An LC circuit is a loop with an ideal capacitor (C) and inductor (L) and no resistance; it oscillates because energy swaps between the capacitor’s electric field (½CV²) and the inductor’s magnetic field (½LI²). If the capacitor starts charged (q = Qmax, I = 0), the charge obeys the SHM equation d²q/dt² = −(1/LC) q, so the circuit oscillates with angular frequency ω = 1/√(LC) and period T = 2π√(LC) (CED 13.6.A.2–3). Use energy conservation (½CVmax² = ½LImax²) to get the maximum current Imax = Vmax√(C/L) or Imax = Qmax/√(LC) (CED 13.6.A.1). Charge and current are 90° out of phase: when |q| is max, I = 0; when I is max, q = 0. For practice problems and a focused study guide on LC circuits, see the Topic 13.6 study guide (https://library.fiveable.me/ap-physics-c-e-m/unit-6/6-circuits-with-capacitors-and-inductors-lc-circuits/study-guide/nTgyGcr23xjTIU5I). For more practice, try the AP Physics C E&M problem bank (https://library.fiveable.me/practice/ap-physics-c-e-m).

Think of the LC circuit as an energy pendulum: the capacitor stores electric potential energy (½CV²) when charged, and the inductor stores magnetic energy (½LI²) when current flows. If you start with the capacitor charged and no current, that energy must go somewhere—so charge flows, building current and transferring energy into the inductor. When the capacitor is uncharged, all energy is in the inductor and the current keeps flowing, recharging the capacitor with opposite sign. This back-and-forth energy exchange continues (neglecting resistance), producing simple harmonic motion governed by d²q/dt² = −(1/LC) q, so q(t) oscillates and the current is 90° out of phase with charge. The angular frequency is ω = 1/√(LC), just like a mass-spring frequency. For AP exam framing, you should be able to derive the differential equation from Kirchhoff’s loop rule and use energy conservation to find the max current (CED 13.6.A.1–A.3). Review Topic 13.6 on Fiveable (study guide: https://library.fiveable.me/ap-physics-c-e-m/unit-6/6-circuits-with-capacitors-and-inductors-lc-circuits/study-guide/nTgyGcr23xjTIU5I) and practice problems (https://library.fiveable.me/practice/ap-physics-c-e-m).

Think of a capacitor as stored electric potential energy in the electric field between its plates, and an inductor as stored kinetic-like energy in the magnetic field of the current. For a capacitor, energy = ½ C V^2 (or ½ Q^2/C); when it’s charged, that field stores energy. For an inductor, energy = ½ L I^2; when current flows, the magnetic field stores energy. In an ideal LC circuit those two energies exchange back and forth with no loss: when the capacitor is fully charged all energy is ½CV^2 and current (and inductor energy) is zero; later all that energy can be in the inductor as ½LI^2. That exchange follows dq²/dt² = −(1/LC) q so charge oscillates SHM with ω = 1/√(LC) and charge and current are 90° out of phase (CED 13.6.A). Use conservation of energy to find things like the max current. For more review and practice tied to the CED, see the Topic 13.6 study guide (https://library.fiveable.me/ap-physics-c-e-m/unit-6/6-circuits-with-capacitors-and-inductors-lc-circuits/study-guide/nTgyGcr23xjTIU5I) and lots of practice problems (https://library.fiveable.me/practice/ap-physics-c-e-m).

LC circuits vs RC circuits—quick, exam-focused differences you should know: - Behavior: An ideal LC (capacitor + inductor) oscillates: charge and current undergo simple harmonic motion with d²q/dt² = −(1/LC) q and angular frequency ω = 1/√(LC). An RC (resistor + capacitor) circuit does not oscillate; it shows exponential charging/discharging with time constant τ = RC. - Energy: In an ideal LC circuit energy swaps between capacitor (½CV²) and inductor (½LI²), conserved (no loss). In RC circuits energy is dissipated as heat in the resistor; the capacitor’s stored energy decays. - Phase: In LC oscillations charge and current are 90° out of phase. In RC transients voltages/currents follow exponential curves with no steady sinusoidal resonance. - Damping/resonance: Adding resistance to an LC makes an RLC—you get damped oscillations and a resonance frequency. Pure RC never resonates. These are exactly the CED points for Topic 13.6 (SHM, ω=1/√(LC), energy exchange). For more review and AP-style practice, see the Topic 13.6 study guide (https://library.fiveable.me/ap-physics-c-e-m/unit-6/6-circuits-with-capacitors-and-inductors-lc-circuits/study-guide/nTgyGcr23xjTIU5I) and thousands of practice problems (https://library.fiveable.me/practice/ap-physics-c-e-m).

