AP Physics C: Electricity and Magnetism

Magnetic Forces and Fields

Magnetic Fields Created by Current-Carrying Wires

Learning Objectives

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

• Explain how electric currents create magnetic fields.

• Determine the direction of magnetic fields around current-carrying wires.

• Apply the Right-Hand Rule for magnetic fields.

• Calculate magnetic fields produced by long straight wires.

• Analyze magnetic fields produced by circular loops and solenoids.

• Solve AP Physics C problems involving magnetic fields generated by currents.

Introduction

Currents as Sources of Magnetic Fields

Previously, we learned that magnetic fields exert forces on moving charges and current-carrying wires.

An equally important fact is that electric currents themselves produce magnetic fields.

This discovery was made in 1820 by Hans Christian Ørsted when he observed that a compass needle deflected near a current-carrying wire.

This observation revealed a fundamental connection between electricity and magnetism.

Magnetic Field Around a Straight Wire

Experimental Observation

Consider a long straight wire carrying current (I).

The magnetic field produced by the wire forms concentric circles around the wire.

The field lines:

• Circle the wire.

• Become weaker farther from the wire.

• Have no beginning or end.

Unlike electric field lines, magnetic field lines always form closed loops.

Right-Hand Rule for Magnetic Fields

To determine the magnetic field direction:

1. Point your thumb in the direction of conventional current.

2. Curl your fingers around the wire.

3. Your fingers indicate the direction of the magnetic field.

This rule is extremely important in AP Physics C.

Direction Examples

Current Out of the Page

Represent the current by:

$$
\odot
$$

The magnetic field circles counterclockwise.

Current Into the Page

Represent the current by:

$$
\otimes
$$

The magnetic field circles clockwise.

Magnitude of the Magnetic Field

Long Straight Wire

The magnetic field produced by a long straight wire is:

$$
B=\frac{\mu_0 I}{2\pi r}
$$

where:

• \(B\) = magnetic field

• \(I\) = current

• \(r\) = distance from the wire

• \(\mu_0\) = permeability of free space

Permeability of Free Space

The constant:

$$
\mu_0=4\pi\times10^{-7};T\cdot m/A
$$

is a fundamental constant in electromagnetism.

Dependence on Distance

Notice:

$$
B\propto\frac{1}{r}
$$

As distance increases:

• Magnetic field decreases.

• The decrease is slower than the electric field of a point charge.

Compare:

Electric field:

$$
E\propto\frac{1}{r^2}
$$

Magnetic field of a wire:

$$
B\propto\frac{1}{r}
$$

Superposition of Magnetic Fields

Principle of Superposition

When multiple currents are present:

The total magnetic field equals the vector sum of individual magnetic fields.

Mathematically:

$$
\vec{B}_{total}=\sum \vec{B}
$$

This principle is essential for solving multi-wire problems.

Force Between Parallel Wires

Magnetic Interaction

A current-carrying wire creates a magnetic field.

A nearby wire carrying current experiences a magnetic force due to that field.

Thus:

• Wire 1 produces a field.

• Wire 2 experiences a force.

The interaction is mutual.

Same Current Direction

If two parallel wires carry currents in the same direction:

They attract.

Opposite Current Directions

If two parallel wires carry currents in opposite directions:

They repel.

Force per Unit Length

Important Equation

For two long parallel wires:

$$
\frac{F}{L}=\frac{\mu_0 I_1 I_2}{2\pi d}
$$

where:

• \(F\) = force

• \(L\) = wire length

• \(I_1\) = first current

• \(I_2\) = second current

• \(d\) = separation distance

Historical Importance

This interaction was once used to define the ampere.

Magnetic Field of a Circular Loop

Circular Current

A circular current loop produces a magnetic field at its center.

The magnitude is:

$$
B=\frac{\mu_0 I}{2R}
$$

where:

• \(R\) = radius of the loop

Multiple Turns

For (N) turns:

$$
B=\frac{\mu_0 N I}{2R}
$$

The field increases proportionally with the number of turns.

Direction of the Loop Field

Right-Hand Rule

Curl your fingers in the direction of current.

