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LC oscillation differential equation

Short answer

LC oscillation is periodic charge and current oscillation in an inductor-capacitor circuit. Applying Kirchhoff's law yields L d²q/dt² + q/C = 0, or d²q/dt² + ω₀²q = 0 with ω₀=1/√(LC). Solution: q = q₀ cos(ω₀t+φ). It is analogous to mass-spring: m↔L, k↔1/C, displacement↔charge.

What is LC oscillation? Derive the differential equation of free oscillation and compare its solution with the mass spring system.

LC oscillation is the periodic oscillation of charge and current in a circuit that contains only an inductor L and a charged capacitor C (no resistance), as energy passes back and forth between the electric field of C and the magnetic field of L.

Differential equation

At time t let the capacitor charge be q and current i = dq/dt.

  1. Voltage across the capacitor is q/C; emf induced across the inductor is −L di/dt.
  2. Kirchhoff's loop rule (no resistance, no source): L di/dt + q/C = 0.
  3. Using i = dq/dt:
L d²q/dt² + q/C = 0
d²q/dt² + ω0² q = 0 ,    ω0 = 1/√(LC)
  1. Solution: q = q0 cos(ω0 t + φ). Frequency f = 1/(2π√(LC)).
  2. Current: i = −ω0 q0 sin(ω0t + φ), a quarter cycle ahead of q.

Comparison with the mass-spring system

Spring-mass: m d²x/dt² + kx = 0, x = x0 cos(ωt + φ), ω = √(k/m).

Mass-springLC circuit
xq
v = dx/dti = dq/dt
mL
k1/C
ω = √(k/m)ω0 = 1/√(LC)
KE ½mv²magnetic energy ½Li²
PE ½kx²electric energy q²/2C
Total ½kx0²Total q0²/2C

Both equations have the same form and the same solution, so the LC circuit is the electrical analogue of SHM: inductance is like inertia and 1/C is like the spring stiffness.

✓ Answer checked by Claude Opus 5.5 (high effort)

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