AP Physics C: Electricity and Magnetism

Magnetic Forces and Fields

The Magnetic Force on a Current-Carrying Wire

Learning Objectives

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

• Explain why a current-carrying wire experiences a magnetic force.

• Calculate the magnetic force on a straight wire in a magnetic field.

• Determine the direction of the magnetic force using the right-hand rule.

• Analyze the motion of current-carrying conductors in magnetic fields.

• Understand the operation of electric motors.

• Solve AP Physics C problems involving magnetic forces on wires.

Introduction

From Moving Charges to Current-Carrying Wires

In the previous lesson, we learned that a moving charge experiences a magnetic force:

$$
\vec{F}_B=q\vec{v}\times\vec{B}
$$

A current in a wire consists of many moving charges.

Therefore, a current-carrying wire placed in a magnetic field also experiences a magnetic force.

This idea forms the basis of:

• Electric motors

• Loudspeakers

• Galvanometers

• Electromagnetic actuators

Magnetic Force on a Straight Wire

Fundamental Equation

Consider a straight wire carrying current (I) in a uniform magnetic field (\vec{B}).

The magnetic force on the wire is:

$$
\vec{F}=I\vec{L}\times\vec{B}
$$

where:

• \(\vec{F}\) = magnetic force on the wire

• \(I\) = current

• \(\vec{L}\) = length vector of the wire

• \(\vec{B}\) = magnetic field

The direction of (\vec{L}) is the direction of conventional current.

Magnitude of the Force

The magnitude is:

$$
F=ILB\sin\theta
$$

where:

• \(I\) = current

• \(L\) = length of wire in the field

• \(B\) = magnetic field strength

• \(\theta\) = angle between the current direction and the magnetic field

Special Cases

Parallel Wire and Field

If:

$$
\theta=0^\circ
$$

then:

$$
F=ILB\sin0^\circ=0
$$

No magnetic force acts on the wire.

Antiparallel Wire and Field

If:

$$
\theta=180^\circ
$$

then:

$$
F=ILB\sin180^\circ=0
$$

Again, the force is zero.

Perpendicular Wire and Field

If:

$$
\theta=90^\circ
$$

then:

$$
F=ILB
$$

The magnetic force is maximum.

Origin of the Force

Microscopic Explanation

Inside the wire, charge carriers move with drift velocity.

Each moving charge experiences a magnetic force:

$$
\vec{F}_B=q\vec{v}\times\vec{B}
$$

The sum of all these microscopic forces produces a net force on the wire itself.

Physical Interpretation

The magnetic field pushes on the moving charges.

The charges collide with atoms in the wire.

As a result, the entire wire experiences a force.

Direction of the Magnetic Force

Right-Hand Rule for Wires

To determine the direction of force:

1. Point your fingers in the direction of current.

2. Curl them toward the magnetic field.

3. Your thumb points in the direction of the force.

This follows:

$$
\vec{F}=I\vec{L}\times\vec{B}
$$

Important Note

Always use:

• Conventional current direction

• Not electron flow direction

when applying the right-hand rule.

Common Field Symbols

Magnetic Field Out of the Page

Represented by:

$$
\odot
$$

The dot represents the tip of an arrow coming toward you.

Magnetic Field Into the Page

Represented by:

$$
\otimes
$$

The cross represents the tail of an arrow moving away from you.

Force on a Wire Segment

Length in the Magnetic Field

Only the portion of wire actually inside the magnetic field experiences force.

If only a section of wire is in the field:

Use that length in:

$$
F=ILB\sin\theta
$$

Practical Example

If a 2.0 m wire exists but only 0.50 m lies within the field:

Use:

$$
L=0.50m
$$

in calculations.

Torque on a Current Loop

Opposite Forces

Consider a rectangular current loop in a magnetic field.

Two sides experience forces in opposite directions.

These forces create a torque.

Motor Principle

The resulting torque causes the loop to rotate.

This principle forms the basis of electric motors.

The torque is:

$$
\tau=N I A B \sin\theta
$$

where:

• \(N\) = number of turns

• \(I\) = current

• \(A\) = loop area

• \(B\) = magnetic field

• \(\theta\) = angle between the area vector and field

Electric Motors

How Motors Work

An electric motor converts:

Electrical Energy

into

Mechanical Energy

using magnetic forces on current-carrying conductors.

Key Components

A simple motor contains:

• Current loop

• Magnetic field

• Power source

• Commutator

The magnetic forces generate continuous rotation.

Example 1

Force on a Straight Wire

A wire carries:

$$
I=5.0A
$$

through a magnetic field:

$$
B=0.40T
$$

The wire length in the field is:

$$
L=0.60m
$$

The wire is perpendicular to the field.

Find the magnetic force.

Solution

Use:

$$
F=ILB
$$

Substitute:

$$
F=(5.0)(0.40)(0.60)
$$

$$
F=1.2N
$$

Answer

$$
F=1.2N
$$

Example 2

Wire at an Angle

A wire carries:

$$
I=10A
$$

through a magnetic field:

$$
B=0.25T
$$

The length is:

$$
L=0.80m
$$

The angle between the current and field is:

$$
30^\circ
$$

Find the magnetic force.

Solution

Use:

$$
F=ILB\sin\theta
$$

Substitute:

$$
F=(10)(0.80)(0.25)\sin30^\circ
$$

Since:

$$
\sin30^\circ=0.5
$$

$$
F=1.0N
$$

Answer

$$
F=1.0N
$$

Example 3

Finding Current

A wire experiences a force:

$$
F=3.0N
$$

while in a magnetic field:

$$
B=0.50T
$$

The wire length is:

$$
L=1.5m
$$

The wire is perpendicular to the field.

Find the current.

Solution

Use:

$$
F=ILB
$$

Solve for current:

$$
I=\frac{F}{LB}
$$

Substitute:

$$
I=\frac{3.0}{(1.5)(0.50)}
$$

$$
I=4.0A
$$

Answer

$$
I=4.0A
$$

Common AP Exam Mistakes

Mistake 1

Forgetting the angle factor.

The complete equation is:

$$
F=ILB\sin\theta
$$

not simply:

$$
F=ILB
$$

unless the wire is perpendicular to the field.

Mistake 2

Using electron flow direction.

Always use conventional current when applying the right-hand rule.

Mistake 3

Using the total wire length.

Only the section inside the magnetic field contributes to the force.

AP Free-Response Strategy

Draw Current and Field Directions

Before calculating:

• Draw the current.

• Draw the magnetic field.

• Determine the force direction.

This avoids most sign errors.

Check Special Cases

Always ask:

• Parallel?

• Perpendicular?

• General angle?

This often simplifies the calculation immediately.

Connect to Motors

When current loops appear:

Think about:

• Opposing forces

• Torque

• Rotation

Many AP conceptual questions focus on motor operation.

Summary

Key Takeaways

• A current-carrying wire experiences a magnetic force because its charges are moving.

• The magnetic force on a wire is:

$$
\vec{F}=I\vec{L}\times\vec{B}
$$

• The magnitude is:

$$
F=ILB\sin\theta
$$

• Maximum force occurs when:

$$
\theta=90^\circ
$$

• No force occurs when:

$$
\theta=0^\circ
$$

or

$$
\theta=180^\circ
$$

• Force direction is determined using the right-hand rule.

• Current loops experience torque:

$$
\tau=N I A B \sin\theta
$$

• Magnetic forces on wires form the operating principle behind electric motors and many electromagnetic devices.

• Understanding magnetic forces on wires is essential before studying magnetic fields produced by currents and electromagnetic induction.