LC Oscillations

So far every circuit considered has included an external AC source continuously driving the current. This section instead asks: what happens if a capacitor, already charged to some potential, is disconnected from any source and connected directly to an ideal (zero-resistance) inductor, with no driving emf at all?

Initially (Fig. 13.16a), the fully-charged capacitor holds all of the circuit's energy in its electric field, $U_E=\dfrac12\dfrac{q_0^2}{C}$, with zero current and hence zero energy in the inductor's magnetic field. Once the loop is closed, the capacitor begins to discharge through the inductor (Fig. 13.16b): a growing current builds up a magnetic field in the inductor, so energy is progressively transferred from the capacitor's electric field into the inductor's magnetic field. At the instant the capacitor becomes fully discharged (Fig. 13.16c), the current reaches its maximum value $I_0$, and ALL the original electrostatic energy has been converted into magnetic energy, $U_B=\dfrac12 LI_0^2=U_E$ (energy conservation, since no resistance is present to dissipate any of it). The process then continues in reverse: as the magnetic flux through the inductor now begins to fall, an induced emf (Lenz's law) keeps the current flowing in the SAME direction a little longer, which recharges the capacitor -- but now with the OPPOSITE polarity to its initial state (Fig. 13.16d-e). Once the capacitor is again fully charged (in this reversed polarity), the whole process repeats itself in the opposite direction, and the cycle continues indefinitely. …