# Magnetic fields

> IB Physics SL · IB Physics SL (2025 syllabus)
> Source: https://www.owlsprep.com/study/ib-physics-sl-u4-magnetic-fields/

This sub-topic introduces core magnetic field concepts, force calculations for moving charges and current-carrying wires, and rules to find field and force directions. It is a foundational topic for electromagnetism exam questions.

**Prerequisites:** [Electric fields fundamentals](https://www.owlsprep.com/study/ib-physics-sl-u4-electric-fields/)

## Learning objectives

- Define magnetic fields and magnetic flux density
- Calculate magnetic force on moving charges and current-carrying wires
- Apply right-hand rules to find directions of forces and fields
- Describe magnetic fields around common current-carrying conductors

## Magnetic Field Basics

A magnetic field is a force field that exists around magnetic materials and moving electric charges. Unlike electric fields (which act on all charges), magnetic fields only exert a force on moving charges. Field lines are used to visualize fields: they run from north to south outside a magnet, never cross, and closer lines mean a stronger field.

**Magnetic Flux Density** — A measure of magnetic field strength, defined as force per unit current per unit length on a wire placed perpendicular to the field. SI unit is the tesla (T).

*Notation:* B

*Example:* A strong bar magnet has $B \approx 1$ T near its surface.

$$B = \frac{F}{Il}$$

**Worked example:** A 0.5 m long wire carrying 2 A current is placed perpendicular to a uniform magnetic field, and experiences a force of 0.4 N. Calculate $B$.

1. Use the definition of magnetic flux density for a perpendicular wire:
2. $$B = \frac{F}{Il}$$
3. Substitute the given values: $F = 0.4$ N, $I = 2$ A, $l = 0.5$ m:
4. $$B = \frac{0.4}{(2)(0.5)} = 0.4$$
5. Final answer with unit: $B = 0.4$ T

## Magnetic Force on Moving Charges

When a charged particle moves through a magnetic field, it experiences the magnetic component of the Lorentz force. The magnitude of the force depends on the particle's charge, speed, field strength, and the angle between the particle's velocity and the magnetic field.

$$F = qvB \sin\theta$$

Where $\theta$ is the angle between the velocity vector $\vec{v}$ and the magnetic flux density vector $\vec{B}$. Force is maximum when $\theta = 90^\circ$ ($\sin 90^\circ = 1$) and zero when $\theta = 0^\circ$ (velocity parallel to field).

> **Right-Hand Rule for Force Direction**
>
> Point your index finger along velocity, middle finger along the magnetic field, and your thumb points to the force direction for positive charges. Reverse the direction for negative charges.

**Worked example:** An electron moving at $2 \times 10^6$ m/s enters a uniform 0.1 T magnetic field perpendicular to its velocity. Calculate the magnitude of the force on the electron. ($e = 1.6 \times 10^{-19}$ C)

1. Velocity is perpendicular to $B$, so $\theta = 90^\circ$, $\sin 90^\circ = 1$, so $F = qvB$:
2. $$F = (1.6 \times 10^{-19})(2 \times 10^6)(0.1)$$
3. Calculate the result:
4. $$F = 3.2 \times 10^{-14} \text{ N}$$
5. Direction is opposite to the right-hand rule result because the electron has negative charge.

## Force on Current-Carrying Wires

A current in a wire is a flow of moving charges, so the net magnetic force on all individual charges adds up to a total force on the entire wire. The formula for total force is derived directly from the Lorentz force law for individual charges.

$$F = BIl \sin\theta$$

Where $I$ is current, $l$ is wire length, and $\theta$ is the angle between the current direction and the magnetic field.

**Check your understanding**

Test your understanding:

1. When is the magnetic force on a current-carrying wire in a uniform field equal to zero?

   - When current is parallel to $B$
   - When current is perpendicular to $B$
   - When $B = 0$
   - Both A and C

   *Answer:* Both A and C

   *Why:* Correct! $\sin 0^\circ = 0$ when current is parallel to $B$, so $F = 0$, and $F$ is also zero if there is no magnetic field.

