AP Physics C: Electricity and Magnetism

Electromagnetic Induction

Faraday’s Law

Learning Objectives

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

• Define magnetic flux.

• State and apply Faraday’s Law of Induction.

• Calculate induced EMF from changing magnetic flux.

• Understand the significance of the negative sign in Faraday’s Law.

• Analyze situations involving changing magnetic fields, changing areas, and changing orientations.

• Solve AP Physics C problems involving electromagnetic induction.

Introduction

The Discovery of Electromagnetic Induction

In 1831, Michael Faraday discovered that changing magnetic fields can produce electric currents.

This discovery established a deep connection between electricity and magnetism and became one of the foundations of modern electrical technology.

Today, Faraday’s Law explains the operation of:

• Electric generators

• Transformers

• Induction cooktops

• Wireless charging systems

• Power plants

Electromagnetic Induction

What is Electromagnetic Induction?

Electromagnetic induction is the production of an electromotive force (EMF) caused by a changing magnetic environment.

A current can be induced by:

• Changing magnetic field strength

• Changing loop area

• Changing loop orientation

• Moving conductors through magnetic fields

The common feature is a change in magnetic flux.

Magnetic Flux

Definition

Magnetic flux measures the amount of magnetic field passing through a surface.

It is defined as:

$$
\Phi_B=\vec{B}\cdot\vec{A}
$$

Magnitude Form

The magnitude is:

$$
\Phi_B=BA\cos\theta
$$

where:

• (B) = magnetic field strength

• (A) = area of the surface

• (\theta) = angle between (\vec{B}) and the area vector

Units of Magnetic Flux

The SI unit of magnetic flux is:

$$
T\cdot m^2
$$

which is also called a:

$$
\text{weber (Wb)}
$$

Understanding the Area Vector

Area Direction

The area vector is perpendicular to the surface.

For a flat loop:

• The vector points normal to the plane.

• Either normal direction may be chosen consistently.

The angle in the flux equation is measured relative to the area vector.

Special Cases of Magnetic Flux

Maximum Flux

When the magnetic field is parallel to the area vector:

$$
\theta=0^\circ
$$

then:

$$
\Phi_B=BA
$$

Flux is maximum.

Zero Flux

When:

$$
\theta=90^\circ
$$

then:

$$
\Phi_B=0
$$

No magnetic field passes through the surface.

Faraday’s Law

Fundamental Statement

Faraday’s Law states:

The induced EMF equals the negative rate of change of magnetic flux.

Mathematically:

$$
\boxed{\mathcal{E}-\frac{d\Phi_B}{dt}}
$$

Multiple Loops

For a coil containing (N) turns:

$$
\boxed{\mathcal{E}-N\frac{d\Phi_B}{dt}}
$$

Each loop contributes equally to the total induced EMF.

Meaning of the Negative Sign

Lenz’s Law

The negative sign represents:

$$
\text{Lenz’s Law}
$$

Lenz’s Law states:

The induced current always opposes the change in magnetic flux that created it.

Physical Interpretation

Nature resists changes in magnetic flux.

If flux increases:

• Induced current produces a magnetic field that opposes the increase.

If flux decreases:

• Induced current produces a magnetic field that opposes the decrease.

Ways to Change Magnetic Flux

Method 1: Change Magnetic Field Strength

Since:

$$
\Phi_B=BA\cos\theta
$$

changing (B) changes the flux.

Examples:

• Moving a magnet closer to a loop

• Increasing current in an electromagnet

Method 2: Change Area

Changing the area enclosed by a loop changes flux.

Examples:

• Sliding conducting rods

• Expanding loops

Method 3: Change Orientation

Rotating a loop changes:

$$
\theta
$$

and therefore changes flux.

This principle is used in electric generators.

Faraday’s Law and Motional EMF

Connection

Consider a rod moving through a magnetic field.

The area changes with time.

Flux becomes:

$$
\Phi_B=BABLx
$$

Derivation

Differentiating:

$$
\frac{d\Phi_B}{dt}=BL\frac{dx}{dt}
$$

Since:

$$
\frac{dx}{dt}=v
$$

we obtain:

$$
\mathcal{E}=BLv
$$

Thus motional EMF is a special case of Faraday’s Law.

