6.3 RL Circuits

Inductors resist current change

An inductor develops an EMF opposing changes in current: EMF = -L(dI/dt), where L is inductance in henries. In an RL circuit with a resistor and inductor, this self-induced EMF delays how fast current can build up or decay, unlike resistors alone which respond instantly.

Charging an RL circuit

When a battery is connected, Kirchhoff's voltage law gives ε = IR + L(dI/dt). Solving this differential equation yields I(t) = (ε/R)(1 - e^(-t/τ)), where the time constant τ = L/R. Current rises exponentially toward its max value ε/R, starting at zero because the inductor initially opposes any current change.

Decaying current

If the battery is removed and the circuit is shorted, current decays as I(t) = I₀e^(-t/τ), with τ = L/R again. The inductor's stored energy (U = ½LI²) drives current through the resistor until it dissipates. Larger L slows the decay/growth; larger R speeds it up.

Key terms

Inductor EMF formula

EMF = -L(dI/dt); opposes changes in current, with L the inductance (henries).

RL time constant

τ = L/R; time for current to reach ~63% of its final value (charging) or decay to ~37% (discharging).

Charging current in RL circuit

I(t) = (ε/R)(1 - e^(-t/τ)); starts at 0, approaches ε/R as t → ∞.

Decaying current in RL circuit

I(t) = I₀e^(-t/τ); current decreases exponentially as stored magnetic energy dissipates through R.

Energy stored in an inductor

U = ½LI²; represents energy stored in the inductor's magnetic field.

Try an AP-style question

What is the time constant of an RL circuit with L = 2 H and R = 4 Ω?

1. 0.5 s
2. 2 s
3. 0.125 s
4. 8 s