**Worked example:** A 1.2 m wire carrying 3 A current is at $30^\circ$ to a 0.2 T magnetic field. Find the force on the wire.

1. Use the force formula for current-carrying wires:
2. $$F = BIl \sin\theta$$
3. Substitute values: $\sin 30^\circ = 0.5$
4. $$F = (0.2)(3)(1.2)(0.5) = 0.36$$
5. Final answer: $F = 0.36$ N, direction perpendicular to both current and $B$.

## Magnetic Fields from Currents

All moving charges produce magnetic fields, so any current flowing through a conductor creates a magnetic field around it. The shape and direction of the field depend on the geometry of the conductor.

- **Straight wire**: Concentric circular field lines around the wire, direction given by the right-hand grip rule.
- **Solenoid (long coil)**: Uniform magnetic field inside the coil, shape identical to a bar magnet, direction given by right-hand grip rule.
- **Flat circular coil**: Field perpendicular to the plane of the coil at its center.

**Right-Hand Grip Rule** — A mnemonic to find the direction of the magnetic field produced by a current-carrying conductor.

*Example:* For straight wires: thumb points along current, fingers curl in field direction. For solenoids: fingers curl along current, thumb points to the north end.

**Worked example:** A current flows vertically upwards through a straight wire directly in front of you. What is the direction of the magnetic field at a point to the right of the wire?

1. Point the thumb of your right hand upwards, aligned with the current direction.
2. Curl your fingers around the wire to follow the direction of the magnetic field lines.
3. At a point to the right of the wire, your fingers point into the plane of the page.
4. Conclusion: Magnetic field direction is into the page.

## Common pitfalls

- **Wrong:** Claiming magnetic force acts on a stationary charge in a magnetic field
  - Why it fails: Magnetic force depends on velocity: if $v=0$, then $F=0$
  - Correct: Remember magnetic fields only exert force on charges moving relative to the field
- **Wrong:** Using the right-hand rule direction directly for negative charges
  - Why it fails: The right-hand rule is defined for positive charges; negative charge force is opposite
  - Correct: Always reverse the direction from the right-hand rule for negative charges
- **Wrong:** Omitting the $\sin\theta$ term, assuming $F = BIl$ or $F = qvB$ always
  - Why it fails: Force is only maximum when velocity/current is perpendicular to $B$
  - Correct: Always check the angle and include $\sin\theta$ in your calculation
- **Wrong:** Mixing up the right-hand grip rule for straight wires and solenoids
  - Why it fails: The rule is applied differently for each conductor geometry
  - Correct: Straight wire: thumb = current, fingers = field. Solenoid: fingers = current, thumb = north pole

## Cheatsheet

| Quantity/Rule | Formula/Description | Notes |
| --- | --- | --- |
| Magnetic flux density | $B = \frac{F}{Il}$ | Perpendicular wire, unit: T |
| Force on moving charge | $F = qvB \sin\theta$ | $\theta$ = angle $\vec{v}$ & $\vec{B}$ |
| Force on current wire | $F = BIl \sin\theta$ | $\theta$ = angle $I$ & $\vec{B}$ |
| Force direction (positive) | Right hand: index = $v$, middle = $B$, thumb = $F$ | Reverse for negative charges |
| Straight wire field | Thumb along current, fingers = field direction | Concentric circular field lines |
| Solenoid field | Fingers along current, thumb = north pole | Uniform field inside solenoid |

## What's next

Magnetic fields are a core component of electromagnetism, which accounts for roughly 20% of the IB Physics SL exam marks. Mastering the basics here will make it much easier to understand electromagnetic induction, the next key topic in unit 4. You can also deepen your understanding by reviewing the circular motion of charged particles in magnetic fields, a common exam application of this topic. Next, explore the following related topics to build your knowledge.

- [Electric Fields](https://www.owlsprep.com/study/ib-physics-sl-u4-electric-fields/)
- [Nuclear and quantum physics](https://www.owlsprep.com/study/ib-physics-sl-u5-overview/)

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