Transient and Oscillatory Behavior with Inductors: AP Physics C E&M Notes

LR circuit

A circuit (often series) containing a resistor R and inductor L, showing time-dependent (transient) current behavior because the inductor resists changes in current.

Transient (in circuits)

The time-dependent adjustment period after a switch is opened/closed, before the circuit reaches steady state.

Faraday’s law (circuit context)

A changing current changes magnetic flux, inducing an emf; in inductors this leads to a voltage that opposes changes in current (Lenz’s law).

Inductor voltage-current relation

The voltage across an inductor is proportional to the rate of change of current: V_L = L(dI/dt) (sign depends on chosen polarity).

Ohm’s law (resistor relation)

The voltage across a resistor is proportional to current: V_R = IR.

Inductor current continuity

Current through an inductor cannot change instantaneously; I(0+) equals I(0−).

Inductor “open-circuit at t = 0+” idea

Right after switching, if initial current is zero, it remains zero at t=0+, so the inductor can behave like an open circuit initially.

Inductor “short-circuit in steady DC” idea

Long after switching in DC, dI/dt = 0 so V_L = 0, meaning an ideal inductor behaves like a wire (no voltage drop).

Kirchhoff’s loop rule (KVL)

The algebraic sum of potential changes around a closed loop is zero; used to derive LR and LC differential equations.

Series LR differential equation (with battery)

From KVL: L\frac{dI}{dt} + RI = \mathcal{E} (equivalently \mathcal{E} − IR − L\frac{dI}{dt} = 0).

RL time constant (tau)

The characteristic timescale for LR transients: $\tau = \frac{L}{R}$.

Steady-state current in an LR circuit

For a DC source after a long time: I_\infty = \mathcal{E}/R (inductor drop goes to zero).

LR current growth (switch closing, I(0)=0)

Current rises exponentially: I(t) = I_\infty(1 − e^{−t/ au}).

“63% rule” for LR growth

After one time constant au, current reaches 1 − e^{−1} \approx 0.632 (about 63%) of its final value.

Resistor voltage during LR growth

V_R(t) = I(t)R = \mathcal{E}(1 − e^{−t/ au}).

Inductor voltage during LR growth

V_L(t) = \mathcal{E}e^{−t/ au}; initially equals the battery voltage and decays to 0.

Initial LR switching voltages (t = 0+)

If I(0)=0, then V_{R(0+)} = 0 and V_{L(0+)} = \mathcal{E} (entire battery voltage appears across the inductor).

LR current decay (battery removed, initial I0)

In a closed R–L loop with no source: I(t) = I_0 e^{−t/ au}, with au = \frac{L}{R}.

Inductor stored energy

Energy stored in an inductor’s magnetic field: U_L = (1/2)LI^2.

Inductor power relation

Instantaneous power into/out of an inductor: P_L = V_L I (sign indicates storing vs releasing energy).

LC circuit

An ideal circuit with only an inductor L and capacitor C; it oscillates because energy exchanges between electric (capacitor) and magnetic (inductor) storage without dissipation.

Capacitor voltage-charge relation

Voltage across a capacitor: V_C = \frac{q}{C}, where q(t) is the capacitor charge (with sign).

Ideal LC differential equation for charge

From KVL and I = \frac{dq}{dt}: L\frac{d^2q}{dt^2} + \frac{1}{C}q = 0, which is simple harmonic motion.

LC angular frequency

The oscillation angular frequency for an ideal LC circuit: \omega = \frac{1}{\sqrt{LC}}.

LC period and frequency

Period T = 2\pi\sqrt{LC} and frequency f = \frac{1}{2\pi\sqrt{LC}}; charge and current are out of phase by \frac{\pi}{2} (a quarter cycle).