AP Physics C: Electricity and Magnetism

Electromagnetic Induction

Inductance

Learning Objectives

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

• Define inductance and self-inductance.

• Explain how changing current induces EMF in a circuit.

• Apply the formula for inductors in circuits.

• Analyze energy stored in magnetic fields.

• Solve RL circuit problems involving inductors.

• Understand the physical meaning of inductance in AP Physics C contexts.

Introduction

What is Inductance?

Inductance is the property of a circuit that resists changes in current due to the magnetic field created by that current.

When current changes in a conductor:

• The magnetic field changes.

• The magnetic flux changes.

• An EMF is induced.

This induced EMF always opposes the change in current.

Self-Inductance

Core Idea

A changing current in a circuit produces a changing magnetic field, which induces an EMF in the same circuit.

This is called self-induction.

The induced EMF is proportional to the rate of change of current:

$$
\mathcal{E}_L=L \frac{dI}{dt}
$$

Meaning of Symbols

• \(\mathcal{E}_L\) = induced EMF (inductor EMF)

• \(L\) = inductance

• \(I\) = current

• \(\frac{dI}{dt}\) = rate of change of current

Inductance

Definition

Inductance is defined as:

$$
L=\frac{N\Phi_B}{I}
$$

where:

• \(N\) = number of turns

• \(\Phi_B\) = magnetic flux per turn

• \(I\) = current

Units

The SI unit of inductance is:

$$
\text{henry (H)}
$$

where:

$$
1H = 1 \frac{V \cdot s}{A}
$$

Physical Meaning of Inductance

Resistance to Change in Current

Inductance does not oppose current itself.

It opposes change in current.

• Increasing current → induced EMF opposes increase

• Decreasing current → induced EMF opposes decrease

This behavior is a direct consequence of Lenz’s Law.

The Inductor

Symbol and Behavior

An inductor is represented in circuits as:

• Coil symbol

It stores energy in a magnetic field when current flows.

Energy Stored in an Inductor

Derivation Idea

Power delivered to an inductor is:

$$
P = I \mathcal{E}_L
$$

Substituting:

$$
\mathcal{E}_L = L \frac{dI}{dt}
$$

gives:

$$
P = L I \frac{dI}{dt}
$$

Integrating over time:

Energy Expression

The energy stored in an inductor is:

$$
U_L = \frac{1}{2} L I^2
$$

Interpretation

Energy is stored in the magnetic field produced by the current.

This is analogous to:

• Capacitor energy storage in electric fields

Comparison:

• Capacitor:
  $$
  U = \frac{1}{2} C V^2
  $$

• Inductor:
  $$
  U = \frac{1}{2} L I^2
  $$

RL Circuits

Basic Structure

An RL circuit consists of:

• Resistor (R)

• Inductor (L)

• Battery (V)

Current Growth in RL Circuits

Switching On the Circuit

When the switch is closed:

• Current does not instantly reach maximum.

• Inductor resists change in current.

Governing Equation

Using Kirchhoff’s loop rule:

$$
V – IR – L\frac{dI}{dt} = 0
$$

Solution Behavior

Current increases according to:

$$
I(t) = \frac{V}{R}\left(1 – e^{-tR/L}\right)
$$

Time Constant

The RL time constant is:

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

Interpretation

• Large (L) → slow current change

• Large (R) → fast current stabilization

Current Decay in RL Circuits

Switching Off the Circuit

When the battery is removed:

Current decays as:

$$
I(t) = I_0 e^{-tR/L}
$$

Physical Meaning

The inductor tries to maintain current flow.

It releases stored magnetic energy into the circuit.

Magnetic Field Energy Density (Conceptual)

Energy in Space

Energy stored in an inductor corresponds to energy stored in its magnetic field.

Energy density:

$$
u_B \propto B^2
$$

This connects inductors to field theory in electromagnetism.

Example 1

Induced EMF in an Inductor

An inductor has:

$$
L = 0.50H
$$

Current increases at:

$$
\frac{dI}{dt} = 3.0A/s
$$

Find induced EMF.

Solution

Use:

$$
\mathcal{E}_L = -L \frac{dI}{dt}
$$

Substitute:

$$
\mathcal{E}_L = -(0.50)(3.0)
$$

$$
\mathcal{E}_L = -1.5V
$$

Answer

$$
\mathcal{E}_L = -1.5V
$$

Example 2

Energy Stored in an Inductor

An inductor has:

$$
L = 2.0H
$$

Current is:

$$
I = 4.0A
$$

Find stored energy.

Solution

Use:

$$
U_L = \frac{1}{2} L I^2
$$

Substitute:

$$
U_L = \frac{1}{2}(2.0)(16)
$$

$$
U_L = 16J
$$

Answer

$$
U_L = 16J
$$

Example 3

RL Time Constant

A circuit has:

$$
L = 0.30H, \quad R = 6.0\Omega
$$

Find time constant.

Solution

Use:

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

Substitute:

$$
\tau = \frac{0.30}{6.0}
$$

$$
\tau = 0.050s
$$

Answer

$$
\tau = 0.050s
$$

Common AP Exam Mistakes

Mistake 1

Confusing inductance with resistance.

• Resistance dissipates energy.

• Inductance stores energy.

Mistake 2

Forgetting negative sign in:

$$
\mathcal{E}_L = -L\frac{dI}{dt}
$$

The sign reflects Lenz’s Law.

Mistake 3

Assuming current changes instantaneously in RL circuits.

In reality:

$$
I(t)
$$

changes exponentially.

AP Free-Response Strategy

Identify Inductive Effects

Ask:

• Is current changing?

• Is magnetic field changing?

If yes → inductance is involved.

Use Time Constant First

Always compute:

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

before solving time-dependent problems.

Connect to Energy

If current is present in an inductor:

$$
U_L = \frac{1}{2} L I^2
$$

This often appears in conceptual FRQs.

Summary

Key Takeaways

• Inductance measures resistance to changes in current.

$$
\mathcal{E}_L = -L \frac{dI}{dt}
$$

• Inductance is defined as:

$$
L = \frac{N\Phi_B}{I}
$$

• Unit:

$$
1H = 1 \frac{V \cdot s}{A}
$$

• Energy stored in an inductor:

$$
U_L = \frac{1}{2} L I^2
$$

• RL circuits have exponential behavior:

$$
I(t) = \frac{V}{R}(1 – e^{-tR/L})
$$

• Time constant:

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

• Inductors store energy in magnetic fields and oppose changes in current, forming the basis of transformers, filters, and many electromagnetic systems.