Rucete ✏ AP Physics C In a Nutshell
16. Inductors
This chapter introduces the concept of self-inductance, energy storage in inductors, and the behavior of RL and LC circuits.
Qualitative Understanding of Self-Inductance
• A changing current through a solenoid induces an EMF opposing the change according to Faraday’s and Lenz’s laws.
• Increasing current induces an EMF that opposes the increase; decreasing current induces an EMF that opposes the decrease.
• Inductors stabilize current flow by resisting changes through induced EMF.
Quantitative Definition of Self-Inductance
• The induced EMF is proportional to the rate of change of current: |EMF| = L|dI/dt|.
• Self-inductance (L) depends only on the geometry and physical properties of the inductor.
• For a solenoid: L = μ₀πn²lR², where n is turns per length, l is length, R is radius.
• Unit of inductance: henry (H), where 1 H = 1 (V·s)/A.
Energy Stored in an Inductor
• Increasing current stores energy in the inductor by working against the induced EMF.
• Energy stored: U = ½LI², depending only on the final current.
• Energy conservation ensures that energy input to increase current is stored in the magnetic field of the inductor.
Circuit Problems with Inductors: Key Observations
• Current through an inductor cannot change abruptly; it must be continuous.
• At steady-state (constant current), the voltage across an inductor is zero, behaving like a short circuit.
• Kirchhoff’s loop rule still applies to circuits with inductors.
RL Circuits (Resistor and Inductor Circuits)
• RL circuits behave analogously to RC circuits, with current replacing charge and inductance replacing capacitance.
• Time constant for RL circuits: τ = L/R, where L is inductance and R is resistance.
Current Growth in an RL Circuit
• When a battery is connected to a resistor and inductor, current starts at zero and increases toward a final steady-state value.
• Kirchhoff’s loop equation: Vbattery − IR − L(dI/dt) = 0.
• Solving the differential equation yields: I(t) = (V/R)(1 − e^(−t/τ)).
• Initially, current is zero; eventually, current reaches V/R, as in Ohm’s law.
Current Decay in an RL Circuit
• When a current-carrying RL circuit is disconnected from the battery, current decreases exponentially to zero.
• Kirchhoff’s loop equation during decay: IR + L(dI/dt) = 0.
• Solution: I(t) = I₀e^(−t/τ), where I₀ is the initial current.
• Inductor acts to sustain current flow as it decays.
Voltage Across an Inductor in RL Circuits
• Initially, voltage across the inductor equals the battery voltage (opposing current growth).
• Over time, as current stabilizes, the voltage across the inductor decreases to zero.
• Voltage behavior matches exponential decay: V(t) = V₀e^(−t/τ) during decay phase.
Time Constants and Behavior
• The time constant τ = L/R determines how quickly the circuit responds to changes.
• In one time constant, the current grows to about 63% of its final value during charging, or decays to about 37% during discharging.
Curve Sketching for RL Circuits
• Current growth and decay are both exponential curves.
• Voltage across the inductor decreases exponentially during both growth and decay phases.
LC Circuits (Inductor and Capacitor Circuits)
• An LC circuit consists of a capacitor connected to an inductor without resistance.
• Energy oscillates between the electric field of the capacitor and the magnetic field of the inductor.
• Kirchhoff’s loop rule leads to a second-order differential equation describing simple harmonic motion.
Mathematical Analysis of LC Circuits
• Starting from Kirchhoff’s rule: Vcapacitor + Vinductor = 0.
• Charge and current are related: I = −dQ/dt.
• The differential equation becomes: d²Q/dt² + (1/LC)Q = 0.
• Solution for charge: Q(t) = Qmax cos(ωt + φ), where ω = 1/√(LC).
• Current is given by I(t) = (Qmax√(1/LC))sin(ωt + φ).
Energy Conservation in LC Circuits
• Total energy remains constant, alternating between capacitor and inductor.
• Utotal = Ucapacitor + Uinductor at all times.
• Maximum capacitor energy when charge is maximum; maximum inductor energy when current is maximum.
Comparison to Mass-Spring Systems
• LC circuits are directly analogous to mass-spring simple harmonic motion systems.
• Inductance L ↔ mass m; 1/C ↔ spring constant k.
• Angular frequency ω = 1/√(LC), just as ω = √(k/m) for springs.