LC circuit in AP Physics C: Electricity and Magnetism

In AP Physics C: E&M, an LC circuit is a capacitor connected to an inductor with no resistance. Charge and current oscillate sinusoidally at angular frequency ω = 1/√(LC), and energy swaps between the capacitor's electric field and the inductor's magnetic field.

What is LC circuit?

An LC circuit is just a charged capacitor (C) hooked up to an inductor (L), with no resistor in the loop. The capacitor starts dumping its charge through the inductor. The inductor resists that change in current, so the current builds up gradually. By the time the capacitor is empty, the current is at its maximum and all the energy sits in the inductor's magnetic field. The inductor then keeps the current flowing, which recharges the capacitor with the opposite polarity. Then the whole thing runs in reverse. With no resistance, nothing is lost, so it sloshes back and forth forever.

If you apply Kirchhoff's loop rule, you get d²q/dt² + q/(LC) = 0. That's the exact same form as the equation for a mass on a spring. So the charge is q(t) = Q₀cos(ωt + φ), with ω = 1/√(LC) and period T = 2π√(LC). The energy split comes from U_C = q²/(2C) and U_L = ½LI², and their sum stays constant at Q₀²/(2C).

Why LC circuit matters in AP® Physics C: E&M

LC circuits are the focus of Topic 13.6 (Circuits with Capacitors and Inductors) in Unit 13: Electromagnetic Induction. This is where the course pulls everything together. You use the capacitor energy from Unit 10, the loop rule from Unit 11, and the self-inductance and back emf from earlier in Unit 13, all in one system. It's also the cleanest place on the E&M exam to show calculus-based reasoning. You set up a second-order differential equation, recognize it as simple harmonic motion, and read off the frequency. Expect it to be tested through energy conservation, the differential equation, and frequency calculations.

How LC circuit connects across the course

Simple Harmonic Motion (Unit 13)

An LC circuit is a mass-spring system made of electricity. Charge q plays the role of position x, current I plays velocity v, inductance L acts like mass (it resists changes in motion), and 1/C acts like the spring constant k. Swap those into ω = √(k/m) and you get ω = 1/√(LC) for free.

Inductor and Self-Inductance (Unit 13)

The inductor's back emf, ε = −L(dI/dt), is what makes oscillation possible. It fights any change in current, so the current can't stop instantly when the capacitor empties. That 'electrical inertia' carries the charge past zero and recharges the capacitor the other way.

Capacitor Energy Storage (Unit 10)

The starting energy in an LC circuit is just the capacitor energy you learned in Unit 10, U = ½CV² = Q²/(2C). Set that equal to ½LI²max and you can find the peak current without touching a differential equation.

Kirchhoff's Loop Rule (Unit 11)

The LC differential equation comes straight from the loop rule. The capacitor voltage q/C and the inductor voltage L(dI/dt) must add to zero around the loop. Write I = dq/dt and the SHM equation drops right out.

Is LC circuit on the AP® Physics C: E&M exam?

LC circuits show up in a few predictable ways.

• Max current from energy conservation. A typical stem gives a 5.0 μF capacitor charged to 12 V and connected to a 2.0 mH inductor, then asks for the peak current. Set ½CV² = ½LI², so I = V√(C/L) = 12√(5.0×10⁻⁶ / 2.0×10⁻³) = 0.60 A.
• Angular frequency. With 5.0 μF and 2.0 H, ω = 1/√(LC) = 1/√(1.0×10⁻⁵) ≈ 316 rad/s. Notice that the initial voltage doesn't matter for frequency, just like amplitude doesn't change a spring's period.
• Picking the differential equation. You may need to identify d²q/dt² + q/(LC) = 0 as the correct equation. Watch for distractors with a first-derivative term (that's damping from a resistor) or the wrong sign.
• Energy sharing at a phase angle. If q(t) = Q₀cos(ωt + φ), the energy is split equally when q²/(2C) is half the total. That means cos²(ωt + φ) = ½, so the phase is π/4 (or 3π/4, 5π/4, ...).

No released FRQ has used the term verbatim, but FRQs on circuits with inductors reward the same skills. You derive an equation from the loop rule, justify the behavior with energy, and sketch how q or I varies with time.

LC circuit vs RC and RL circuits

RC and RL circuits have a resistor, and their behavior is exponential. Charge or current rises or decays toward a final value with a time constant (τ = RC or τ = L/R), and it never reverses. An LC circuit has no resistor, so nothing dissipates energy and the behavior is sinusoidal. The charge swings back and forth past zero at ω = 1/√(LC). Here's a quick check. If the equation is first-order (only dq/dt), it's exponential. If it's second-order with no first-derivative term, it oscillates.

Key things to remember about LC circuit

• An LC circuit oscillates because the inductor keeps current flowing after the capacitor empties, which recharges the capacitor with the opposite polarity.

• The governing equation is d²q/dt² + q/(LC) = 0, which has the same form as simple harmonic motion, so the charge varies as q(t) = Q₀cos(ωt + φ).

• The angular frequency is ω = 1/√(LC) and the period is T = 2π√(LC), and neither depends on the initial charge or voltage.

• Total energy stays constant, so Q₀²/(2C) = ½LI²max, which gives the maximum current as Imax = ωQ₀ = V√(C/L).

• Current is maximum when the capacitor's charge is zero, and current is zero when the charge is maximum, so q and I are 90° out of phase.

• Energy is split equally between the capacitor and inductor at a phase angle of π/4 (and every quarter-cycle after that), not at the halfway point in time between extremes of energy.

Frequently asked questions about LC circuit

What is an LC circuit in AP Physics C E&M?

It's a capacitor connected to an inductor with no resistance, covered in Topic 13.6 of Unit 13. Charge and current oscillate sinusoidally at ω = 1/√(LC) as energy moves back and forth between the capacitor's electric field and the inductor's magnetic field.

Does the frequency of an LC circuit depend on how much the capacitor is charged?

No. The angular frequency is ω = 1/√(LC), so it depends only on L and C. Charging the capacitor to 24 V instead of 12 V doubles the maximum current but leaves the frequency unchanged, just like a bigger amplitude doesn't change a spring's period.

Is current maximum when the capacitor is fully charged in an LC circuit?

No, it's the opposite. When the capacitor holds its maximum charge Q₀, the current is zero. The current peaks at Imax = ωQ₀ at the instant the capacitor's charge passes through zero.

How is an LC circuit different from an RC or RL circuit?

RC and RL circuits include a resistor, so they decay or grow exponentially with a time constant (RC or L/R) and never oscillate. An LC circuit has no resistor, so it oscillates sinusoidally forever with no energy loss.

How is an LC circuit like a mass on a spring?

Both obey the same differential equation. Charge maps to position, current to velocity, L to mass, and 1/C to the spring constant, so ω = √(k/m) becomes ω = 1/√(LC). Capacitor energy q²/(2C) matches spring energy ½kx², and inductor energy ½LI² matches kinetic energy ½mv².