---
title: "AP Physics C: E&M 12.2: Magnetism and Moving Charges"
description: "Review AP Physics C: E&M Topic 12.2, including magnetism and moving charges, magnetic fields from moving charges, Lorentz force, right-hand rule, crossed fields, velocity selectors, and the Hall effect."
canonical: "https://fiveable.me/ap-physics-c-e-m/unit-12/2-magnetism-and-moving-charges/study-guide/aujVCr641dSEbfts"
type: "study-guide"
subject: "AP Physics C: E&M"
unit: "Unit 12 – Magnetic Fields & Electromagnetism"
lastUpdated: "2026-06-09"
---

# AP Physics C: E&M 12.2: Magnetism and Moving Charges

## Summary

Review AP Physics C: E&M Topic 12.2, including magnetism and moving charges, magnetic fields from moving charges, Lorentz force, right-hand rule, crossed fields, velocity selectors, and the Hall effect.

## Guide

A moving [charge](/ap-physics-c-e-m/unit-10/2-redistribution-of-charge-between-conductors/study-guide/3zelmsMupFfJh7VP "fv-autolink") creates a magnetic field, and an external magnetic field exerts a force on a charge moving through it. The [magnetic force](/ap-physics-c-e-m/key-terms/magnetic-force "fv-autolink") is $\vec{F}_B = q(\vec{v} \times \vec{B})$, so it points perpendicular to the velocity and tends to bend charges into circular or helical paths without doing work.

## Why This Matters for the AP Physics C: E&M Exam

[Unit 12](/ap-physics-c-e-m/unit-12 "fv-autolink") is one of the heaviest weighted parts of the exam, and this topic gives you the core tools for everything that follows. Both multiple-choice and free-response questions expect you to predict the direction of a magnetic force or field using the [right-hand rule](/ap-physics-c-e-m/key-terms/right-hand-rule "fv-autolink"), derive expressions for circular motion, and explain how changing one quantity (like speed or charge) affects another (like radius or force). You will also need to reason about experimental setups such as velocity selectors and the Hall effect. On free-response, naming a rule is not enough; you have to apply the right-hand rule and the cross-product relationship explicitly to earn credit.

## Key Takeaways

- A single moving charge produces a magnetic field whose strength depends on the charge, its speed, the distance to the point, and the angle between velocity and the position vector.
- The magnetic force on a charge is $\vec{F}_B = q(\vec{v} \times \vec{B})$, with magnitude $F_B = |q|vB\sin\theta$.
- Magnetic force is always perpendicular to velocity, so it changes direction but not speed and does zero work.
- In crossed electric and magnetic fields, the forces add independently as $\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$.
- A velocity selector balances electric and magnetic forces so only charges with $v = E/B$ pass straight through.
- The Hall effect produces a measurable [voltage](/ap-physics-c-e-m/key-terms/voltage "fv-autolink") that reveals the sign and density of [charge carriers](/ap-physics-c-e-m/unit-10/1-electrostatics-with-conductors/study-guide/4Vb5LzwBQm2HSChq "fv-autolink").

## Magnetic Field of a Moving Charge

A moving electric charge generates a magnetic field around it, much like a current-carrying wire does.

The field magnitude depends on the charge, its speed, the distance $r$ from the charge to the point you care about, and the angle $\theta$ between the velocity vector and the position vector pointing from the charge to that point. In proportional form:

$$B \propto \frac{qv\sin\theta}{r^2}$$

So the field is zero along the line of motion ($\theta = 0^\circ$ or $180^\circ$) and is largest when the velocity and position vectors are perpendicular ($\theta = 90^\circ$).

The direction of $\vec{B}$ at any point is perpendicular to both the velocity and the position vector from the charge to that point. Use the [cross product](/ap-physics-c-e-m/key-terms/cross-product "fv-autolink") $\vec{v} \times \hat{r}$ with the right-hand rule: point your fingers along $\vec{v}$, curl them toward $\hat{r}$, and your thumb points along $\vec{B}$ for a positive charge. For a negative charge, reverse the direction. This means the field lines wrap around the velocity vector as concentric circles.

## Force on a Moving Charge in a Magnetic Field

When a charged particle moves through a magnetic field, it feels a force perpendicular to both its velocity and the field. That perpendicular push is what bends charges into circular or helical paths.

