12.1 Magnetic Fields

AP Physics 2 · Unit 12 Magnetism and Electromagnetism

Magnetism feels like a completely new force, but it's actually the same electromagnetic interaction you've been studying — just with moving charges instead of stationary ones.

Learning Outcomes

Describe what a magnetic field is and how it differs from an electric field
Draw magnetic field lines for a bar magnet and a current-carrying wire
Use the right-hand rule to determine the direction of a magnetic field around a straight wire
Calculate the magnetic field strength around a long straight wire using $B = \frac{\mu_0 I}{2\pi r}$
Explain why magnetic field lines always form closed loops (no magnetic monopoles)

Part 1: What Is a Magnetic Field?

You've all held a magnet. You know that magnets exert forces on each other — north repels north, north attracts south, and there's an invisible something stretching between them.

That "something" is the magnetic field, symbol $\vec{B}$.

Just like the electric field $\vec{E}$ told you the force per unit charge at every point in space, the magnetic field tells you about the magnetic force that a moving charge would experience at each point.

But here's the key difference — and I want you to remember this:

[write on board: Electric fields act on any charge. Magnetic fields act only on MOVING charges.]

A stationary charge in a magnetic field feels nothing. No force. Zero. The charge has to be moving for the magnetic field to push on it. We'll get to the exact force formula in the next section, but for now, just remember: magnetic fields only care about charges in motion.

[pause]

Part 2: Magnetic Field Lines

Magnetic field lines are our way of visualizing the field, just like we did with electric field lines. But there's one crucial difference:

[draw a bar magnet on the board with field lines]

Magnetic field lines always form closed loops. They come out of the north pole, curve around through space, enter the south pole, and continue through the magnet from south back to north. Always loops. Never start or end on a point.

This is fundamentally different from electric field lines, which start on positive charges and end on negative charges. Why the difference? Because magnetic monopoles don't exist. You can't have a "north charge" by itself. Every magnet, no matter how many times you cut it, always has both a north and a south pole. Cut a bar magnet in half, you get two smaller magnets, each with its own N and S.

[write on board: No magnetic monopoles → field lines form closed loops]

[pause]

The rules for drawing field lines:

Direction: from N to S outside the magnet; from S to N inside the magnet
Density: closer lines = stronger field; spread-out lines = weaker field
Tangent: the field direction at any point is along the tangent to the line at that point
Never cross: magnetic field lines never intersect each other

Part 3: The Magnetic Field Around a Current-Carrying Wire

Now here's where it gets interesting. In 1820, a Danish physicist named Hans Christian Ørsted was giving a lecture, and he noticed that a compass needle near a wire deflected when he turned on an electric current. This was the first evidence that electricity and magnetism are connected.

A current-carrying wire produces a magnetic field that circles around the wire. Here's what it looks like:

[draw a straight wire with circular field lines around it, viewed end-on]

The field forms concentric circles around the wire. The direction is given by the right-hand rule:

[gesture — grab an imaginary wire with your right hand]

Point your right thumb in the direction of the current. Your fingers curl in the direction of the magnetic field. That's it. Right thumb = current, fingers = field.

[pause]

The strength of the magnetic field at a distance $r$ from a long straight wire carrying current $I$ is:

[write on board: $B = \frac{\mu_0 I}{2\pi r}$]

Where:

$I$ is the current in amperes ($\mathrm{A}$)
$r$ is the perpendicular distance from the wire in meters ($\mathrm{m}$)
$\mu_0 = 4\pi \times 10^{-7}\ \mathrm{T \cdot m/A}$ is the permeability of free space

The unit of magnetic field is the tesla ($\mathrm{T}$). One tesla is a very strong field — a refrigerator magnet is about $0.005\ \mathrm{T}$, and the Earth's magnetic field is only about $50\ \mu\mathrm{T}$.

Notice what the formula tells you:

More current → stronger field. Double the current, double the field. Linear.
Farther away → weaker field. The field falls off as $1/r$, not $1/r^2$. So it decreases, but not as fast as, say, the electric field from a point charge.

[pause]

Part 4: Magnetic Field Between Two Parallel Wires

Here's a cool application. Two parallel wires, both carrying current. Each wire produces a magnetic field, and that field exerts a force on the other wire (since the charges in the other wire are moving).

If the currents flow in the same direction, the wires attract. If the currents flow in opposite directions, the wires repel.

[draw two parallel wires on the board, showing field from wire 1 at the location of wire 2]

Why? Use the right-hand rule. Wire 1 produces a magnetic field at the location of wire 2. The direction of that field depends on the current direction in wire 1. Then the force on wire 2 depends on the current direction in wire 2 and the field from wire 1.

