LC Oscillator: Charge Sloshing Between a Capacitor and a Coil

What you see. A capacitor (two aluminium plates with a dielectric) wired in a loop with an air-cored coil on a ferrite rod and a resistor. Yellow dots are charge carriers moving round the loop (their speed is the current); the plates glow red and blue with the charge they hold, the dielectric brightens with the electric field and the coil and ferrite glow with the magnetic field. Real circuits oscillate hundreds of times a second, so the cycle is slowed to a few seconds on screen (slider); the physics runs in true circuit time.

The equation. Kirchhoff's loop rule gives L q'' + R q' + q/C = Vs(t), the same equation as a mass on a spring (L is the mass, 1/C the spring constant, R the friction). Without loss the charge oscillates at ω0 = 1/√(LC); with resistance the amplitude decays as e^−Rt/2L and the ringing frequency drops to √(ω0² − (R/2L)²); above R = 2√(L/C) the charge no longer oscillates. The energy q²/2C in the electric field and Li²/2 in the magnetic field trade places twice per cycle, and every joule that leaves is counted as heat ∫R i² dt in the resistor. Driven by Vd sin ωt, the steady current is Vd/√(R² + (ωL − 1/ωC)²): a resonance peak at ω₀ whose half-power width is f₀/Q with Q = ω₀L/R.

Honest simplifications. Ideal lumped elements: no coil resistance or self-capacitance, no capacitor leakage, no radiation, a perfect dielectric, a linear (non-saturating) ferrite core. The positions of the charge carriers are a visual aid (about 40 markers, not electrons), and the dielectric is only drawn as a field indicator. The drive is switched on at t = 0 so a start-up transient appears before the steady state.

Checks. (1) period of the lossless oscillator from zero crossings against 2π√(LC); (2) RK4 against the closed-form damped solution; (3) energy ledger of a driven circuit (source work = stored energy + heat); (4) steady driven current against the impedance formula and the −3 dB bandwidth against f₀/Q.