AP Physics C E&M: LC Circuits

An LC Circuit, consisting of just an inductor and a capacitor, is the electromagnetic equivalent of a frictionless mass on a spring. Understanding its oscillatory behavior is crucial because it forms the heart of every radio tuner, forms the basis for understanding more complex AC circuits, and provides a perfect illustration of the deep analogies between different branches of physics. Mastering this topic requires you to move seamlessly between differential equations, energy principles, and physical intuition.

The Setup and Qualitative Behavior

Consider a simple loop containing an inductor with inductance $L$ and a capacitor with capacitance $C$. Assume the wires have negligible resistance. If you charge the capacitor initially to some maximum charge $Q_{max}$ and then close the circuit, you initiate electromagnetic oscillations.

The process is cyclical: The charged capacitor begins to discharge, creating a current in the inductor. As current builds, energy stored in the capacitor's electric field decreases while energy in the inductor's magnetic field increases. When the capacitor is fully discharged, the current is at its maximum. However, the inductor, opposing any change in current, then acts to sustain the current, which proceeds to re-charge the capacitor with opposite polarity. This energy sloshes back and forth indefinitely in an ideal, lossless system. This continuous, periodic transfer of energy between the electric field of the capacitor and the magnetic field of the inductor is the hallmark of an LC oscillation.

Deriving the Governing Differential Equation

We derive the equation of motion by applying Kirchhoff's loop rule: the sum of the potential differences around the closed loop is zero. We need expressions for the voltage across each element.

1. Capacitor: The voltage across a capacitor is $V_C=q/C$, where $q$ is the instantaneous charge on the capacitor.
2. Inductor: The voltage across an inductor is $V_L=-L\frac{di}{dt}$. The negative sign indicates it opposes the change in current.

Applying the loop rule: $V_L+V_C=0$. Therefore: $$-L\frac{di}{dt}+\frac{q}{C}=0$$

Since current $i$ is the rate of flow of charge, $i=\frac{dq}{dt}$. This means $\frac{di}{dt}=\frac{d^{2}q}{d{t}^2}$. Substituting this in gives: $$-L\frac{d^{2}q}{d{t}^2}+\frac{q}{C}=0$$

Rearranging into standard form yields the defining differential equation for an ideal LC circuit: $$\frac{d^{2}q}{d{t}^2}+\frac{1}{LC}q=0$$

Solving the Equation and Finding Resonance

The equation $\frac{d^{2}q}{d{t}^2}+\frac{1}{LC}q=0$ is mathematically identical to the equation for simple harmonic motion (SHM), $\frac{d^{2}x}{d{t}^2}+{\omega }^{2}x=0$. By direct comparison, we identify the angular frequency (or resonant frequency) of oscillation as: $$\omega =\frac{1}{\sqrt{LC}}$$ The corresponding frequency in Hertz (cycles per second) is $f=\omega /(2\pi )$.

The general solution to this differential equation is a sinusoidal function: $$q(t)=Q_{max}\cos (\omega t+\varphi )$$ where $Q_{max}$ is the maximum charge (the amplitude) and $\varphi$ is the phase constant determined by initial conditions. For the common case where $q=Q_{max}$ at $t=0$, we have $\varphi =0$, so $q(t)=Q_{max}\cos (\omega t)$.

The current is the derivative of charge: $$i(t)=\frac{dq}{dt}=-\omega Q_{max}\sin (\omega t)=-I_{max}\sin (\omega t)$$ where $I_{max}=\omega Q_{max}$ is the maximum current. Notice the current is $90^{\circ }$ out of phase with the charge; current is maximum when the charge is zero, and vice-versa.

The Mechanical Analogy to SHM

The power of the LC circuit analysis lies in its perfect analogy to a mass-spring system. This analogy provides immense intuitive leverage.

When you solve a problem, ask: "What is the q here (like x), and what is the dq/dt (like v)?" This transforms an abstract circuit problem into a more intuitive mechanical one.

Tracking the Energy Exchange

The total energy in the ideal, lossless LC circuit is constant and is conserved between the capacitor and the inductor. At any time t: $$U_{total}=U_E+U_B=\frac{1}{2C}[q(t){]}^2+\frac{1}{2}L[i(t){]}^2$$ Substituting our solutions $q(t)=Q_{max}\cos (\omega t)$ and $i(t)=-\omega Q_{max}\sin (\omega t)$: $$U_{total}=\frac{1}{2C}(Q_{max}^2{\cos }^2(\omega t))+\frac{1}{2}L({\omega }^2{Q}_{max}^2{\sin }^2(\omega t))$$ Since ${\omega }^2=1/(LC)$, the second term becomes $\frac{1}{2C}{Q}_{max}^2{\sin }^2(\omega t)$. Factoring gives: $$U_{total}=\frac{Q_{max}^2}{2C}({\cos }^2(\omega t)+{\sin }^2(\omega t))=\frac{Q_{max}^2}{2C}$$ This confirms energy conservation. Graphically, $U_E$ and $U_B$ are sinusoidal functions that are $90^{\circ }$ out of phase, so their sum is always a horizontal line. At the instant the capacitor is fully charged, all energy is electric ($U_E=Q_{max}^2/(2C),U_B=0$). A quarter-period later, the capacitor is discharged and current is max, so all energy is magnetic ($U_B=L{I}_{max}^2/2,U_E=0$).

Common Pitfalls

1. Confusing maximum charge and maximum current timing. A classic exam trap is asking for the current when the charge is maximum. Because they are $90^{\circ }$ out of phase, when $q=Q_{max}$, $i=0$. Conversely, when $i=I_{max}$, $q=0$. Remember the energy is all in one form at those extremes.
2. Misapplying the phase constant. The general solution is $q(t)=Q_{max}\cos (\omega t+\varphi )$. If the problem states, "The capacitor is fully charged at $t=0$," then $\varphi =0$. If it says, "The circuit is closed but the capacitor is initially uncharged and a battery charges it through the inductor," the initial conditions are different, leading to a sine solution. Always carefully translate the initial physical state into $q(0)$ and $i(0)$ to solve for $\varphi$.
3. Forgetting the negative sign in Faraday's Law for the inductor. The loop rule equation is $-L(di/dt)+q/C=0$. Dropping the negative sign leads to an exponential, not oscillatory, solution. The negative sign is the source of the restoring "force" that creates oscillation.
4. Misstating the resonant frequency formula. The formula is $\omega =1/\sqrt{LC}$, not $1/(LC)$ or $\sqrt{LC}$. A quick units check can save you: $\sqrt{H\cdot F}$ yields $\sqrt{s^2}=s$, so $1/\sqrt{LC}$ has units of $1/s$ (radians/second), which is correct for angular frequency.

Summary

• An ideal LC circuit oscillates with a resonant angular frequency given by $\omega =1/\sqrt{LC}$, driven by the continuous exchange of energy between the capacitor's electric field and the inductor's magnetic field.
• The charge on the capacitor obeys the SHM differential equation $\frac{d^{2}q}{d{t}^2}+\frac{1}{LC}q=0$, with solutions like $q(t)=Q_{max}\cos (\omega t)$.
• The mechanical analogy is powerful: inductance $L$ analogs to mass $m$ (inertia), and inverse capacitance $1/C$ analogs to spring constant $k$ (restoring force).
• Energy is conserved: $U_{total}=\frac{q^2}{2C}+\frac{1}{2}L{i}^2=\text{constant}$. The energy graphs for $U_E$ and $U_B$ are out of phase sinusoids.
• On exams, pay close attention to initial conditions to determine the phase of oscillation, and remember that current and charge are a quarter-cycle ($90^{\circ }$) out of phase.