LC Oscillations

An LC circuit (inductor + capacitor, no resistance) oscillates indefinitely once given an initial charge or current — energy bouncing between E-field of C and B-field of L.

Charge on capacitor: q(t) = q₀ cos(ω₀ t). Current: i(t) = −q₀ω₀ sin(ω₀ t).

Angular frequency: ω₀ = 1/√(LC). Period T = 2π√(LC).

Energy: U_E = q²/(2C) in capacitor; U_B = ½LI² in inductor. Total U_tot = q₀²/(2C) is CONSTANT.

Energy oscillates at 2ω₀ — it transfers between C and L twice per cycle.

Direct mechanical analogue: SHM of a mass on a spring. Q ↔ x, I ↔ v, L ↔ m, 1/C ↔ k, energy U_B ↔ KE, U_E ↔ PE.

Adding any resistance turns it into damped LCR oscillation — amplitude decays exponentially.