Inductance & Circuit Oscillations Explained | Chapter 30 of University Physics

Chapter 30 introduces the concept of inductance—how changing currents and magnetic fields interact to store energy—and examines the dynamic behavior of RL, LC, and LRC circuits. You’ll see how mutual and self-inductance drive real-world applications from smoothing filters to oscillators.

Watch the full video summary here for step-by-step derivations and demonstrations.

Mutual Inductance

When the current in one coil changes, it induces an emf in a nearby coil. This mutual inductance M depends on coil geometry and core material:

• Emf induced in coil 2: ℰ₂ = –M·(di₁/dt)
• Emf induced in coil 1: ℰ₁ = –M·(di₂/dt)
• Definition: M = (N₂·Φ₂)/i₁ = (N₁·Φ₁)/i₂

Self-Inductance & Inductors

A changing current in a coil induces an emf in itself, characterized by the self-inductance L:

• ℰ = –L·(di/dt)
• L = N·Φ/i, where N is the turn count and Φ the magnetic flux per turn
• Inductors resist rapid current changes, making them vital in DC smoothing and AC filtering.

Magnetic Energy & Energy Density

Energy stored in an inductor’s magnetic field is:

U = ½ L·I². In terms of field intensity, the energy density in vacuum is:

u = B²/(2μ₀), and in materials u = B²/(2μ). This stored energy powers ignition coils and pulse circuits.

R-L Circuits

Current Growth (Charging)

When a source E is applied to an R and L in series:

i(t) = (E/R) [1 – e^(–t/τ)], with time constant τ = L/R. The current rises gradually toward its steady value.

Current Decay (Discharging)

When the source is removed, energy in the inductor dissipates through the resistor:

i(t) = I₀·e^(–t/τ), where I₀ is the initial current.

L-C Circuits

Combining an inductor and capacitor yields an LC oscillator, with energy swapping between electric and magnetic fields. The natural angular frequency is:

ω = 1/√(L·C), analogous to a mass-spring system, producing sinusoidal current and voltage oscillations.

L-R-C Series Circuits

Adding resistance introduces damping. Solve via Kirchhoff’s loop rule for three regimes:

• Underdamped (R² < 4L/C): oscillations decay with ω′ = √(1/LC – R²/4L²)
• Critically damped (R² = 4L/C): fastest return with no oscillation
• Overdamped (R² > 4L/C): slow, non-oscillatory return

Practical Applications

• Transformers: rely on mutual inductance to step voltages up or down.
• Switch-mode power supplies: exploit LC resonance for efficient voltage conversion.
• Radio tuners: use variable inductance to select frequencies.

Conclusion

Chapter 30 highlights how inductance underpins energy storage and dynamic response in electrical circuits. From smoothing filters to oscillators and transformers, mastering L-based analysis opens doors to advanced electronics design and signal processing.

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