LC and RLC oscillations

Connect a charged capacitor directly to an inductor and something happens with no battery anywhere in the circuit: the capacitor discharges through the inductor, building up a magnetic field as it does, and once the capacitor is empty that field's collapse drives current onward, recharging the capacitor with the opposite polarity. The process then reverses and repeats. This is oscillation with no external driver needed, and it is the electrical exact match of a mass on a spring: the capacitor's electric field storing energy the way a stretched spring does, the inductor's magnetic field storing energy the way a moving mass does, and the two trading that energy back and forth every cycle. Nothing forces this rhythm; it is the natural, resonant behaviour of the L and C connected together.

The resonant frequency follows from the same mathematics as the mechanical oscillator, f = 1/(2π√(LC)). Both L and C sit under the square root in the denominator, so increasing either one — not just one of them — slows the oscillation and lowers the frequency, which surprises anyone expecting more inductance to mean more "drive". A larger inductor opposes changes in current more strongly, and a larger capacitor holds more charge at a given voltage; both effects make each half-cycle take longer.

A real circuit always has some resistance, and resistance changes everything about the story even though it appears in neither the formula for frequency above nor in the LC circuit's own energy storage. Resistance dissipates energy as heat every cycle, exactly as friction drains energy from a swinging pendulum, so a real RLC circuit's oscillation amplitude decays over time — damped oscillation, the electrical counterpart of a shock absorber. Push it periodically at its own resonant frequency, as a radio receiver does when tuning to a station, and the response grows sharply, the same resonance behaviour met in mechanical driven oscillators.