The magnetic force is:

$$\vec{F}_{B}=q(\vec{v} \times \vec{B})$$

Where:
- $\vec{F}_{B}$ is the magnetic force vector (newtons)
- $q$ is the charge of the moving object (coulombs)
- $\vec{v}$ is the velocity vector (meters per second)
- $\vec{B}$ is the magnetic field vector (teslas)

The magnitude is:

$$F_B = |q|vB\sin\theta$$

where $\theta$ is the angle between the velocity and field. The force is maximum when the charge moves perpendicular to the field ($\theta = 90^\circ$) and zero when it moves parallel to it ($\theta = 0^\circ$).

One key consequence: because the force is always perpendicular to the velocity, it can never speed the charge up or slow it down. It only changes direction, so the magnetic force does no work and the particle's speed stays constant. When a charge moves perpendicular to a uniform field, this perpendicular force acts as a centripetal force and the charge travels in a circle.

## Combined Electric and Magnetic Fields

In many real setups, a charge sits in both an electric and a magnetic field at once. Each field exerts its own force independently, and the total force is the vector sum:

$$\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$$

A few things to keep in mind:
- The [electric force](/ap-physics-c-e-m/unit-8/1-electric-charge-and-electric-force/study-guide/vbxIAJB9gM4zK3F7 "fv-autolink"), $q\vec{E}$, points along the field and can speed up or slow down the charge.
- The magnetic force bends the path but does no work.
- Depending on the strengths and directions, the combination can produce straight-line, curved, or complex trajectories.

When the fields are set up so the electric and magnetic forces cancel, a charge can travel in a straight line through crossed fields. This is the idea behind a velocity selector: only charges with speed $v = E/B$ pass through undeflected, since that is when $|q|E = |q|vB$. This principle shows up in instruments like mass spectrometers.

## The Hall Effect

The Hall effect connects moving charges and magnetic fields in a practical, measurable way. When [current](/ap-physics-c-e-m/unit-11/4-electric-power/study-guide/u2cRqQTlthIAJtwp "fv-autolink") flows through a [conductor](/ap-physics-c-e-m/unit-11/3-resistance-resistivity-and-ohms-law/study-guide/TnRPkql9C75GQe0d "fv-autolink") sitting in a magnetic field, the charge carriers feel a magnetic force.

Here is the sequence:

1. Current (moving charges) flows through a conductor.
2. A magnetic field is applied perpendicular to the current.
3. Charge carriers feel a magnetic force perpendicular to both their motion and the field.
4. That force pushes charges toward one side of the conductor.
5. The buildup of charge creates an electric field across the conductor's width.
6. At equilibrium, the electric force balances the magnetic force.

The resulting voltage across the conductor is the Hall voltage. Measuring it lets you determine:
- The sign of the charge carriers (positive or negative)
- The density of charge carriers in the material
- The strength of the magnetic field

The Hall effect is used in magnetic field sensors, current measurement, and characterizing semiconductors.

## How to Use This on the AP Physics C: E&M Exam

### Problem Solving

- For force problems, find the direction with the right-hand rule first, then the magnitude with $F_B = |q|vB\sin\theta$. For a negative charge, flip the direction you got from $\vec{v} \times \vec{B}$.
- For circular motion, set the magnetic force equal to the centripetal force: $|q|vB = \dfrac{mv^2}{r}$, which gives $r = \dfrac{mv}{|q|B}$. Practice solving this for radius, speed, charge, or mass.
- For crossed fields, treat the electric and magnetic forces separately, then add them as vectors.

### Free Response

- When you predict whether a quantity increases, decreases, or stays the same, justify it with the functional relationship, not just an equation name. For example, explain that doubling speed doubles the force, which keeps the radius the same since both scale with $v$.
- State the right-hand rule reasoning explicitly. Saying "by the right-hand rule, $\vec{v} \times \vec{B}$ points in $+y$, and the charge is negative, so the force points in $-y$" earns more than just naming the rule.

### Common Trap

- Do not assume the magnetic force changes a charge's speed or [kinetic energy](/ap-physics-c-e-m/key-terms/kinetic-energy "fv-autolink"). It never does, because it is always perpendicular to velocity.

## Common Misconceptions

- The magnetic force does no work. Because it is always perpendicular to velocity, it changes direction but not speed, so kinetic energy stays constant.
- A magnetic field is not always perpendicular to velocity in general; only the force is. The angle $\theta$ between $\vec{v}$ and $\vec{B}$ can be anything, which is why $\sin\theta$ appears in the magnitude.
- The right-hand rule gives the direction of $\vec{v} \times \vec{B}$ for a positive charge. For an [electron](/ap-physics-c-e-m/unit-11/1-electric-current/study-guide/9YRMrkv1PVy23BzH "fv-autolink") or other negative charge, the force points the opposite way.
- A moving charge produces no field along its own line of motion. The field is zero at $\theta = 0^\circ$ and largest perpendicular to the velocity.
- The Hall voltage depends on the sign of the carriers, so its polarity tells you whether the carriers are positive or negative, not just that current is flowing.

