An LC circuit connects a charged capacitor to an inductor with no resistance, so charge oscillates in simple harmonic motion with angular frequency ω = 1/√(LC) while energy swaps between the capacitor's electric field and the inductor's magnetic field.
An LC circuit is just two components in a loop, a capacitor (C) and an inductor (L). Charge up the capacitor, close the switch, and something cool happens. The capacitor dumps its charge through the inductor, building up a current and a magnetic field. Then the inductor's induced emf keeps pushing charge, recharging the capacitor with the opposite polarity. The energy sloshes back and forth between the capacitor's electric field (U = q²/2C) and the inductor's magnetic field (U = ½LI²), forever, since there's no resistor to burn anything off.
Here's the line that should make it click. An LC circuit is a mass-spring oscillator wearing an electrical costume. Apply Kirchhoff's loop rule and you get d²q/dt² = -(1/LC)q, which is the exact same differential equation as simple harmonic motion with q playing the role of position. The solution is q(t) = Q₀cos(ωt + φ) with ω = 1/√(LC). The inductor acts like the mass (it resists changes in current the way mass resists changes in velocity), and 1/C acts like the spring constant.
LC circuits live in Topic 13.6 (Circuits with Capacitors and Inductors) in the electromagnetic induction unit of AP Physics C: E&M. This topic is where the whole course loops back on itself. You need capacitance and energy storage from earlier units, induced emf and inductance from Unit 13, and the entire SHM toolkit from Physics C: Mechanics. The exam loves it for exactly that reason. One LC problem can test whether you can write a loop equation, recognize a differential equation as SHM, and use energy conservation, all in a single setup. If you can solve a mass-spring problem, you can solve an LC problem. The physics transfers one-to-one.
Keep studying AP® Physics C: E&M Unit 13
Simple harmonic motion (Physics C: Mechanics Unit 7)
The LC circuit obeys d²q/dt² = -(1/LC)q, the same equation as d²x/dt² = -(k/m)x. Every SHM result you know carries over directly. Charge behaves like position, current behaves like velocity, and the inductor plays the role of inertia.
Angular frequency (Unit 13)
For an LC circuit, ω = 1/√(LC). Notice what's not in that formula. The frequency doesn't depend on how much charge you start with, just like a mass-spring system's frequency doesn't depend on amplitude.
Inductors and RL circuits (Unit 13)
An inductor paired with a resistor gives exponential decay of current. Swap the resistor for a capacitor and you get oscillation instead. Comparing the two cases is a classic way the exam checks whether you understand what each component actually does.
Energy stored in capacitors and inductors (Units 10 and 13)
LC problems run on energy bookkeeping. The total energy q²/2C + ½LI² stays constant, so the capacitor's maximum energy equals the inductor's maximum energy. That single equation, ½CV₀² = ½LI_max², solves a huge fraction of LC questions.
LC circuits show up mostly as calculation and concept multiple-choice questions. Typical stems give you L, C, and an initial voltage or charge, then ask for the angular frequency (ω = 1/√(LC)), the maximum current (set ½CV₀² = ½LI_max² and solve), or the correct differential equation for the charge. For example, a 5.0 μF capacitor charged to 12 V and connected to a 2.0 mH inductor gives I_max = V₀√(C/L) = 0.60 A. Conceptual versions ask about phase, like at what point the energy is split equally between the two components (when cos²(ωt + φ) = ½, so the phase is π/4). On an FRQ, the natural ask is a derivation. Be ready to write the loop rule, arrange it into SHM form, identify ω by matching it to d²q/dt² = -ω²q, and sketch q(t) and I(t) showing they're a quarter cycle out of phase. No released FRQ has used the term verbatim, but the derive-the-differential-equation move is a standard Physics C skill.
An RC circuit has a resistor, so charge decays exponentially toward a final value and energy gets dissipated as heat. An LC circuit has no resistance, so nothing dissipates. The charge oscillates sinusoidally forever and total energy stays constant. The math is completely different too. RC gives a first-order equation with e^(-t/RC) solutions, while LC gives a second-order equation with cosine solutions. If a problem hands you a time constant, you're in RC or RL land. If it hands you a frequency, you're in LC land.
An LC circuit oscillates with angular frequency ω = 1/√(LC), and this frequency does not depend on the initial charge or voltage.
Kirchhoff's loop rule gives d²q/dt² = -(1/LC)q, the same form as simple harmonic motion, so q(t) = Q₀cos(ωt + φ).
Total energy is conserved in an ideal LC circuit, constantly trading between the capacitor's electric energy (q²/2C) and the inductor's magnetic energy (½LI²).
Maximum current happens when the capacitor is fully discharged, and you find it from energy conservation: ½CV₀² = ½LI_max², so I_max = V₀√(C/L).
Charge and current are a quarter cycle out of phase, so when one is at its maximum, the other is zero.
The energy is split equally between the capacitor and inductor at a phase angle of π/4 (and every quarter cycle after), since that's where cos² and sin² both equal one half.
It's a circuit with just a capacitor and an inductor, where charge oscillates back and forth in simple harmonic motion at ω = 1/√(LC). Energy alternates between the capacitor's electric field and the inductor's magnetic field, with the total staying constant.
No, an ideal LC circuit conserves energy forever because there's no resistance to dissipate it as heat. Real circuits always have some resistance, which makes them RLC circuits with damped oscillations, but ideal LC is what the AP exam tests.
An RC circuit decays exponentially with time constant τ = RC because the resistor burns off energy. An LC circuit oscillates sinusoidally at ω = 1/√(LC) because nothing dissipates energy. Exponential decay versus endless oscillation is the giveaway.
Use energy conservation. All the capacitor's initial energy becomes the inductor's energy at peak current, so ½CV₀² = ½LI_max². For a 5.0 μF capacitor charged to 12 V with a 2.0 mH inductor, that gives I_max = 12√(5.0×10⁻⁶/2.0×10⁻³) = 0.60 A.
Both obey the same differential equation. The inductor acts like the mass because it resists changes in current the way mass resists changes in velocity, and 1/C acts like the spring constant. Charge maps to position and current maps to velocity, so every SHM result transfers directly.
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