31.11: Oscillations In An LC Circuit

An idealized LC circuit of zero resistance can oscillate without any source of emf by shifting the energy stored in the circuit between the electric and magnetic fields. In such an LC circuit, if the capacitor contains a charge q before the switch is closed, then all the energy of the circuit is initially stored in the electric field of the capacitor. This energy is given by

When the switch is closed, the capacitor begins to discharge, producing a current in the circuit. The current, in turn, creates a magnetic field in the inductor. The net effect of this process is a transfer of energy from the capacitor, with its diminishing electric field, to the inductor, with its increasing magnetic field. When the capacitor is completely discharged and all the energy is stored in the inductor's magnetic field, the current in the inductor is at its maximum value. At this instant, the energy stored in the inductor is given by

At an arbitrary time, the capacitor charge and current varies with time. Therefore the total energy U in the circuit is given by

Since there is no resistance in the circuit, no energy is lost through Joule heating; the energy in circuit remains conserved. After reaching the maximum current in the inductor, the current continues to transport charge between the capacitor plates, thereby recharging the capacitor. Since the inductor resists a change in current, current continues to flow, even though the capacitor is discharged. This continued current causes the capacitor to charge with opposite polarity. If there is no energy dissipation, charge on the capacitor plates continues to change polarity indefinitely, causing electrical oscillations. The angular frequency of these oscillations in the circuit is given by

Consider an LC circuit connecting a charged capacitor with an inductor. When the circuit is closed, the capacitor discharges through the inductor, transferring energy from the electric field to the magnetic field.

The current continues to flow towards the discharged capacitor as the inductor resists a change in current through it. This continued current charges the capacitor with opposite polarity, increasing the capacitor's electric field while decreasing the inductor's magnetic field.

The charged capacitor again discharges, converting electrical energy to magnetic energy. On re-charging the capacitor, energy flows back to the capacitor, and the initial state of the circuit is restored.

If there is no energy dissipation, the charges on the capacitor continue to change polarity indefinitely, called electrical oscillations.

Here, the charge on the capacitor and the current through the inductor varies sinusoidally with time.

Initially, when the charge on the capacitor is at maximum, the current in the inductor is zero. As time progresses, the charge becomes zero on the capacitor, and the current becomes maximum in the inductor. With time, the process reverses and repeats itself.