Start with conservation of energy: in an ideal LC circuit energy just oscillates between the capacitor and the inductor. If the capacitor starts with voltage V0 (or charge Q0 = C V0), its initial energy is Uc = ½ C V0² (or ½ Q0²/C). At the instant current is maximum the capacitor’s energy is zero and all energy is magnetic: Ul = ½ L Imax². Set them equal and solve for Imax. Two equivalent forms: - Imax = V0 · sqrt(C / L) - Imax = Q0 / sqrt(L C) So if you know V0, plug into the first; if you know initial charge Q0, use the second. This result follows EK = EP exchange and matches the LC angular frequency ω = 1/√(LC) and the 90° phase difference between q and i (CED 13.6.A.1–A.3). For more review and worked examples see the Topic 13.6 study guide (https://library.fiveable.me/ap-physics-c-e-m/unit-6/6-circuits-with-capacitors-and-inductors-lc-circuits/study-guide/nTgyGcr23xjTIU5I) and try practice problems (https://library.fiveable.me/practice/ap-physics-c-e-m).

Start with the differential equation the CED gives: d²q/dt² = −(1/LC) q. That’s exactly the form of simple harmonic motion, d²x/dt² = −ω² x, so ω² = 1/(LC) and ω = 1/√(LC). Physically it comes from how capacitor and inductor store energy and push each other: the capacitor’s voltage (and charge q) produces an emf that drives current through the inductor, and the inductor’s changing current produces a back-emf that changes the capacitor’s charge. Mathematically those two laws (q = C V and V_L = L di/dt, plus i = dq/dt and Kirchhoff’s loop rule) combine to give that second-order ODE with coefficient 1/(LC). Energy view: the capacitor’s ½C V² and the inductor’s ½L i² swap back and forth at frequency ω = 1/√(LC). This is exactly what Topic 13.6 expects you to know for the AP (see the study guide: https://library.fiveable.me/ap-physics-c-e-m/unit-6/6-circuits-with-capacitors-and-inductors-lc-circuits/study-guide/nTgyGcr23xjTIU5I).

Think of an LC circuit like a mass on a spring but with charge and current instead of position and velocity. A charged capacitor has electric potential energy (½CV² or Q²/(2C)). When you connect it to an inductor, that energy converts to magnetic energy in the inductor (½LI²) and back—it oscillates. Mathematically the capacitor charge q(t) satisfies d²q/dt² = −(1/LC) q, which is the simple harmonic oscillator equation. So q(t) = Q cos(ωt) and I(t) = dq/dt = −ωQ sin(ωt) with ω = 1/√(LC) and period T = 2π√(LC). Note charge and current are 90° out of phase (current zero when charge is max). Use energy conservation to get the max current: ½C V0² = ½L Imax² ⇒ Imax = Q0/√(LC) (since Q0 = CV0). This is exactly what the AP C CED expects (see 13.6.A.2–A.3). For a quick topic review check the LC study guide (https://library.fiveable.me/ap-physics-c-e-m/unit-6/6-circuits-with-capacitors-and-inductors-lc-circuits/study-guide/nTgyGcr23xjTIU5I) and try practice problems (https://library.fiveable.me/practice/ap-physics-c-e-m).

