Energy Trading Between Capacitor and Inductor

Picture a charged capacitor connected to an inductor. The capacitor starts full, with maximum voltage and zero current. As it discharges, current builds in the inductor, which fights the change in current by building a magnetic field, storing energy there as (1/2)LI^2. The capacitor's energy, (1/2)q²/C, drops as the inductor's energy rises. When the capacitor is fully empty, q = 0, every bit of energy has moved into the magnetic field, so the current is at its absolute maximum. Then the inductor, which hates losing current, keeps pushing charge along, recharging the capacitor with the opposite polarity. The cycle reverses and repeats forever.

The total energy U = q²/(2C) + Li²/2 never changes in an ideal LC loop. That's just conservation of energy applied to two different storage mechanisms: one electric, one magnetic. Because the total is fixed, you can always solve for one variable given the other. Splitting the energy evenly between capacitor and inductor means the capacitor holds half the original energy, so q = Q0/√2, not Q0/2 — the square root comes from the square in the energy formula.

This trading is the electrical twin of a mass on a spring swapping kinetic and potential energy, and it oscillates at angular frequency omega = 1/√LC. Real circuits have resistance, though, and that changes everything: a resistor bleeds energy out of the loop as heat every cycle, so q and i no longer oscillate with constant amplitude. The oscillation becomes damped, decaying toward zero, which sets up the driven and damped RLC behavior that follows.