Circuits with Resistors and Inductors (LR Circuits)

Electromagnetic induction is the reason a changing magnetic field can create an electric effect in a circuit ⚡. In this lesson, students, you will learn how a resistor and an inductor work together in an LR circuit, how current changes over time, and why the inductor resists sudden changes in current. This topic connects directly to electromagnetic induction because an inductor creates a back emf when the current through it changes.

What You Will Learn

By the end of this lesson, students, you should be able to:

Explain what an inductor, resistor, back emf, and time constant are.
Describe how current changes in an LR circuit after a switch is closed or opened.
Use the equations for growing and decaying current in an LR circuit.
Connect the behavior of LR circuits to Faraday’s law and electromagnetic induction.
Apply circuit reasoning to solve AP Physics C style problems with resistors and inductors.

What Is an LR Circuit?

An LR circuit is a circuit that contains a resistor with resistance $R$ and an inductor with inductance $L$. The letter $L$ stands for inductance, measured in henries $(\text{H})$.

A resistor converts electric energy into thermal energy. It opposes current with a voltage drop of $IR$. An inductor stores energy in a magnetic field when current flows through it. The inductor does not just “let current happen instantly.” Instead, it resists changes in current.

That resistance to change comes from induction. When current in the inductor changes, the magnetic field around the coil changes too. A changing magnetic field produces an induced emf, which opposes the change in current. This is described by Lenz’s law.

If the inductor has self-inductance $L$, then the induced emf is

$$\varepsilon_L = -L\frac{dI}{dt}$$

The minus sign matters. It shows that the induced emf opposes the change in current.

Why This Matters in Real Life 🚗

LR behavior appears in motors, relays, transformers, electronics, and many electromagnetic devices. For example, when power is switched to a motor, the current does not jump instantly to its final value. The inductor in the motor windings resists the change, so the current rises gradually.

Current Growth in an LR Circuit

Consider a simple series circuit with a battery of emf $\varepsilon$, a resistor $R$, and an inductor $L$. Suppose the switch is closed at time $t=0$.

At the instant right after the switch closes, the inductor strongly opposes the increase in current. Since the current cannot change instantly in an ideal inductor, the current starts at $I=0$ and then grows over time.

Using Kirchhoff’s loop rule,

$$\varepsilon - IR - L\frac{dI}{dt} = 0$$

This differential equation gives the current as a function of time:

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

where the time constant is

$$\tau=\frac{L}{R}$$

The time constant tells you how quickly the current changes. A larger inductance $L$ means slower change. A larger resistance $R$ means faster change.

Interpreting the Equation

At $t=0$, $I(0)=0$.
As $t\to \infty$, $I(t)\to \frac{\varepsilon}{R}$.
After one time constant $t=\tau$, the current reaches about $63\%$ of its final value.

This is similar to how a capacitor charges in an RC circuit, but here the inductor is the energy-storage element instead of the capacitor.

Example: Switching On a Circuit

Suppose $\varepsilon=12\text{ V}$, $R=6\ \Omega$, and $L=3\text{ H}$.

First find the time constant:

$$\tau=\frac{L}{R}=\frac{3}{6}=0.5\text{ s}$$

The final current is

$$I_{\infty}=\frac{\varepsilon}{R}=\frac{12}{6}=2\text{ A}$$

So the current is

$$I(t)=2\left(1-e^{-t/0.5}\right)$$

After $0.5\text{ s}$,

$$I(0.5)=2\left(1-e^{-1}\right)\approx 1.26\text{ A}$$

That is about $63\%$ of the final value.

Current Decay in an LR Circuit

Now imagine the battery is removed and the loop still contains the resistor and inductor. The current cannot stop instantly. The magnetic field in the inductor stores energy, and that energy keeps current flowing for a short time.

With no battery in the loop, Kirchhoff’s loop rule gives

$$-IR - L\frac{dI}{dt}=0$$

The current decays according to

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

where $I_0$ is the initial current and $\tau=\frac{L}{R}$.

This exponential decay means the current gets smaller and smaller but never drops to zero instantly in the ideal mathematical model.

