Topic 13.5 Notes – Circuits with Resistors and Inductors (LR Circuits)

1. What an LR Circuit Is and What the Inductor Does

An LR circuit typically has a battery $\varepsilon$, a resistor $R$, and an inductor $L$ in series.

The defining property of an inductor is:

$$V_L = L \frac{dI}{dt}$$

The inductor creates an induced emf that opposes any change in current. That “opposes change” language is straight from Lenz’s law. If current tries to increase, the inductor pushes back. If current tries to decrease, it pushes to keep it flowing.

Apply Kirchhoff’s loop rule around the circuit:

$$\varepsilon = IR + L\frac{dI}{dt}$$

This differential equation governs everything about the circuit’s time behavior.

What each element does physically

• Resistor
  • Dissipates energy as heat
    $P = I^2 R$
  • If the inductor releases stored energy, it ends up here.
• Inductor
  • Stores magnetic energy
    $$U_L = \frac{1}{2} L I^2$$
  • When current decreases, this stored energy flows into the resistor and is dissipated.

At the instant a switch is closed or opened, the inductor’s induced emf is equal in magnitude and opposite in direction to the applied voltage across its branch. That’s why current can’t jump instantly in an inductor.

2. The Time Constant

Every LR circuit has a time constant:

$$\tau = \frac{L}{R_{\text{eq}}}$$

Units are seconds.

This tells you how fast the circuit responds. Think of it as electrical inertia.

• Larger $L$ → slower change (harder to change current).
• Larger $R$ → faster change (energy drains faster).

What τ actually represents

• It is the time the circuit would take to reach steady state if it kept changing at its initial rate.
• If current starts at 0 (switch just closed):
  • After one $\tau$, current is about 63% of its final value.
• If current starts at $I_0$ and decays:
  • After one $\tau$, current is about 37% of $I_0$.
• After about 5τ, the circuit is essentially at steady state.

Those 63% and 37% numbers show up constantly in AP questions.

3. Current and Voltage During Transient Behavior

Right after a switch is closed or opened, the circuit is in the transient regime. Current, voltage across the inductor, and stored energy all change exponentially. The graph below shows both the growth and decay behaviors on the same axes, with the one-time-constant markers highlighted.

Current growth and decay in an LR circuit

a. Current Growth (Switch Closed, Initial Current = 0)

$$I(t) = \frac{\varepsilon}{R}\left(1 - e^{-t/\tau}\right)$$

At $t = 0$:

• $I = 0$
• $V_L = \varepsilon$
• Inductor behaves like an open circuit.

As time passes:

• Current rises toward $\varepsilon / R$
• $V_L \to 0$
• Eventually all battery voltage is across the resistor.

b. Current Decay (Battery Removed)

$$I(t) = I_0 e^{-t/\tau}$$

Now the inductor acts like a temporary battery.

• It provides emf to keep current flowing.
• Magnetic energy $\frac{1}{2}LI^2$ decreases.
• That energy is dissipated as heat in the resistor.

Everything approaches an asymptote determined by the initial conditions.

4. Steady State Behavior

After a long time $(t \gg \tau)$:

$$\frac{dI}{dt} = 0$$

So:

• $V_L = 0$
• The inductor behaves like a wire with zero resistance
• The circuit reduces to simple resistor analysis

Final current in a series LR circuit:

$$I_{\text{final}} = \frac{\varepsilon}{R}$$

Stored energy becomes constant:

$$U_L = \frac{1}{2} L I_{\text{final}}^2$$

Here’s the full picture:

On exams, they love switching between “just after the switch closes” and “long after the switch closes.” Your entire answer changes based on that phrase.