Short on time? This page is the recap. For the derivations, Figure 1 and the worked examples, use the full study guide. This topic is in the calculus-based course.
Recap
- A charged capacitor connected across an inductor (no resistor) oscillates: energy moves between the electric field and the magnetic field.
- The total energy is constant, so the maximum current follows from energy conservation alone.
- The loop rule, q/C + L dI/dt = 0 with I = dq/dt, gives the simple harmonic equation for q.
- The period depends only on L and C, not on the starting charge.
- Mass–spring analogy: q ↔ x, I ↔ v, L ↔ m, 1/C ↔ k.
- If the oscillation starts from an inductor current I₀ with the capacitor empty, use q = Q_max sin ωt and I = I₀ cos ωt instead; the capacitor’s voltage can then be much larger than the emf that first set up I₀.
Key relationships
| Quantity | Result |
|---|---|
| Energy (constant) | q²/(2C) + ½LI² = Q₀²/(2C) |
| Maximum current | I_max = Q₀/√(LC) = ωQ₀ |
| Maximum charge from a starting current I₀ | Q_max = I₀√(LC) |
| Differential equation | d²q/dt² = −q/(LC) |
| Angular frequency | ω = 1/√(LC) (rad/s) |
| Period and frequency | T = 2π√(LC); f = 1/(2π√(LC)) |
| Starting fully charged | q = Q₀ cos ωt; I = −ωQ₀ sin ωt |
| Energies | U_C = (Q₀²/2C)cos² ωt; U_L = (Q₀²/2C)sin² ωt; each repeats every T/2 |
| Charged by a battery ℰ | Q₀ = Cℰ; I_max = ℰ√(C/L) |
Assumptions behind the results
- No resistance anywhere, so no energy is dissipated.
- A single capacitor (or equivalent capacitance) and a single inductor.
- The switch connects the elements instantly and adds no resistance.
Mistakes to avoid
- Putting I_max at full charge. The current is zero when |q| is greatest.
- Confusing ω and f. f = ω/(2π).
- Thinking the period depends on Q₀. It does not.
- Wrong energy period. U_C and U_L repeat every T/2.
- Factor-of-change slips. Decide whether Q₀ or the voltage is fixed before scaling.
- Using √(L/C) as a time. Only √(LC) is in seconds.
Quick self-check
- L = 0.10 H and C = 10 μF. What is ω? (1.0 × 10³ rad/s)
- For that circuit, what is T? (6.3 ms)
- Q₀ = 2.0 × 10⁻⁴ C in the same circuit. What is I_max? (0.20 A)
- The capacitor starts fully charged. When is all the energy first in the inductor? (At T/4)
- The same circuit at T/8: what fraction of the energy is in the capacitor? (One half)
Next: practice questions.