit, how long after closing the switch will the capacitor be fully charged again? |

After closing the switch, the capacitor will be fully charged again in a time equal to ** T/2**, or half of the oscillation period. In this time, the voltage on the capacitor will drive a current in the circuit and this current will increase from zero to a maximum value. At the instant the current is a maximum value, the charge on the capacitor and the voltage across the capacitor will be zero. This means there is nothing to drive the current and the current will begin to decrease. The decreasing current in the inductor will induce an

*emf*that will help to drive the current, which will charge the capacitor again. When the capacitor is recharged, it will have a polarity that is opposite to what it started with. This will occur at exactly one half of the total oscillation period, or

*T*/2.

The capacitor will discharge again, the current will increase to a maximum value when the capacitor is fully discharged, and then the capacitor will recharge. At this point the capacitor will be fully charged with a polarity equal to what it started with and the oscillation will start all over again.

Practice 10.5.4 |
---|

A fully charged capacitor is connected in series with an inductor, as shown in the figure. When the switch is closed to complete the circuit, the capacitor can discharge through the circuit. This LC oscillator has an oscillation frequency f. If you replaced this inductor with one having twice the inductance, what would happen to the oscillation frequency of the LC oscillator? |

If you replaced the inductor with one having twice the inductance, **the oscillation frequency would decrease by a factor of **. The natural frequency of oscillation in an LC circuit is and is measured in units of Hz, or cycles per second. Doubling the inductance results in a factor of in the denominator of the calculation.

Practice 10.5.5 |
---|

A fully charged capacitor is connected in series with an inductor, as shown in the figure. When the switch is closed to complete the circuit, the capacitor can discharge through the circuit. This LC oscillator has an oscillation frequency f. If you replaced this capacitor with one having half the capacitance, what would happen to the oscillation frequency of the LC oscillator? |

If you replaced the capacitor with one having half the capacitance, **the oscillation frequency would increase by a factor of **. The natural frequency of oscillation in an LC circuit is and is measured in units of Hz, or cycles per second. Halving the capacitance results in a factor of in the denominator, which is equivalent to multiplying by a factor of .

Practice 10.5.6 |
---|

In a certain oscillating LC circuit the total energy is converted from electric energy in the capacitor to magnetic energy in the inductor in 1.10 μs. What is the period of oscillation? |

**Discuss!**

What is the angular frequency of the oscillations in this circuit?

An LC circuit with a 20.0 nF capacitor needs to have an angular frequency of 2500 rad/s for a certain application. What inductance is needed in the circuit?

What are the energies of the capacitor and inductor when the capacitor is fully charged? Also, what is the total energy for the circuit?

In an oscillating LC circuit with L = 25 mH and C = 5.0 μF, the current is initially a maximum. How long will it take before the capacitor is fully charged for the first time?

An oscillating LC circuit consisting of a 1.1 nF capacitor and a 5.0 mH coil has a maximum voltage of 6.2 V.

(a) What is the maximum charge on the capacitor?

(b) What is the maximum current through the circuit?

(c) What is the maximum energy stored in the magnetic field of the coil?