---
title: "LC oscillation differential equation"
subject: "Engineering Physics"
chapter: 1
key: "semester_1/engineering_physics:lc-oscillation-differential-equation"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/lc-oscillation-differential-equation
site: BCT Engineering
---

# LC oscillation differential equation

Engineering Physics · Chapter 1: Oscillation

- Asked in: 2074 Asoj, 5 marks
- Video: [PHYS 102 | LC Circuits 1 - Setting up the Differential Equation for an LC Circuit](https://www.youtube.com/watch?v=1DzcTE5uKVg) — Professor Hafner
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/lc-oscillation-differential-equation
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-1

## Question

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

## 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.

## Answer

**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:
```text
L d²q/dt² + q/C = 0
d²q/dt² + ω0² q = 0 ,    ω0 = 1/√(LC)
```
4. Solution: q = q0 cos(ω0 t + φ). Frequency f = 1/(2π√(LC)).
5. 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-spring | LC circuit |
|---|---|
| x | q |
| v = dx/dt | i = dq/dt |
| m | L |
| k | 1/C |
| ω = √(k/m) | ω0 = 1/√(LC) |
| KE ½mv² | magnetic energy ½Li² |
| PE ½kx² | electric energy q²/2C |
| Total ½kx0² | Total 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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is LC oscillation?
   A. Oscillation with a resistor dissipating energy
   B. Oscillation driven by AC source
   C. Oscillation of charge and current with energy exchange between C and L
   D. Steady DC current
   Answer: C. Ideal L and C, no R. (Key point: LC oscillation)

2. Free LC oscillation obeys?
   A. d2q/dt2 + 2gamma dq/dt + w0^2 q = 0
   B. d2q/dt2 - w0^2 q = 0
   C. d2q/dt2 = w0 q
   D. d2q/dt2 + w0^2 q = 0, w0 = 1/sqrt(LC)
   Answer: D. Kirchhoff loop rule. (Key point: Equation)

3. The current i = dq/dt relative to q?
   A. A quarter cycle ahead
   B. In phase
   C. 180 degrees out of phase
   D. A full cycle ahead
   Answer: A. i = -w0 q0 sin(w0 t + phi). (Key point: Current phase)

4. In the analogy, k corresponds to?
   A. C
   B. 1/C
   C. L
   D. R
   Answer: B. Inductance is inertia; 1/C is stiffness. (Key point: Analogy)

## Related in Engineering Physics

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