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(dI/dt) + RI = 𝓔 (equivalently 𝓔 − IR − L(dI/dt) = 0).

RL time constant (tau)

The characteristic timescale for LR transients: τ = L/R.

Steady-state current in an LR circuit

For a DC source after a long time: I_∞ = 𝓔/R (inductor drop goes to zero).

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

Current rises exponentially: I(t) = I_∞(1 − e^(−t/τ)).

“63% rule” for LR growth

After one time constant τ, current reaches 1 − e^(−1) ≈ 0.632 (about 63%) of its final value.

Resistor voltage during LR growth

V_R(t) = I(t)R = 𝓔(1 − e^(−t/τ)).

Inductor voltage during LR growth

V_L(t) = 𝓔e^(−t/τ); initially equals the battery voltage and decays to 0.

Initial LR switching voltages (t = 0+)

If I(0)=0, then VR(0+) = 0 and VL(0+) = 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) = I0 e^(−t/τ), with τ = 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: PL = VL 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 = q/C, where q(t) is the capacitor charge (with sign).

Ideal LC differential equation for charge

From KVL and I=dq/dt: L(d²q/dt²) + (1/C)q = 0, which is simple harmonic motion.

LC angular frequency

The oscillation angular frequency for an ideal LC circuit: ω = 1/√(LC).

LC period and frequency

Period T = 2π√(LC) and frequency f = 1/(2π√(LC)); charge and current are out of phase by π/2 (a quarter cycle).