When you connect a charged capacitor to an ideal (lossless) inductor, the energy oscillates back and forth between the capacitor’s electric field and the inductor’s magnetic field. Initially the capacitor stores Uc = 1/2 C V0^2. As the capacitor discharges, current builds in the inductor and magnetic energy Ul = 1/2 L I^2 grows. Energy is conserved, so at the moment the capacitor is fully discharged all energy is in the inductor. Using conservation of energy you get the maximum current: Imax = V0·sqrt(C/L) (The 1/2 factors cancel.) The charge on the capacitor obeys d^2q/dt^2 = −q/(LC), so the circuit undergoes SHM with ω = 1/√(LC) and charge/current 90° out of phase. In a real circuit resistive losses cause damping and the oscillation decays. For AP exam practice, this is exactly Topic 13.6: use energy conservation for Imax and the differential equation/ω = 1/√(LC) (see the Topic 13.6 study guide on Fiveable: https://library.fiveable.me/ap-physics-c-e-m/unit-6/6-circuits-with-capacitors-and-inductors-lc-circuits/study-guide/nTgyGcr23xjTIU5I). For more practice problems, check Fiveable’s practice page (https://library.fiveable.me/practice/ap-physics-c-e-m).

Start with Kirchhoff’s loop rule for an ideal LC loop (no R): the sum of voltage drops around the loop is zero. The voltage across the inductor is L(di/dt) and across the capacitor is q/C, so L (di/dt) + q/C = 0. Use the relation between current and charge: i = dq/dt, so di/dt = d^2q/dt^2. Substitute: L d^2q/dt^2 + q/C = 0. Rearrange: d^2q/dt^2 = −(1/LC) q. That’s the SHM equation in the CED (13.6.A.2). The solution is q(t) = Q cos(ωt + φ) with ω = 1/√(LC) (CED 13.6.A.3). For more review and practice on LC circuits, check the Topic 13.6 study guide (https://library.fiveable.me/ap-physics-c-e-m/unit-6/6-circuits-with-capacitors-and-inductors-lc-circuits/study-guide/nTgyGcr23xjTIU5I) and practice problems (https://library.fiveable.me/practice/ap-physics-c-e-m).

An LC circuit oscillates because energy keeps swapping between the capacitor’s electric field and the inductor’s magnetic field, and each element “pushes back” when the other tries to stop the transfer. Start with a charged capacitor: its energy (½CV²) drives charge flow, producing current through the inductor. The inductor stores that energy as magnetic energy (½LI²) and, because an inductor resists changes in current, it produces an induced emf (−L di/dt) that keeps the current going even after the capacitor’s voltage crosses zero. That inertia of the current makes the circuit overshoot and charge the capacitor with opposite sign, so the current reverses. Mathematically you get d²q/dt² = −q/(LC), simple harmonic motion with ω = 1/√(LC) and a 90° phase difference between charge and current. For AP E&M, this is treated using conservation of energy and the differential equation (CED 13.6.A.1–A.3). For a clear study summary and practice problems, see the LC topic study guide (https://library.fiveable.me/ap-physics-c-e-m/unit-6/6-circuits-with-capacitors-and-inductors-lc-circuits/study-guide/nTgyGcr23xjTIU5I) and the unit page (https://library.fiveable.me/ap-physics-c-e-m/unit-13).

LC circuits matter because they’re the basic oscillators and resonant elements used in lots of real devices—and they directly map to the AP CED ideas (energy exchange, SHM, ω = 1/√(LC)). Real-world applications: - Radio and TV tuners / RF filters: an LC “tank” selects one frequency (f = 1/(2π√(LC))) so your receiver picks the station you want. - Oscillators and clocks in electronics: stable AC sources for timing and signal generation. - Signal processing (band-pass / band-stop filters) in audio and communications. - MRI and some pulse circuits: use resonant coils to store and transfer electromagnetic energy. Why you need to know it for AP: LC circuits are solved with conservation of energy (½CV² ↔ ½LI²), lead to d²q/dt² = −q/(LC) and SHM, and show the 90° phase difference between charge and current—concepts the exam tests (useful for multiple-choice and free-response). For more review and practice, check the Topic 13.6 study guide (https://library.fiveable.me/ap-physics-c-e-m/unit-6/6-circuits-with-capacitors-and-inductors-lc-circuits/study-guide/nTgyGcr23xjTIU5I) and tons of practice problems (https://library.fiveable.me/practice/ap-physics-c-e-m).

