Circuits with Capacitors and Inductors (LC Circuits)

When you hear the phrase LC circuit, think of a system that can store and trade energy back and forth ⚡. The $L$ stands for an inductor, which stores energy in a magnetic field, and the $C$ stands for a capacitor, which stores energy in an electric field. In this lesson, students, you will learn how these parts work together to make oscillations, why the motion repeats, and how this topic connects to electromagnetic induction.

What an LC Circuit Does

An LC circuit is a circuit with only an inductor and a capacitor, usually with very small resistance. The key idea is that energy moves between two forms:

energy in the capacitor’s electric field
energy in the inductor’s magnetic field

A charged capacitor starts with electric potential energy. When the circuit is closed, charge begins to flow. That current creates a magnetic field in the inductor. Because of self-induction, the inductor resists changes in current. This resistance does not stop the current; instead, it helps create a back-and-forth motion. The current can keep going even after the capacitor is momentarily uncharged, because the inductor keeps the current moving 🔄.

The result is oscillation: the charge on the capacitor, current in the circuit, and stored energy all change with time in a repeating way.

For an ideal LC circuit, the total energy stays constant:

$$U = U_C + U_L = \frac{q^2}{2C} + \frac{1}{2}LI^2$$

Here, $q$ is the charge on the capacitor, $C$ is capacitance, $L$ is inductance, and $I$ is the current.

How the Oscillation Works Step by Step

Let’s walk through one cycle using a simple example.

1. The capacitor begins fully charged

Suppose the capacitor has charge $+Q$ on one plate and $-Q$ on the other. At this instant:

the electric field in the capacitor is strong
the current is $I=0$
energy is mainly stored in the capacitor

The capacitor wants to discharge, so charge flows through the circuit.

2. Current increases and the inductor stores energy

As charge moves, the current grows. The inductor creates a magnetic field, and energy starts building in that field. The inductor opposes sudden changes in current, so the current does not jump instantly to its maximum.

The voltage across an inductor is

$$V_L = L\frac{dI}{dt}$$

This relationship shows why inductors matter in changing circuits: if the current is changing quickly, the induced voltage is large.

3. The capacitor becomes momentarily uncharged

At the moment the capacitor’s charge reaches $q=0$:

the electric field in the capacitor is zero
the current is at its maximum
the magnetic energy in the inductor is greatest

Even though the capacitor is uncharged, the current keeps flowing because the magnetic field in the inductor is still active.

4. The capacitor recharges with opposite polarity

The inductor keeps the current going, so charge continues moving and builds up the opposite polarity on the capacitor. Now the capacitor stores energy again, but with reversed charge. Eventually the current slows to zero, and the capacitor is fully charged with opposite sign.

Then the whole process repeats. This is why LC circuits can oscillate like a swinging pendulum: energy moves back and forth between two storage forms 🌟.

Mathematical Model of an Ideal LC Circuit

To describe the motion, we use a charge function $q(t)$ for the capacitor. The current is the time rate of change of charge:

$$I(t)=\frac{dq}{dt}$$

Using Kirchhoff’s loop rule for an ideal LC circuit gives

$$V_L + V_C = 0$$

Substituting the inductor and capacitor voltages gives

$$L\frac{dI}{dt} + \frac{q}{C} = 0$$

Since $I=\frac{dq}{dt}$, this becomes

$$L\frac{d^2q}{dt^2} + \frac{q}{C} = 0$$

$$\frac{d^2q}{dt^2} + \frac{1}{LC}q = 0$$

This is the same type of equation as simple harmonic motion. So the charge oscillates sinusoidally:

$$q(t)=Q\cos(\omega t + \phi)$$

where the angular frequency is

$$\omega = \frac{1}{\sqrt{LC}}$$

and the period is

$$T = 2\pi\sqrt{LC}$$

The frequency is

$$f=\frac{1}{2\pi\sqrt{LC}}$$

This tells you something important: larger $L$ or larger $C$ makes the oscillation slower.

Energy Changes During One Full Cycle

Energy in an ideal LC circuit never disappears; it only changes form.

Capacitor energy

The energy stored in a capacitor is

$$U_C=\frac{q^2}{2C}$$

This is largest when $|q|$ is largest.

Inductor energy

The energy stored in an inductor is

$$U_L=\frac{1}{2}LI^2$$

This is largest when the current is largest.

During the cycle:

when $q$ is maximum, $I=0$, so $U_C$ is maximum and $U_L=0$
when $q=0$, $I$ is maximum, so $U_L$ is maximum and $U_C=0$
at intermediate times, both forms share the total energy

A useful real-world picture is a playground swing 🎢. At the ends of the swing, speed is zero and gravitational potential energy is maximum. At the bottom, speed is maximum and potential energy is minimum. LC circuits behave similarly, except the energy shifts between electric and magnetic fields.