Same direction → field from wire 1 points into the page at wire 2's location (or out, depending on geometry), and the force ends up pointing toward wire 1. Attraction.

Opposite direction → the force flips. Repulsion.

This is actually how the ampere is defined — the force between two parallel wires carrying current. So this isn't just a textbook exercise; it's the foundation of SI electrical units.

[pause]

Part 5: Magnetic Fields vs. Electric Fields — A Comparison

Let me put the two side by side so you can see what's the same and what's different:

Property	Electric Field $\vec{E}$	Magnetic Field $\vec{B}$
Source	Electric charge (stationary or moving)	Moving charge / current
Acts on	Any charge	Only moving charges
Direction of force	Parallel to $\vec{E}$	Perpendicular to both $\vec{v}$ and $\vec{B}$
Field lines	Start on +, end on −	Closed loops (no monopoles)
Unit	$\mathrm{N/C}$ or $\mathrm{V/m}$	Tesla ($\mathrm{T}$)
Falls off (point source)	$1/r^2$ (Coulomb)	$1/r^2$ (for a moving point charge)
Falls off (long wire)	$1/r$ (Gauss)	$1/r$ (Ampère)

The key takeaway: electric force is along the field, magnetic force is perpendicular to both the velocity and the field. This means magnetic forces never do work on a charge — they change direction but not speed. We'll explore this in detail in the next section.

[pause]

Worked Example

Problem: A long straight wire carries a current of $5.0\ \mathrm{A}$.

(a) What is the magnetic field strength at a distance of $2.0\ \mathrm{cm}$ from the wire?

(b) How far from the wire would the field equal the Earth's magnetic field, approximately $5.0 \times 10^{-5}\ \mathrm{T}$?

Solution

(a) Use $B = \frac{\mu_0 I}{2\pi r}$:

$$B = \frac{(4\pi \times 10^{-7})(5.0)}{2\pi(0.020)} = \frac{2 \times 10^{-6}}{0.020} = 5.0 \times 10^{-5}\ \mathrm{T}$$

That's about the same as Earth's magnetic field. So a 5-amp current at 2 cm produces a field comparable to the Earth's. Not huge, but definitely measurable with a compass — which is exactly what Ørsted did.

(b) Set $B = 5.0 \times 10^{-5}\ \mathrm{T}$ and solve for $r$:

$$r = \frac{\mu_0 I}{2\pi B} = \frac{(4\pi \times 10^{-7})(5.0)}{2\pi(5.0 \times 10^{-5})} = \frac{2 \times 10^{-6}}{5.0 \times 10^{-5}} = 0.040\ \mathrm{m} = 4.0\ \mathrm{cm}$$

So at 4 cm, the wire's field equals Earth's field. At distances closer than 4 cm, the wire dominates. Farther away, Earth's field dominates. This is why a compass needs to be close to a current-carrying wire to show a deflection.

Common Mistakes

Using the left hand instead of the right hand. It's always the right hand for magnetic fields and forces in this course.
Forgetting that magnetic fields only affect moving charges. A stationary charge in a magnetic field feels no magnetic force.
Thinking magnetic field lines start and end on poles. They don't — they form closed loops. There are no magnetic monopoles.
Mixing up the $1/r$ dependence. The field around a long wire falls off as $1/r$, not $1/r^2$. This is different from the electric field of a point charge.
Confusing tesla with gauss. $1\ \mathrm{T} = 10^4\ \mathrm{G}$. The AP exam uses tesla.

Summary

Magnetic field $\vec{B}$ describes the magnetic force per unit moving charge at each point in space
Only moving charges feel magnetic forces; stationary charges don't
Magnetic field lines always form closed loops — no magnetic monopoles exist
Right-hand rule: thumb = current direction, fingers = field direction around a wire
Field around a long straight wire: $B = \frac{\mu_0 I}{2\pi r}$, falls off as $1/r$
Parallel wires with same-direction currents attract; opposite currents repel
Magnetic force is always perpendicular to both $\vec{v}$ and $\vec{B}$ — it changes direction, not speed

Homework

A long straight wire carries $12\ \mathrm{A}$. Calculate the magnetic field at $r = 3.0\ \mathrm{cm}$ and $r = 10\ \mathrm{cm}$. By what factor does the field decrease when the distance triples?
Two parallel wires are separated by $5.0\ \mathrm{cm}$. Wire 1 carries $8.0\ \mathrm{A}$ to the right, wire 2 carries $6.0\ \mathrm{A}$ to the right. What is the magnetic field from wire 1 at the location of wire 2? (We'll calculate the actual force in the next section.)
Sketch the magnetic field lines around a bar magnet and around a current-carrying loop. In what way are they similar? In what way are they different from electric field lines around a dipole?