RL circuit in AP Physics C: E&M

An RL circuit is a circuit with a resistor and inductor in series with a voltage source, where the inductor's opposition to changing current makes the current grow or decay exponentially with time constant τ = L/R instead of jumping instantly to its final value.

What is RL circuit?

An RL circuit (the CED calls it an LR circuit) is what you get when you put a resistor and an inductor in series with a battery and a switch. The inductor's whole job is to fight changes in current, so the moment you close the switch, the current can't jump straight to ε/R. Instead it climbs exponentially toward that value, following I(t) = (ε/R)(1 − e^(−t/τ)), where the time constant τ = L/R sets how fast the climb happens.

The two limiting cases are the part you'll use constantly. Right after the switch closes (t = 0), the current is still zero, so the inductor behaves like an open switch and the entire battery emf appears across it, giving an initial rate of change dI/dt = ε/L. After a long time (steady state), the current stops changing, the inductor's induced emf drops to zero, and it behaves like a plain wire. Disconnect the battery and let the inductor drive current through the resistor, and you get the mirror image, exponential decay I(t) = I₀e^(−t/τ).

Why RL circuit matters in AP® Physics C: E&M

RL circuits live in Topic 13.5, Circuits with Resistors and Inductors (LR Circuits), at the end of the electromagnetic induction unit. They're the payoff of the whole unit, because they take Faraday's law and self-inductance and turn them into a circuit you can actually analyze with Kirchhoff's loop rule. Writing the loop equation ε − IR − L(dI/dt) = 0 and solving (or at least interpreting) that differential equation is exactly the kind of reasoning Physics C rewards. RL circuits also close a loop with Unit 11, since they're the magnetic twin of RC circuits, and the exam loves asking you to reason about t = 0 and t → ∞ behavior in both.

How RL circuit connects across the course

Energy stored in an inductor (Unit 13)

While current builds in an RL circuit, the battery isn't just heating the resistor. It's also banking energy in the inductor's magnetic field, U = ½LI². When the battery disconnects, that stored energy is what keeps current flowing as it decays through the resistor.

Steady state (Units 11 & 13)

Steady state is the long-time limit where nothing changes anymore. In an RL circuit that means dI/dt = 0, the inductor's emf vanishes, and it acts like an ideal wire. Compare that to a capacitor, which acts like an open switch at steady state. The two are exact opposites.

RC circuits (Unit 11)

RL circuits are RC circuits with the roles flipped. Both produce exponential curves with a time constant, but the inductor resists changes in current while the capacitor resists changes in voltage. If you understand one, you can predict the other by swapping which quantity is forbidden from jumping.

Faraday's law and self-inductance (Unit 13)

The L(dI/dt) term in the loop equation is just Faraday's law applied to the inductor's own changing flux. Self-induced emf opposes the change in current (Lenz's law), which is physically why the current curve is smooth and exponential rather than instantaneous.

Is RL circuit on the AP® Physics C: E&M exam?

Multiple-choice questions test the limiting cases hard. Expect stems asking for the current, the voltage across the inductor, or dI/dt right after the switch closes versus after a long time. Calculation questions give you values like a 12 V battery, a 30 Ω resistor, and τ = 0.05 s, then ask for L (use L = τR, so 1.5 H) and the initial dI/dt (use ε/L). Decay setups are also common, like finding the induced emf when an inductor carrying I₀ discharges through a resistor, where the answer is I₀R e^(−t/τ). On the free-response side, RL circuits show up as a 'derive the differential equation' task. You write Kirchhoff's loop rule with the L(dI/dt) term, separate variables or identify the solution form, and sketch I versus t showing exponential approach to ε/R. Always check your graph at t = 0 and t → ∞ first, because those endpoints are usually worth points on their own.

RL circuit vs RC circuit

Both produce exponential behavior, but they guard different quantities. An inductor won't let current change instantly, so in an RL circuit the current starts at zero (inductor acts like an open switch) and the inductor becomes a wire at steady state. A capacitor won't let its voltage change instantly, so in an RC circuit current starts at its maximum and the capacitor becomes an open switch at steady state. The time constants are also inverted in structure, τ = L/R for RL but τ = RC for RC. Mixing those up is one of the most common point-losers.

Key things to remember about RL circuit

• The time constant of an RL circuit is τ = L/R, and after one time constant the current reaches about 63% of its final value during growth.

• Right after the switch closes, the current is zero, the inductor acts like an open switch, and the full battery emf appears across it, so the initial rate of change of current is dI/dt = ε/L.

• At steady state the current is constant at ε/R, dI/dt = 0, and the inductor behaves like an ideal wire with no voltage across it.

• When the source is removed, the current decays exponentially as I(t) = I₀e^(−t/τ), powered by the energy U = ½LI² that was stored in the inductor's magnetic field.

• The governing equation comes from Kirchhoff's loop rule, ε − IR − L(dI/dt) = 0, and deriving or solving it is a classic Physics C free-response task.

• Remember the contrast with RC circuits, because inductors fight changes in current while capacitors fight changes in voltage, and their t = 0 and steady-state behaviors are exact opposites.

Frequently asked questions about RL circuit

What is an RL circuit in AP Physics C?

It's a circuit with a resistor and inductor in series with a voltage source, covered in Topic 13.5. Because the inductor opposes changes in current, the current grows or decays exponentially with time constant τ = L/R rather than changing instantly.

Does current flow immediately when you close the switch in an RL circuit?

No. At t = 0 the inductor's induced emf exactly cancels the battery, so the current starts at zero and climbs as I(t) = (ε/R)(1 − e^(−t/τ)). What's not zero is the rate of change, which starts at its maximum value of ε/L.

How is an RL circuit different from an RC circuit?

An inductor blocks sudden changes in current, while a capacitor blocks sudden changes in voltage. So an RL circuit starts with zero current and the inductor ends up acting like a wire, while an RC circuit starts with maximum current and the capacitor ends up acting like an open switch. Also τ = L/R for RL versus τ = RC for RC.

How do you find the time constant of an RL circuit?

Divide inductance by resistance, τ = L/R. For example, if a circuit with a 30 Ω resistor has τ = 0.05 s, the inductance is L = τR = 1.5 H. With a 12 V battery, the initial dI/dt is ε/L = 8 A/s.

What happens to the energy in an inductor when the battery is disconnected?

The energy stored in the magnetic field, U = ½LI², drives a decaying current I(t) = I₀e^(−t/τ) through the resistor, where it's dissipated as heat. The induced emf during this decay is I₀R e^(−t/τ), a common multiple-choice answer form.