Example: Switching Off a Circuit

If $I_0=4\text{ A}$, $L=2\text{ H}$, and $R=8\ \Omega$, then

$$\tau=\frac{2}{8}=0.25\text{ s}$$

So the current is

$$I(t)=4e^{-t/0.25}$$

After $0.25\text{ s}$,

$$I(0.25)=4e^{-1}\approx 1.47\text{ A}$$

The current has fallen to about $37\%$ of its starting value.

Energy in an Inductor

An inductor stores energy in its magnetic field. The energy stored is

$$U=\frac{1}{2}LI^2$$

This formula is important because it shows that energy increases with the square of current. If current doubles, the stored energy becomes four times larger.

In a growing LR circuit, the battery supplies energy. Some energy is converted to thermal energy in the resistor, and some is stored in the inductor’s magnetic field. In a decaying LR circuit, the magnetic energy stored in the inductor is released and eventually dissipates as heat in the resistor.

Real-World Idea

When a circuit with a large inductor is opened, the inductor can produce a large voltage spike because it tries to keep the current from changing suddenly. That is why inductive devices can create sparks or require protective components.

How to Think About LR Circuits on the AP Exam

students, AP Physics C problems often ask you to combine circuit rules with calculus-based reasoning. Here is the main strategy:

Use Kirchhoff’s loop rule to write the equation for the circuit.
Identify whether the current is increasing or decreasing.
Use the correct exponential form for $I(t)$.
Use $\tau=\frac{L}{R}$ to compare how quickly different circuits respond.
Remember that the inductor opposes changes in current, not the current itself.

Common Misunderstanding

A resistor limits current based on voltage and resistance. An inductor limits changes in current based on magnetic induction. These are not the same thing.

For a resistor,

$$V=IR$$

For an inductor,

$$\varepsilon_L=-L\frac{dI}{dt}$$

A resistor depends on the current at that instant. An inductor depends on how fast the current is changing.

Connecting LR Circuits to Electromagnetic Induction

This lesson belongs to electromagnetic induction because the inductor works through Faraday’s law. Faraday’s law states that the induced emf is related to the rate of change of magnetic flux:

$$\varepsilon=-\frac{d\Phi_B}{dt}$$

In a coil, the magnetic flux $\Phi_B$ changes when the current changes. So the inductor generates an emf that opposes the change. This is called self-induction.

That is the key connection: a changing current creates a changing magnetic field, and that changing magnetic field creates an induced emf in the same circuit.

Example Connection to a Coil

If a coil carries increasing current, the magnetic field around the coil increases. The flux through the coil increases, so the induced emf acts to reduce the increase. This is why the current rises gradually rather than instantly.

Conclusion

LR circuits show how electricity and magnetism work together in a time-dependent way. The resistor converts electrical energy to heat, while the inductor stores energy in a magnetic field and resists changes in current. The result is exponential growth or decay of current, controlled by the time constant $\tau=\frac{L}{R}$.

students, this topic is important because it ties together circuit analysis, differential equations, and electromagnetic induction. If you understand why the inductor opposes changes in current, you understand the core idea behind LR circuits and a major piece of AP Physics C Electricity and Magnetism ⚡

Study Notes

An LR circuit contains a resistor with resistance $R$ and an inductor with inductance $L$.
The inductor produces a back emf given by $\varepsilon_L=-L\frac{dI}{dt}$.
The time constant is $\tau=\frac{L}{R}$.
When a battery is connected, current grows as $I(t)=\frac{\varepsilon}{R}\left(1-e^{-t/\tau}\right)$.
When the battery is removed, current decays as $I(t)=I_0e^{-t/\tau}$.
After one time constant, growing current reaches about $63\%$ of its final value.
After one time constant, decaying current falls to about $37\%$ of its initial value.
The energy stored in an inductor is $U=\frac{1}{2}LI^2$.
Inductors are explained by Faraday’s law, $\varepsilon=-\frac{d\Phi_B}{dt}$, so LR circuits are a major part of electromagnetic induction.