13.6 Circuits with Capacitors and Inductors (LC Circuits)

An LC circuit pairs a charged capacitor with an inductor, and the two trade energy back and forth so charge and current oscillate like simple harmonic motion. The charge follows the equation d^2q/dt^2 = -(1/LC)q, which gives an angular frequency of omega = 1/sqrt(LC).

Why This Matters for the AP Physics C: E&M Exam

LC circuits are where induction meets oscillation, and the AP Physics C: E&M exam likes to test whether you can connect circuit behavior to the math of simple harmonic motion. Both the multiple-choice and free-response sections ask you to identify and analyze functional relationships, so expect to reason about how omega, the period, and the maximum current change when L or C changes.

This topic rewards you for working across representations. You may need to sketch or interpret graphs of charge and current versus time, set up the differential equation from Kirchhoff's loop rule, and use energy conservation to solve for an unknown. The mass-spring analogy (L behaves like mass, 1/C behaves like the spring constant) helps you predict behavior quickly, which is useful when a question wants comparison rather than a full calculation.

Key Takeaways

• In an ideal LC circuit, total energy stays constant and shifts between the capacitor's electric field and the inductor's magnetic field.
• Set (1/2)C V_max^2 equal to (1/2)L I_max^2 to find the maximum current, because all the energy is in the inductor when the capacitor is fully discharged.
• The charge obeys d^2q/dt^2 = -(1/LC)q, the same form as simple harmonic motion, so q(t) is sinusoidal.
• The angular frequency is omega = 1/sqrt(LC), and the period is T = 2pisqrt(LC). Larger L or C means slower oscillation.
• Current is the derivative of charge, so i(t) is sinusoidal and 90 degrees out of phase with q(t): when charge is maximum, current is zero, and vice versa.
• If there are several capacitors, combine them into an equivalent capacitance first, then apply the standard LC relationships.

Properties of Capacitor-Inductor Circuits

Conservation of Energy in LC Circuits

In an LC circuit, energy continuously transfers back and forth between the capacitor and inductor while the total energy stays constant. That energy exchange is the heart of how the circuit oscillates.

• When the capacitor is fully charged, all energy exists as electric potential energy in the field between the capacitor plates.
• As the capacitor discharges, current flows through the inductor, building a magnetic field.
• Energy gradually moves from the capacitor to the inductor's magnetic field.
• Once the capacitor is fully discharged, all energy resides in the inductor's magnetic field.
• The process reverses, with the inductor's collapsing magnetic field pushing charge back onto the capacitor.

The maximum current in the inductor can be found using energy conservation:

$U_L = \frac{1}{2}LI_{max}^2$ (energy stored in inductor)

$U_C = \frac{1}{2}CV_{max}^2$ (energy stored in capacitor)

Since energy is conserved: $\frac{1}{2}LI_{max}^2 = \frac{1}{2}CV_{max}^2$

Solving for the maximum current: $I_{max} = V_{max}\sqrt{\frac{C}{L}}$

LC Circuits with Multiple Capacitors

If a problem includes multiple capacitors, replace them with an equivalent capacitance before analyzing the LC oscillation. The key LC ideas stay the same: energy conservation, the differential equation $\frac{d^2 q}{dt^2} = -\frac{1}{LC}q$, and the angular frequency $\omega = \frac{1}{\sqrt{LC}}$.

For example, if a circuit contains two capacitors in parallel (with capacitances $C_1$ and $C_2$) connected to a single inductor $L$, first find the equivalent capacitance $C_{eq} = C_1 + C_2$. Then every standard LC relationship applies using $C_{eq}$ in place of $C$: the circuit oscillates with angular frequency $\omega = \frac{1}{\sqrt{LC_{eq}}}$, and energy conservation gives $\frac{1}{2}LI_{max}^2 = \frac{1}{2}C_{eq}V_{max}^2$. A larger equivalent capacitance leads to a lower angular frequency and a longer period.

Simple Harmonic Motion in LC Circuits

The charge on the capacitor follows the same math as a mass on a spring, which makes the LC circuit a clean example of simple harmonic motion in an electrical system.

Apply Kirchhoff's loop rule around the circuit: $L\frac{dI}{dt} + \frac{q}{C} = 0$

Since current is the rate of change of charge ($I = \frac{dq}{dt}$), substitute and rearrange: $\frac{d^{2}q}{dt^{2}} = -\frac{1}{LC}q$

This has the same form as the equation for simple harmonic motion: $\frac{d^{2}x}{dt^{2}} = -\omega^2x$

The solution gives charge as a function of time: $q(t) = q_{max}\cos(\omega t + \phi)$

Where:

• $q_{max}$ is the maximum charge on the capacitor
• $\omega$ is the angular frequency of oscillation
• $\phi$ is the phase constant set by initial conditions

Find the current by differentiating the charge equation: $I(t) = \frac{dq}{dt} = -\omega q_{max}\sin(\omega t + \phi)$

Notice the sine and cosine: charge and current are 90 degrees out of phase. When charge is at its maximum, current is zero, and when charge is zero, current is at its maximum.

Angular Frequency of LC Circuits

The angular frequency tells you how quickly energy oscillates between the capacitor and inductor. It depends only on the values of inductance and capacitance.