Use a simple LC tank and measure the oscillating voltage with an oscilloscope. Setup - Components: known C (capacitor), known L (inductor with low R), a SPDT switch, a resistor for controlled damping (optional), oscilloscope with probe. - Charge the capacitor (use a battery) with switch in charge position, then flip to connect C and L to start free oscillation. - Measure the voltage across the capacitor (or the inductor) on the oscilloscope. What to record - On the scope, measure the period T directly (time between peaks) or use FFT to find frequency f. Do several trials and average. - Compute theoretical ω = 1/√(LC), f = ω/(2π), and compare to measured f = 1/T. - If damping is noticeable, fit an exponential envelope to get decay rate but still measure period (frequency is nearly unchanged for light damping). AP ties - This is an AP-style experimental design: vary L or C, plot T^2 vs LC (T^2 ∝ LC) and check slope. That aligns with CED Essential Knowledge 13.6.A.2–A.3. For a quick review and practice problems on LC circuits, see the Topic 13.6 study guide (https://library.fiveable.me/ap-physics-c-e-m/unit-6/6-circuits-with-capacitors-and-inductors-lc-circuits/study-guide/nTgyGcr23xjTIU5I) and extra practice (https://library.fiveable.me/practice/ap-physics-c-e-m).

In an ideal LC circuit the capacitor’s charge q(t) oscillates as simple harmonic motion: d²q/dt² = −(1/LC) q, so q(t) = Q_max cos(ωt) with ω = 1/√(LC) (CED 13.6.A.2–3). The current through the inductor is i(t) = dq/dt = −Q_max ω sin(ωt), so current is 90° out of phase with charge (when q is maximum i = 0; when q = 0 i is maximum). You can also get the maximum current from energy conservation: ½ C V_max² (or ½ Q_max²/C) converts to ½ L I_max², giving I_max = Q_max/√(LC) (CED 13.6.A.1). This is exactly the kind of relationship the exam tests—know the differential equation, ω = 1/√(LC), phase difference, and energy argument. For a quick refresher, see the Topic 13.6 study guide (https://library.fiveable.me/ap-physics-c-e-m/unit-6/6-circuits-with-capacitors-and-inductors-lc-circuits/study-guide/nTgyGcr23xjTIU5I) and try practice problems (https://library.fiveable.me/practice/ap-physics-c-e-m).

Because the LC differential equation is linear and homogeneous: d²q/dt² = −(1/LC) q, the charge obeys simple harmonic motion. That equation fixes the angular frequency ω = 1/√(LC), so the period T = 2π√(LC). Notice there's no q (amplitude) in that formula—the initial charge only sets the amplitude (maximum q) and therefore the maximum current, not the restoring “strength” or rate of oscillation. Physically: the capacitor and inductor trade energy (½CV² ↔ ½LI²). A larger initial charge means more energy and bigger swings of q and I, but the rate at which energy swaps is set solely by L and C. Because the restoring term (voltage from the capacitor) is proportional to q, the motion stays linear and the frequency is amplitude-independent. (This is exactly what the CED expects: derive d²q/dt² = −(1/LC) q and ω = 1/√(LC).) For more review and practice on LC circuits, see the Topic 13.6 study guide (https://library.fiveable.me/ap-physics-c-e-m/unit-6/6-circuits-with-capacitors-and-inductors-lc-circuits/study-guide/nTgyGcr23xjTIU5I) and thousands of practice problems (https://library.fiveable.me/practice/ap-physics-c-e-m).

If you hook a charged capacitor to an ideal (no-resistance) inductor you’ll see an oscillation: charge on the capacitor and current in the inductor swap back and forth like simple harmonic motion. Energy alternates between the capacitor (½CV²) and the inductor (½LI²). Start with capacitor charge Q0 (voltage V0): by energy conservation ½C V0² = ½L Imax² so Imax = V0·√(C/L) (or Imax = Q0/√(LC)). The charge obeys d²q/dt² = −(1/LC) q, so ω = 1/√(LC) and T = 2π√(LC). Charge and current are 90° out of phase (when Q is max, I = 0). In a real lab you’ll see damping (resistance, radiative loss) so amplitude decays. These are exactly the points AP expects: energy conservation, the SHM differential equation, and ω = 1/√(LC). For a quick refresher, check the Topic 13.6 study guide (https://library.fiveable.me/ap-physics-c-e-m/unit-6/6-circuits-with-capacitors-and-inductors-lc-circuits/study-guide/nTgyGcr23xjTIU5I) and try practice problems (https://library.fiveable.me/practice/ap-physics-c-e-m).