LC Oscillations
q(t) = q₀cos(ω₀t). Energy bounces between capacitor and inductor.
Key Notes
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.
Formulas
Differential equation
Same form as mass-spring: m·d²x/dt² + kx = 0.
Resonant frequency
Free-oscillation frequency.
Charge & current
Current leads charge by π/2.
Total energy (constant)
Energy bounces but the sum is fixed.
Peak current
Reached when all energy is in the inductor.
Important Points
Pure LC oscillates FOREVER in an ideal circuit. Real circuits have R ⇒ damping ⇒ exponential decay.
Energy is fully in C at t = 0 (max q, zero I) and fully in L at quarter-period (zero q, max I).
Doubling L or C QUADRUPLES the period (T ∝ √(LC)).
LC oscillation is the prototype for every harmonic oscillator — and explains why so much of physics looks like SHM.
The frequency f₀ = 1/(2π√(LC)) matches the resonance frequency of a series LCR circuit — same condition X_L = X_C.
Adding driven AC at f = f₀ to an LCR circuit makes the LC subsystem oscillate at max amplitude — that's resonance.',
LC Oscillations notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 12 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.