LC OscillationsJEE Advanced

Charge sloshing between capacitor and inductor — interactive Physics simulation for IIT-JEE.

Concept

Connect a charged capacitor across an inductor: the charge cannot simply stop — the inductor's back-EMF keeps current flowing, overshooting the capacitor to the opposite polarity. Energy sloshes between electric ( $q^2/2C$ ) and magnetic ( $\tfrac12 Li^2$ ) forms forever (ideally) — the exact electrical twin of a mass-spring oscillator.

Key formula

\omega = \frac{1}{\sqrt{LC}}, \qquad q = q_0\cos\omega t, \qquad \frac{q^2}{2C} + \frac{Li^2}{2} = \frac{q_0^2}{2C}

Derivation

KVL: $L\frac{di}{dt} + \frac{q}{C} = 0$ with $i = dq/dt$ gives $\ddot q = -\frac{1}{LC}q$ — the SHM equation.

Mechanical dictionary: $q \leftrightarrow x$ , $i \leftrightarrow v$ , $L \leftrightarrow m$ (inertia), $1/C \leftrightarrow k$ (stiffness). Current peaks when charge is zero, exactly like speed at the equilibrium point.

Scenarios to explore

LC Oscillations — Charge sloshing at ω = 1/√(LC) — the electrical spring.

Real-world applications

Radio tuners — the tank circuit selects f = 1/2π√(LC).
Oscillators & RF transmitters.
Induction-heating and wireless-charging resonance.

JEE exam tips

q and i quarter-period apart: q = q₀ at t = 0 ⇒ i max at T/4.
Energies equal when q = q₀/√2 — at t = T/8.
Real circuits decay (R > 0): damped, like friction on the spring.

Common mistakes

Peak current and peak charge occurring together — they are 90° out of phase.
Forgetting energy conservation gives i_max = q₀ω instantly.
Confusing LC oscillation (free) with driven LCR resonance.

Exam traps to avoid

ω = 1/√(LC), not √(L/C) — check dimensions if unsure.
Halving C doubles ω²: frequency scales as 1/√C.