LC Oscillations (Qualitative Idea)

Setting up the ideal LC circuit. Imagine an inductor $L$ and a capacitor $C$ connected together in a simple closed loop, with NO resistor and NO external AC source at all -- just the capacitor, initially charged to some voltage, and the inductor. This is the resistance-free, source-free limit of the LCR circuit studied above ($R=0$, no driving voltage).

The oscillation, described qualitatively. At the very first instant, all the circuit's energy sits in the charged capacitor's electric field, and the current is momentarily zero. The charge on the capacitor immediately begins to drive a current through the inductor; as this current builds up, energy starts flowing OUT of the capacitor's electric field and INTO the inductor's magnetic field. By the time the capacitor has fully discharged (its voltage momentarily zero), the current -- and hence the magnetic energy stored in the inductor -- is at its own maximum. The inductor's self-induced back emf now keeps driving this current onward in the SAME direction even though the capacitor is no longer pushing it, which charges the capacitor back up, but with the OPPOSITE polarity to before. Once the capacitor is fully recharged (now with reversed polarity), the current has fallen back to zero and all the energy is once again purely electrical, stored in the capacitor -- and the entire process now repeats in reverse, eventually charging the capacitor back to its ORIGINAL polarity, completing one full cycle. This continuous, self-sustaining exchange between the capacitor's electric field and the inductor's magnetic field is called an LC oscillation, and it takes place at the same natural angular frequency already derived for resonance, $\omega_0=1/\sqrt{LC}$ (Section 7.11), which is why the two topics sit next to each other in the syllabus.

The mechanical analogy. This behaviour is exactly analogous to a frictionless spring-mass system left to oscillate on its own: energy sloshes back and forth between the spring's stored elastic potential energy (like the capacitor's electric energy) and the mass's kinetic energy (like the inductor's magnetic energy), with the total mechanical energy staying constant in the idealised, friction-free case -- the electrical and mechanical pictures share the very same mathematics of simple harmonic motion.

Why a real LC circuit's oscillation dies away. Every real inductor and every real connecting wire has SOME resistance, however small, which was deliberately set to zero in the ideal picture above. This resistance dissipates a little energy as $I^2R$ heat on every cycle, exactly as friction would slowly drain a real spring-mass system's energy -- so a real LC circuit's oscillations gradually shrink in amplitude (are damped) and eventually die out completely, unless energy lost this way is continually replaced. …