From the differential equation $\frac{d^{2}q}{dt^{2}} = -\frac{1}{LC}q$, you can read off: $\omega^2 = \frac{1}{LC}$

Taking the square root: $\omega = \frac{1}{\sqrt{LC}}$

This relationship reveals a few key points:

• Larger capacitance or inductance values produce slower oscillations.
• Smaller capacitance or inductance values produce faster oscillations.
• The frequency depends on the inverse square root of the product LC.

The period of oscillation (time for one full cycle) is: $T = \frac{2\pi}{\omega} = 2\pi\sqrt{LC}$

How to Use This on the AP Physics C: E&M Exam

Problem Solving

• Start energy problems by writing total energy as (1/2)C V_max^2 when the capacitor is fully charged. Set it equal to (1/2)L I_max^2 to solve for the maximum current.
• For frequency and period, plug directly into omega = 1/sqrt(LC) and T = 2pisqrt(LC). Watch your units: capacitance in farads, inductance in henries.
• If a charge equation like q(t) = q_max cos(omega t) is given, read q_max and omega straight off the expression, then use omega to back out L or C.
• When there are multiple capacitors, combine them into C_eq first, then proceed as a single-capacitor LC circuit.

Free Response

• Derive the differential equation from Kirchhoff's loop rule rather than just stating it. Show the substitution I = dq/dt clearly so your logic is easy to follow.
• When asked to compare scenarios, lean on functional dependence. For example, doubling C multiplies the period by sqrt(2), not by 2.
• If asked to sketch q(t) and i(t), make sure the current curve is shifted a quarter period from the charge curve, with current peaking where charge crosses zero.

Common Trap

• Do not assume current is maximum when charge is maximum. They are 90 degrees out of phase, so one peaks while the other is zero.

Practice Problem 1: Conservation of Energy in LC Circuits

Solution

(a) The maximum energy stored in the circuit is initially all in the capacitor: $U_{max} = \frac{1}{2}CV^2 = \frac{1}{2}(5.0 \times 10^{-6} \text{ F})(12 \text{ V})^2 = 3.6 \times 10^{-4} \text{ J}$

(b) To find the maximum current, use energy conservation. When all energy transfers to the inductor: $\frac{1}{2}LI_{max}^2 = 3.6 \times 10^{-4} \text{ J}$

Solving for $I_{max}$: $I_{max} = \sqrt{\frac{2 \times 3.6 \times 10^{-4} \text{ J}}{30 \times 10^{-3} \text{ H}}} = 0.155 \text{ A}$

Alternatively, using the formula $I_{max} = V_{max}\sqrt{\frac{C}{L}}$: $I_{max} = (12 \text{ V})\sqrt{\frac{5.0 \times 10^{-6} \text{ F}}{30 \times 10^{-3} \text{ H}}} = 0.155 \text{ A}$

(c) The frequency of oscillation is: $f = \frac{\omega}{2\pi} = \frac{1}{2\pi\sqrt{LC}} = \frac{1}{2\pi\sqrt{(30 \times 10^{-3} \text{ H})(5.0 \times 10^{-6} \text{ F})}} = 411 \text{ Hz}$

Practice Problem 2: Simple Harmonic Motion in LC Circuits

Solution

(a) From the equation $q(t) = (3.0 \times 10^{-5}) \cos(400t)$, the maximum charge is: $q_{max} = 3.0 \times 10^{-5} \text{ C}$

(b) The angular frequency is the coefficient of t in the cosine function: $\omega = 400 \text{ rad/s}$

(c) The period of oscillation is: $T = \frac{2\pi}{\omega} = \frac{2\pi}{400 \text{ rad/s}} = 0.0157 \text{ s} = 15.7 \text{ ms}$

(d) Using the formula $\omega = \frac{1}{\sqrt{LC}}$, solve for $L$:

$400 \text{ rad/s} = \frac{1}{\sqrt{L(4.0 \times 10^{-6} \text{ F})}}$

Squaring both sides:

$(400)^2 = \frac{1}{L(4.0 \times 10^{-6})}$

So,

$L = \frac{1}{(400)^2(4.0 \times 10^{-6})} = \frac{1}{160000 \times 4.0 \times 10^{-6}} = \frac{1}{0.64} = 1.56 \text{ H}$

Therefore, the inductance is $\boxed{1.56 \text{ H}}$.

Common Misconceptions

• Charge and current do not peak at the same time. They are a quarter cycle apart, so when the capacitor holds maximum charge, the current is zero.
• An ideal LC circuit does not lose energy or fade out. Without a resistor, the oscillation continues at constant amplitude. Damping only appears when resistance is present.
• The angular frequency omega = 1/sqrt(LC) depends only on L and C, not on the initial charge or voltage. Changing how much you charge the capacitor changes the amplitude, not the frequency.
• Energy is not harmed when the capacitor fully discharges. It is stored in the inductor's magnetic field at that instant, then handed back to the capacitor.
• omega is angular frequency in rad/s, not the ordinary frequency f in Hz. Use f = omega/(2*pi) when a question asks for frequency in hertz.
• Doubling C does not double the period. Because the period depends on the square root of LC, doubling C multiplies the period by sqrt(2).

Related AP Physics C: E&M Guides

• 13.2 Electromagnetic Induction
• 13.3 Induced Currents and Magnetic Forces
• 13.4 Inductance
• 13.5 Circuits with Resistors and Inductors (LR Circuits)
• 13.1 Magnetic Flux