Important AP Physics Reasoning Skills

On AP Physics C: Electricity and Magnetism, you need more than memorization. You must connect equations, graphs, and physical meaning.

1. Recognize the analogy to SHM

The equation

$$\frac{d^2q}{dt^2} + \frac{1}{LC}q = 0$$

has the same form as the equation for spring motion:

$$\frac{d^2x}{dt^2} + \omega^2x = 0$$

So charge in an LC circuit behaves like position in a spring system. This helps with graph questions and concept questions.

2. Use energy ideas

If the circuit is ideal, total energy remains constant. If the capacitor starts with energy

$$U_0=\frac{Q^2}{2C}$$

then at any time

$$\frac{q^2}{2C}+\frac{1}{2}LI^2=\frac{Q^2}{2C}$$

This equation is powerful for solving for current when charge is known, or charge when current is known.

3. Understand directions and signs

If the capacitor’s charge is decreasing, then $\frac{dq}{dt}$ has the sign of the current. If the charge is positive on one plate, current direction depends on how you define the circuit direction. In AP problems, always be consistent with your sign convention.

4. Interpret graphs

If $q(t)$ is a cosine curve, then $I(t)=\frac{dq}{dt}$ is a sine curve shifted by one-quarter cycle. That means:

when $q$ is maximum, $I=0$
when $q=0$, $I$ is maximum in magnitude
when $q$ is increasing most quickly, $I$ is positive and largest

This graph relationship often appears in multiple-choice and free-response problems.

Connection to Electromagnetic Induction

LC circuits belong in the topic of electromagnetic induction because the inductor depends on changing magnetic flux. A changing current in the inductor changes the magnetic field, which induces a voltage that opposes the change. That is the heart of Faraday’s law and Lenz’s law.

In symbols, magnetic flux changing with time produces an induced emf:

$$\mathcal{E} = -\frac{d\Phi_B}{dt}$$

The negative sign shows Lenz’s law: the induced effect opposes the change that causes it.

In an inductor, the changing current changes magnetic flux, so the inductor creates a self-induced emf:

$$V_L = L\frac{dI}{dt}$$

This is why the inductor can sustain oscillations. Without induction, the capacitor would simply discharge and stop. The inductor stores the motion of charge in magnetic form and pushes the system toward the opposite state.

So LC circuits are not just isolated circuits. They are a direct example of how electric and magnetic fields interact through time-varying currents and induced emf.

A Practical Example

Imagine a flash-camera circuit or a tuning circuit in a radio 📻. A capacitor can store energy and release it quickly. In a radio tuner, an LC circuit can be designed so that its natural frequency matches a desired station’s signal frequency. Since

$$f=\frac{1}{2\pi\sqrt{LC}}$$

you can adjust $L$ or $C$ to change the resonant frequency.

If $L$ increases, then $f$ decreases. If $C$ increases, then $f$ also decreases. This makes LC circuits useful for selecting signals in electronics.

Conclusion

students, the main idea of an LC circuit is simple but powerful: energy moves back and forth between a capacitor’s electric field and an inductor’s magnetic field. The capacitor provides electric potential energy, the inductor provides magnetic energy and resists changes in current, and together they create oscillations. The mathematics matches simple harmonic motion, with period

$$T=2\pi\sqrt{LC}$$

and frequency

$$f=\frac{1}{2\pi\sqrt{LC}}$$

This topic fits directly into electromagnetic induction because the inductor’s behavior depends on changing current and induced emf. Understanding LC circuits helps you solve AP Physics C problems involving energy, oscillation, and the interaction between electric and magnetic fields.

Study Notes

An LC circuit contains an inductor $L$ and a capacitor $C$.
Energy oscillates between capacitor energy $U_C=\frac{q^2}{2C}$ and inductor energy $U_L=\frac{1}{2}LI^2$.
In an ideal LC circuit, total energy stays constant: $$\frac{q^2}{2C}+\frac{1}{2}LI^2=\text{constant}$$
The charge satisfies $$\frac{d^2q}{dt^2}+\frac{1}{LC}q=0$$
The angular frequency is $$\omega=\frac{1}{\sqrt{LC}}$$
The period is $$T=2\pi\sqrt{LC}$$
The frequency is $$f=\frac{1}{2\pi\sqrt{LC}}$$
The inductor voltage is $$V_L=L\frac{dI}{dt}$$
LC circuits are connected to electromagnetic induction through self-induced emf and Lenz’s law.
Maximum capacitor charge means zero current; maximum current means zero capacitor charge.
Bigger $L$ or bigger $C$ makes the oscillation slower.