#LC Circuits: Your Ultimate Guide ⚡

Hey there, future AP Physics C: E&M master! Let's dive into LC circuits – those awesome oscillating systems that play a huge role in electromagnetism. This guide is designed to be your go-to resource the night before the exam, so let's make every minute count!

#Overview of LC Circuits

LC circuits are all about the interplay between capacitors (which store energy in electric fields) and inductors (which store energy in magnetic fields). They're like a playground for energy, constantly swapping back and forth. Understanding these circuits is key to grasping more complex electrical systems.

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• Energy Transfer: Energy oscillates between the capacitor and the inductor.
• Simple Harmonic Motion: The charge and current in the circuit vary sinusoidally.
• Angular Frequency: The frequency of oscillation depends on the inductance and capacitance.

#Properties of LC Circuits

# Conservation of Energy in LC Circuits

Imagine a swing set – energy goes back and forth between potential and kinetic. In an LC circuit, it's the same, but with electric and magnetic fields! 🔋

• Total Energy is Constant: The total energy in the circuit remains constant, transferring between the capacitor and the inductor.
• Capacitor Fully Charged: All energy is stored as electric potential energy: $$U_C = \frac{1}{2}CV_{max}^2$$
• Energy Transfer: As the capacitor discharges, energy moves to the inductor, becoming magnetic potential energy.
• Inductor Fully Energized: All energy is stored in the inductor's magnetic field: $$U_L = \frac{1}{2}LI_{max}^2$$
• Reversal: The process reverses, energy goes back to the capacitor.

• Maximum Current Calculation:
  • Equate the initial energy in the capacitor to the maximum energy in the inductor: $$\frac{1}{2}CV_{max}^2 = \frac{1}{2}LI_{max}^2$$
  • Solve for the maximum current: $$I_{max} = V_{max}\sqrt{\frac{C}{L}}$$

# Simple Harmonic Motion in LC Circuits

Think of a mass on a spring – it oscillates back and forth. An LC circuit behaves similarly, with charge oscillating sinusoidally.

• Sinusoidal Oscillation: The charge on the capacitor oscillates with simple harmonic motion.
• Restoring Force: The inductor opposes changes in current, acting as the restoring force.
• Inertia: The capacitor resists changes in voltage, providing the inertia.
• Differential Equation: Applying Kirchhoff's loop rule: $$L\frac{dI}{dt} + \frac{q}{C} = 0$$
• Rearranging: Substituting $I = \frac{dq}{dt}$: $$\frac{d^{2}q}{dt^{2}} = -\frac{1}{LC}q$$
• SHM Equation: This matches the form of simple harmonic motion: $$\frac{d^{2}x}{dt^{2}} = -\omega^2x$$
• Solution: Charge as a function of time: $$q(t) = q_{max}\cos(\omega t + \phi)$$
  • $q_{max}$: Maximum charge
  • $\omega$: Angular frequency
  • $\phi$: Phase constant

# Angular Frequency of LC Circuits

The speed of the oscillation! 📈

• Derivation: From the differential equation, we have: $$\omega^2 = \frac{1}{LC}$$
• Angular Frequency: $$\omega = \frac{1}{\sqrt{LC}}$$
• Dependence: $\omega$ depends only on inductance ($L$) and capacitance ($C$).
• Increasing L or C: Decreases $\omega$ (slower oscillation).
• Decreasing L or C: Increases $\omega$ (faster oscillation).
• Period of Oscillation: $$T = \frac{2\pi}{\omega} = 2\pi\sqrt{LC}$$
  • Period is proportional to $\sqrt{LC}$.

#Final Exam Focus

Okay, let's zoom in on what's most important for the exam. Focus on these areas:

• Energy Conservation: Be ready to calculate energy transfers and maximum current/voltage.
• SHM Equations: Know how to relate the differential equation to simple harmonic motion.
• Angular Frequency: Master the formula and how it changes with L and C.

#Common Question Types

• Multiple Choice: Conceptual questions about energy transfer, frequency changes, and SHM.
• Free Response: Deriving equations, calculating values, and explaining circuit behavior.

#Last-Minute Tips

• Review Formulas: Quickly refresh your memory on key equations.
• Practice Problems: Do a few practice problems to get into the groove.
• Stay Calm: Take deep breaths, you've got this!

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Practice Question

Practice Questions

#Multiple Choice Questions

1. An LC circuit has an inductance of 2.0 mH and a capacitance of 5.0 μF. What is the angular frequency of oscillation? (A) 10 rad/s (B) 100 rad/s (C) 1000 rad/s (D) 10000 rad/s

2. In an LC circuit, if the capacitance is doubled and the inductance is halved, what happens to the angular frequency? (A) It doubles (B) It halves (C) It remains the same (D) It is multiplied by $\sqrt{2}$

3. When the current in an LC circuit is at its maximum, what is the energy stored in the capacitor? (A) Maximum (B) Minimum (C) Half the total energy (D) Zero

#Free Response Question

Question:

A series LC circuit consists of a 10 mH inductor and a 25 μF capacitor. Initially, the capacitor is charged to 10 V, and there is no current flowing in the circuit.

(a) Calculate the initial energy stored in the capacitor. (b) Calculate the maximum current in the circuit. (c) Determine the angular frequency of the circuit. (d) Write an expression for the charge on the capacitor as a function of time, assuming the charge is at its maximum value at t=0. (e) If a resistor is added in series to the LC circuit, what effect will it have on the oscillations of charge and current in the circuit? Explain your reasoning.

Answer Key:

(a) Initial energy stored in the capacitor: $$U_C = \frac{1}{2}CV^2 = \frac{1}{2}(25 \times 10^{-6} F)(10 V)^2 = 1.25 \times 10^{-3} J$$ (1 point for correct formula, 1 point for correct answer)

(b) Maximum current in the circuit: $$I_{max} = V_{max}\sqrt{\frac{C}{L}} = 10 \sqrt{\frac{25 \times 10^{-6}}{10 \times 10^{-3}}} = 0.5 A$$ (1 point for correct formula, 1 point for correct answer)

(c) Angular frequency of the circuit: $$\omega = \frac{1}{\sqrt{LC}} = \frac{1}{\sqrt{(10 \times 10^{-3} H)(25 \times 10^{-6} F)}} = 2000 rad/s$$ (1 point for correct formula, 1 point for correct answer)

(d) Charge on the capacitor as a function of time: $$q(t) = q_{max}\cos(\omega t)$$ $$q_{max} = CV_{max} = (25 \times 10^{-6} F)(10 V) = 2.5 \times 10^{-4} C$$ $$q(t) = (2.5 \times 10^{-4} C)\cos(2000t)$$ (1 point for correct formula, 1 point for correct answer)

(e) Effect of adding a resistor: Adding a resistor will cause the oscillations to be damped. The energy will gradually be dissipated as heat in the resistor, causing the amplitude of the oscillations to decrease over time. (1 point for identifying damping, 1 point for correct explanation)

You've got this! Go ace that exam! 💪