Induced Current

Closed Conducting Loop

If the loop has resistance:

$$
R
$$

then the induced current is:

$$
I=\frac{\mathcal{E}}{R}
$$

Combining with Faraday’s Law:

$$
I=\frac{1}{R}\left|\frac{d\Phi_B}{dt}\right|
$$

Example 1

Changing Magnetic Field

A circular loop has area:

$$
A=0.20m^2
$$

A magnetic field increases from:

$$
0.50T
$$

to

$$
1.50T
$$

in:

$$
0.10s
$$

Find the induced EMF.

Solution

Calculate the change in flux:

$$
\Delta\Phi_B=A\Delta B
$$

$$
\Delta\Phi_B=(0.20)(1.0)
$$

$$
\Delta\Phi_B=0.20Wb
$$

Apply Faraday’s Law:

$$
\mathcal{E}=\frac{\Delta\Phi_B}{\Delta t}
$$

$$
\mathcal{E}=\frac{0.20}{0.10}
$$

$$
\mathcal{E}=2.0V
$$

Answer

$$
\mathcal{E}=2.0V
$$

Example 2

Rotating Loop

A loop of area:

$$
0.10m^2
$$

rotates from:

$$
0^\circ
$$

to

$$
90^\circ
$$

in a magnetic field:

$$
B=0.80T
$$

Find the change in flux.

Solution

Initial flux:

$$
\Phi_i=BA\cos0^\circ
$$

$$
\Phi_i=(0.80)(0.10)
$$

$$
\Phi_i=0.080Wb
$$

Final flux:

$$
\Phi_f=BA\cos90^\circ0
$$

Thus:

$$
\Delta\Phi_B=0.080Wb
$$

Answer

$$
\Delta\Phi_B=0.080Wb
$$

Example 3

Multiple-Turn Coil

A coil contains:

$$
N=200
$$

turns.

The magnetic flux changes at a rate:

$$
\frac{d\Phi_B}{dt}=0.015Wb/s
$$

Find the induced EMF.

Solution

Use:

$$
\mathcal{E}=N\frac{d\Phi_B}{dt}
$$

Substitute:

$$
\mathcal{E}=(200)(0.015)
$$

$$
\mathcal{E}=3.0V
$$

Answer

$$
\mathcal{E}=3.0V
$$

Electric Generators

Rotating Coil Generator

A rotating loop in a magnetic field continuously changes:

$$
\theta
$$

and therefore changes magnetic flux.

According to Faraday’s Law:

An alternating EMF is produced.

This is the operating principle behind electric power generation.

Energy Considerations

Conservation of Energy

Faraday’s Law does not create energy.

An external source must perform work to change the magnetic flux.

Examples:

• Rotating a generator

• Moving a conductor

• Operating a turbine

Mechanical energy becomes electrical energy.

Common AP Exam Mistakes

Mistake 1

Using magnetic field instead of magnetic flux.

Remember:

Faraday’s Law depends on:

$$
\Phi_B
$$

not simply (B).

Mistake 2

Ignoring the angle term.

The correct flux expression is:

$$
\Phi_B=BA\cos\theta
$$

Mistake 3

Forgetting the number of turns.

For coils:

$$
\mathcal{E}=-N\frac{d\Phi_B}{dt}
$$

The factor (N) is crucial.

AP Free-Response Strategy

Step 1

Identify what causes the flux change:

• Changing (B)?

• Changing (A)?

• Changing (\theta)?

Step 2

Calculate magnetic flux:

$$
\Phi_B=BA\cos\theta
$$

Step 3

Find the rate of change:

$$
\frac{d\Phi_B}{dt}
$$

Step 4

Apply Faraday’s Law:

$$
\mathcal{E}=-N\frac{d\Phi_B}{dt}
$$

Summary

Key Takeaways

• Magnetic flux is:

$$
\Phi_B=BA\cos\theta
$$

• The SI unit of flux is:

$$
1Wb=1T\cdot m^2
$$

• Faraday’s Law states:

$$
\boxed{\mathcal{E}-\frac{d\Phi_B}{dt}}
$$

• For multiple loops:

$$
\boxed{\mathcal{E}-N\frac{d\Phi_B}{dt}}
$$

• Flux changes can result from:

  • Changing magnetic field strength

  • Changing area

  • Changing orientation

• The negative sign represents Lenz’s Law.

• Motional EMF:

$$
\mathcal{E}=BLv
$$

is a special case of Faraday’s Law.

• Faraday’s Law is the fundamental principle behind electric generators, transformers, and electromagnetic induction.