12.3 Magnetism and Current-Carrying Wires

A current-carrying wire creates a magnetic field that wraps around it in concentric circles, with strength found from $B=\frac{\mu_0}{2\pi}\frac{I}{r}$. When you place a current-carrying wire in an external magnetic field, that field pushes on the wire with a force given by $F_B = I\ell B\sin\theta$.

Why This Matters for the AP Physics 2 Exam

This topic connects moving charges to magnetic effects, which shows up across the magnetism unit on both multiple-choice and free-response questions. You will need to predict how the field or force changes when current, distance, length, or angle changes, and you will need to use the right-hand rule to assign directions in three dimensions. On free-response questions, naming the right-hand rule alone does not earn credit. You have to explain the steps that connect the equation or principle to your claim about magnitude or direction.

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Key Takeaways

• The magnetic field around a long straight wire forms concentric circles tangent to the field vectors, with no part of the field pointing along the wire.
• Field strength scales directly with current and inversely with perpendicular distance: $B=\frac{\mu_0}{2\pi}\frac{I}{r}$.
• Use the right-hand rule (thumb along current, fingers curl in field direction) for a straight wire, and curl fingers with the current for a loop so the thumb gives the axial field.
• Add fields from multiple wires using vector addition, accounting for both magnitude and direction.
• The force on a current-carrying wire is $F_B = I\ell B\sin\theta$, maximum when current and field are perpendicular and zero when parallel.
• The right-hand rule for force uses fingers along the current, palm toward the field, and thumb along the force.

Magnetic Field Produced by a Current-Carrying Wire

A current-carrying wire generates a magnetic field around it. This is one of the core ideas linking moving charges to magnetic effects.

• Magnetic field vectors around a long, straight, current-carrying wire are tangent to concentric circles centered on the wire.
• The field has no component pointing toward, away from, or parallel to the wire itself.

The field strength at any point near a long, straight wire depends on two things: how much current flows and how far the point is from the wire.

• Field magnitude is directly proportional to current (double the current, double the field strength).

• Field magnitude is inversely proportional to the perpendicular distance from the wire's central axis (double the distance, half the field).

• Calculate using: $B=\frac{\mu_{0}}{2 \pi} \frac{I}{r}$

  Where:

  • $B$ = magnetic field strength (in teslas, T)
  • $\mu_0$ = permeability of free space ($4\pi \times 10^{-7}$ T·m/A)
  • $I$ = current in the wire (in amperes, A)
  • $r$ = perpendicular distance from the wire (in meters, m)

To find the direction of a straight wire's magnetic field, use the right-hand rule:

1. Point your thumb in the direction of conventional current.
2. Your fingers curl in the direction of the magnetic field.
3. This gives the circular pattern around the wire.

For a current loop, the magnetic field at the center points along the loop's axis. Curl your fingers in the direction of the current flow, and your thumb points in the field direction at the center.

When two or more current-carrying wires are nearby, find the net field at a point using vector addition of the individual fields. Account for both the magnitude and direction of each contribution.

Force on a Current-Carrying Wire in a Magnetic Field

When a current-carrying wire sits in an external magnetic field, the field can exert a force on it. This is the idea behind electric motors and many other electromagnetic devices.

The force magnitude depends on four factors:

1. Current in the wire (more current means more force)
2. Length of wire inside the magnetic field (longer wire means more force)
3. Magnetic field strength (stronger field means more force)
4. Angle between the current direction and field direction (largest force when perpendicular)

This is quantified by:

$F_{B}=I \ell B \sin \theta$

Where:

• $F_B$ = magnetic force on the wire (in newtons, N)
• $I$ = current in the wire (in amperes, A)
• $\ell$ = length of wire in the magnetic field (in meters, m)
• $B$ = external magnetic field strength (in teslas, T)
• $\theta$ = angle between current and field directions

The force is largest when current and field are perpendicular ($\theta = 90°$, $\sin 90° = 1$) and zero when they are parallel ($\theta = 0°$, $\sin 0° = 0$).

To find the direction of the force on the wire, use this version of the right-hand rule:

1. Point your fingers in the direction of conventional current.
2. Orient your palm toward the magnetic field.
3. Your thumb points in the direction of the force.

This helps you see the three-dimensional relationship between current, field, and force, which are mutually perpendicular in the maximum-force case.

How to Use This on the AP Physics 2 Exam

Problem Solving

• For field problems, identify the current and the perpendicular distance first, then plug into $B=\frac{\mu_0}{2\pi}\frac{I}{r}$. Watch unit conversions, since distances are often given in centimeters.
• For force problems, check the angle between current and field before using $F_B = I\ell B\sin\theta$. A common slip is forgetting the $\sin\theta$ factor when the wire is not perpendicular to the field.
• When several wires contribute, sketch each wire's field at the point, then combine them as vectors rather than just adding numbers.

Free Response

• Show every step from the equation to your conclusion. Saying "the force points right because of the right-hand rule" is not enough to support a stronger score.
• When asked how a quantity changes, state whether it increases, decreases, or stays the same, then tie that to the proportionality in the relevant equation.
• For direction questions, describe how you applied the right-hand rule (which way your thumb, fingers, or palm pointed) so the reader can follow your reasoning.

Common Trap

• Mixing up the two right-hand rules. The field-of-a-wire rule curls fingers around the wire; the force-on-a-wire rule uses palm-push direction. Keep them separate.

Practice Problem 1: Magnetic Field from a Wire

Solution

Use the equation for the field near a long, straight wire:

$B=\frac{\mu_{0}}{2 \pi} \frac{I}{r}$

Given:

• Current, $I = 5.0$ A
• Distance from wire, $r = 10$ cm = 0.10 m
• Permeability of free space, $\mu_0 = 4\pi \times 10^{-7}$ T·m/A

Substituting:

$B=\frac{4\pi \times 10^{-7}}{2 \pi} \frac{5.0}{0.10}$

$B=\frac{4\pi \times 10^{-7}}{2 \pi} \times 50$

$B= 2 \times 10^{-7} \times 50$

$B= 1.0 \times 10^{-5}$ T

The magnetic field at 10 cm from the wire is $1.0 \times 10^{-5}$ T, or 10 μT.

Practice Problem 2: Force on a Current-Carrying Wire

Solution

Use the equation for force on a current-carrying wire:

$F_{B}=I \ell B \sin \theta$

Given:

• Current, $I = 3.0$ A
• Wire length, $\ell = 25$ cm = 0.25 m
• Magnetic field strength, $B = 0.50$ T
• Angle between current and field, $\theta = 30°$

Substituting:

$F_{B}= 3.0 \times 0.25 \times 0.50 \times \sin 30°$

$F_{B}= 3.0 \times 0.25 \times 0.50 \times 0.5$

$F_{B}= 0.1875$ N

The magnetic force on the wire is 0.1875 N, or about 0.19 N.

Common Misconceptions

• The field around a straight wire is not straight or radial. The vectors are tangent to circles centered on the wire, so the field curls around it.
• A bigger distance does not mean a stronger field. Field strength drops as distance grows because it is inversely proportional to $r$.
• The force on a wire is not always $I\ell B$. You must include $\sin\theta$, so the force is zero when the current runs parallel to the field.
• Current, field, and force are not all in the same plane in the maximum-force case. They are mutually perpendicular, which is why direction problems need a careful right-hand rule.
• Naming "the right-hand rule" is not a complete justification on free response. You have to explain how the rule leads to your specific direction.

Related AP Physics 2 Guides

• 12.1 Magnetic Fields
• 12.2 Magnetism and Moving Charges
• 12.4 Electromagnetic Induction and Faraday's Law