Your thumb points in the direction of the magnetic field through the center of the loop.

Solenoids

Definition

A solenoid is a long coil of wire carrying current.

Examples include:

• Electromagnets

• Relays

• MRI machines

• Inductors

Magnetic Field Inside a Solenoid

For an ideal solenoid:

$$
B=\mu_0 n I
$$

where:

• \(n\) = number of turns per meter

• \(I\) = current

Uniform Field

Inside an ideal solenoid:

• Field is nearly uniform.

• Field lines are parallel.

• Magnetic field resembles the electric field between parallel plates.

Comparison of Current Geometries

Long Straight Wire

$$
B=\frac{\mu_0 I}{2\pi r}
$$

Field lines form circles.

Circular Loop

$$
B=\frac{\mu_0 I}{2R}
$$

Field points through the center.

Solenoid

$$
B=\mu_0 n I
$$

Field is approximately uniform inside.

Example 1

Magnetic Field Near a Straight Wire

A long wire carries:

$$
I=8.0A
$$

Find the magnetic field at:

$$
r=0.040m
$$

Solution

Use:

$$
B=\frac{\mu_0 I}{2\pi r}
$$

Substitute:

$$
B=\frac{(4\pi\times10^{-7})(8.0)}{2\pi(0.040)}
$$

Simplify:

$$
B=4.0\times10^{-5}T
$$

Answer

$$
B=4.0\times10^{-5}T
$$

Example 2

Field at the Center of a Loop

A circular loop has:

$$
I=5.0A
$$

and

$$
R=0.10m
$$

Find the magnetic field at the center.

Solution

Use:

$$
B=\frac{\mu_0 I}{2R}
$$

Substitute:

$$
B=\frac{(4\pi\times10^{-7})(5.0)}{2(0.10)}
$$

$$
B=3.14\times10^{-5}T
$$

Answer

$$
B=3.14\times10^{-5}T
$$

Example 3

Solenoid Field

A solenoid has:

$$
n=1500
;turns/m
$$

and

$$
I=2.0A
$$

Find the magnetic field inside.

Solution

Use:

$$
B=\mu_0 n I
$$

Substitute:

$$
B=(4\pi\times10^{-7})(1500)(2.0)
$$

$$
B=3.77\times10^{-3}T
$$

Answer

$$
B=3.77\times10^{-3}T
$$

Common AP Exam Mistakes

Mistake 1

Confusing the right-hand rule for force with the right-hand rule for field direction.

Remember:

• Current → Field direction

• Velocity → Force direction

These are different applications.

Mistake 2

Using the wrong distance dependence.

For a straight wire:

$$
B\propto\frac{1}{r}
$$

not

$$
\frac{1}{r^2}
$$

Mistake 3

Forgetting superposition.

When multiple wires exist:

Add magnetic fields as vectors.

AP Free-Response Strategy

Draw Field Directions First

Before calculating:

• Sketch the current.

• Use the right-hand rule.

• Determine field directions.

This often reveals whether fields add or cancel.

Memorize Three Core Equations

Straight wire:

$$
B=\frac{\mu_0 I}{2\pi r}
$$

Loop center:

$$
B=\frac{\mu_0 I}{2R}
$$

Solenoid:

$$
B=\mu_0 n I
$$

These are among the most frequently tested magnetic field equations in AP Physics C.

Summary

Key Takeaways

• Electric currents create magnetic fields.

• Magnetic field direction is found using the right-hand rule.

• Long straight wires produce circular magnetic field lines.

$$
B=\frac{\mu_0 I}{2\pi r}
$$

• Circular loops produce fields through their centers.

$$
B=\frac{\mu_0 I}{2R}
$$

• Solenoids create nearly uniform magnetic fields.

$$
B=\mu_0 n I
$$

• Parallel currents:

  • Same direction → attract

  • Opposite directions → repel

• Magnetic fields obey superposition.

$$
\vec{B}_{total}=\sum\vec{B}
$$

• Current-generated magnetic fields form the foundation for electromagnets, motors, generators, transformers, and many modern technologies.