Key Concepts of Fundamental LC Oscillations

Why This Matters

LC circuits represent one of the most elegant examples of energy conservation and simple harmonic motion in electromagnetism. When you study these circuits, you're not just learning about inductors and capacitors—you're seeing how energy storage, oscillatory behavior, and differential equations all connect in a system that mirrors mechanical oscillators like mass-spring systems. The AP exam loves testing whether you can move fluidly between these analogies and apply conservation principles to predict circuit behavior.

Don't just memorize the formulas for charge and current oscillations—understand why energy sloshes back and forth between electric and magnetic fields, how the phase relationships emerge from the underlying differential equation, and what each variable represents physically. If an FRQ asks you to derive the oscillation frequency or explain energy transfer, you need to connect the math to the mechanism. You've got this—these concepts click once you see the pattern.

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Energy Storage and Exchange

The foundation of LC oscillations lies in how inductors and capacitors store energy in fundamentally different forms. The capacitor stores energy in an electric field between its plates, while the inductor stores energy in a magnetic field created by current flow. This complementary storage mechanism enables continuous energy exchange.

Energy in the Capacitor

• Electric field energy $U_C = \frac{1}{2}CV^2 = \frac{q^2}{2C}$—maximum when charge is maximum, zero when capacitor is fully discharged
• Voltage across capacitor $V_C(t) = \frac{q(t)}{C}$—directly proportional to instantaneous charge, peaks when energy is fully stored in the electric field
• Initial conditions typically set $Q_0$ as the starting charge—this determines the total energy available for oscillation

Energy in the Inductor

• Magnetic field energy $U_L = \frac{1}{2}LI^2$—maximum when current is maximum, zero when current momentarily stops
• Voltage across inductor $V_L = L\frac{dI}{dt}$—proportional to rate of current change, not to current itself
• Energy transfer occurs as capacitor discharge drives current through the inductor—the inductor "resists" current changes by storing energy magnetically

Conservation of Total Energy

• Total energy remains constant in an ideal LC circuit: $U_{total} = \frac{1}{2}CV^2 + \frac{1}{2}LI^2 = \frac{Q_0^2}{2C}$
• Maximum current occurs when all energy transfers to the inductor: $I_{max} = \frac{Q_0}{\sqrt{LC}}$—derived directly from energy conservation
• No dissipation in the ideal case—real circuits include resistance that causes damping over time

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The Differential Equation and Its Solutions

The oscillatory behavior of LC circuits emerges from a second-order differential equation identical in form to the mass-spring equation. Applying Kirchhoff's loop rule around the circuit yields $\frac{d^2q}{dt^2} = -\frac{q}{LC}$, which is the defining equation for simple harmonic motion.

Angular Frequency and Period

• Angular frequency $\omega = \frac{1}{\sqrt{LC}}$—determined entirely by circuit components, independent of initial conditions
• Resonant frequency $f = \frac{1}{2\pi\sqrt{LC}}$—the natural frequency at which the circuit oscillates without external driving
• Period of oscillation $T = 2\pi\sqrt{LC}$—time for one complete cycle of charge oscillation

Charge Oscillation

• General solution $q(t) = Q_0\cos(\omega t + \phi)$—describes sinusoidal variation of capacitor charge with time
• Amplitude $Q_0$ represents maximum charge—set by initial conditions when circuit begins oscillating
• Phase constant $\phi$ depends on initial conditions—typically zero if capacitor starts fully charged

Current Oscillation

• Current as derivative $I(t) = \frac{dq}{dt} = -\omega Q_0\sin(\omega t + \phi)$—found by differentiating the charge equation
• Maximum current $I_{max} = \omega Q_0 = \frac{Q_0}{\sqrt{LC}}$—occurs when charge passes through zero
• 90° phase lag between current and charge—current peaks when charge is zero, not when charge is maximum

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Phase Relationships and Timing

Understanding when each quantity reaches its maximum or zero is crucial for exam success. The phase relationships in LC circuits follow directly from the mathematical relationship between position (charge) and velocity (current) in simple harmonic motion.

Charge-Current Phase Difference

• Current leads charge by 90°—when $q = Q_0$ (maximum), $I = 0$; when $q = 0$, $I = \pm I_{max}$
• Physical interpretation: current must flow before charge can accumulate—charge is the integral of current over time
• Graphically: current curve is shifted left by $T/4$ relative to charge curve on a time plot

Voltage Phase Relationships

• Capacitor voltage in phase with charge: $V_C = q/C$, so both peak and zero together
• Inductor voltage in phase with current derivative: $V_L = L(dI/dt)$, peaks when current is changing fastest (at $I = 0$)
• Inductor and capacitor voltages are 180° out of phase—when one is at positive maximum, the other is at negative maximum

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Mass-Spring Analogy

The mathematical equivalence between LC circuits and mechanical oscillators provides powerful physical intuition. Every quantity in the LC circuit has a direct analog in the mass-spring system, making it easier to predict behavior and check your reasoning.

Variable Correspondences

• Inductance $L$ corresponds to mass $m$—both represent inertia that resists changes in motion (current or velocity)
• $1/C$ corresponds to spring constant $k$—both represent "stiffness" that provides restoring force proportional to displacement
• Charge $q$ corresponds to position $x$—both are the primary oscillating variables in their respective systems

Energy Correspondences

• Capacitor energy $\frac{q^2}{2C}$ maps to spring potential energy $\frac{1}{2}kx^2$—stored when system is at maximum displacement
• Inductor energy $\frac{1}{2}LI^2$ maps to kinetic energy $\frac{1}{2}mv^2$—stored when system passes through equilibrium
• Angular frequency $\omega = \frac{1}{\sqrt{LC}}$ parallels $\omega = \sqrt{k/m}$—note the reciprocal placement of $C$

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Real-World Considerations

Ideal LC circuits oscillate forever, but real circuits include resistance that dissipates energy. Understanding damping and applications connects textbook physics to practical engineering.

Damping Effects

• Resistance introduces energy loss—oscillation amplitude decreases exponentially over time in RLC circuits
• Quality factor $Q$ measures how "sharp" the resonance is—higher $Q$ means less damping and more sustained oscillations
• Underdamped oscillations still occur if resistance is small—the circuit rings down gradually rather than stopping immediately

Applications in Electronics

• Radio tuning circuits use variable capacitors to adjust $\omega = \frac{1}{\sqrt{LC}}$ and select specific broadcast frequencies
• Oscillators and filters rely on LC resonance to generate or isolate signals at particular frequencies
• Signal processing exploits the frequency-selective nature of LC circuits—only signals near resonance pass through efficiently

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Quick Reference Table

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Self-Check Questions

1. At the instant when the capacitor in an LC circuit holds its maximum charge $Q_0$, what is the current through the inductor, and where is all the energy stored?

2. Compare the phase relationship between $q(t)$ and $I(t)$ in an LC circuit. If you sketch both on the same time axis starting from $t = 0$ with $q(0) = Q_0$, which curve crosses zero first?

3. Using the mass-spring analogy, explain why doubling the inductance $L$ while keeping $C$ constant causes the oscillation period to increase by a factor of $\sqrt{2}$.

4. An FRQ asks you to derive the expression for maximum current $I_{max}$ in terms of $Q_0$, $L$, and $C$. Which conservation principle should you apply, and what equation results?

5. Compare an ideal LC circuit to a real RLC circuit. How does the presence of resistance affect (a) the oscillation frequency and (b) the amplitude over time?