## Practice Problem 1: Magnetic Force on a Moving Charge

> An electron with a charge of -1.6 × 10^-19 C moves with a velocity of 2.0 × 10^6 m/s in the positive x-direction. It enters a region with a uniform magnetic field of 0.50 T pointing in the positive z-direction. Calculate the magnitude and direction of the magnetic force on the electron.

**Solution**

Use $$\vec{F}_{B}=q(\vec{v} \times \vec{B})$$

Given:
- Charge q = -1.6 × 10^-19 C
- Velocity v = 2.0 × 10^6 m/s in the +x direction
- Magnetic field B = 0.50 T in the +z direction

First, find the direction. The cross product $\vec{v} \times \vec{B}$ (with $\vec{v}$ in $+x$ and $\vec{B}$ in $+z$) points in the $-y$ direction. Since the electron has a negative charge, the force flips to the $+y$ direction.

Now the magnitude:
$$F_B = |q|vB\sin\theta$$

The velocity and field are perpendicular ($\theta = 90^\circ$), so $\sin(90^\circ) = 1$.
$$F_B = (1.6 × 10^{-19} \text{ C})(2.0 × 10^6 \text{ m/s})(0.50 \text{ T})(1)$$
$$F_B = 1.6 × 10^{-13} \text{ N}$$

The electron experiences a force of 1.6 × 10^-13 N in the positive y-direction.

## Practice Problem 2: Hall Effect

> A copper strip with thickness 1.5 cm carries a current of 10 A. When placed in a magnetic field of 0.75 T perpendicular to the strip, a Hall voltage of 0.225 μV is measured across the width. Calculate the number density of charge carriers in copper.

**Solution**

For the Hall effect, use:
$$V_H = \frac{IB}{n|q|t}$$

Where:
- $$V_H$$ is the Hall voltage
- $$I$$ is the current
- $$B$$ is the magnetic field
- $$n$$ is the number density of charge carriers
- $$|q|$$ is the magnitude of the carrier charge (for electrons, $|q|$ = 1.6 × 10^-19 C)
- $$t$$ is the thickness of the strip (1.5 cm = 0.015 m)

Solving for $n$:
$$n = \frac{IB}{V_H |q| t}$$

Substituting:
$$n = \frac{(10\,\text{A})(0.75\,\text{T})}{(0.225 \times 10^{-6}\,\text{V})(1.6 \times 10^{-19}\,\text{C})(0.015\,\text{m})}$$

Numerator:
$$(10)(0.75) = 7.5$$

Denominator, step by step:
$$(0.225 \times 10^{-6})(1.6 \times 10^{-19}) = 3.6 \times 10^{-26}$$
$$(3.6 \times 10^{-26})(0.015) = 5.4 \times 10^{-28}$$

Then divide:
$$n = \frac{7.5}{5.4 \times 10^{-28}} = 1.39 \times 10^{28}\,\text{m}^{-3}$$

The number density of charge carriers is:
$$n = 1.39 \times 10^{28}\,\text{m}^{-3}$$

## Related AP Physics C: E&M Guides

- [12.1 Magnetic Fields](/ap-physics-c-e-m/unit-12/1-magnetic-fields/study-guide/hK7HuQSBBDlQOdh5)
- [12.3 Magnetic Fields of Current-Carrying Wires and the Biot-Savart Law](/ap-physics-c-e-m/unit-12/3-magnetic-fields-of-current-carrying-wires-and-the-biot-savart-law/study-guide/9L5jxtAFTJoI6u5v)
- [12.4 Ampère's Law](/ap-physics-c-e-m/unit-12/4-amperes-law/study-guide/RURo66Hv1aueyDWX)

## Vocabulary

- **Hall effect**: The phenomenon where a potential difference is created across a conductor perpendicular to both the direction of current flow and an applied magnetic field.
- **Lorentz force**: The force exerted on a moving charged object by a magnetic field, given by F_B = q(v × B).
- **charged object**: An object that possesses electric charge and can experience forces from electric and magnetic fields.
- **conductor**: A material that allows electric charge to move through it, with resistivity that typically increases with temperature.
- **cross-product**: A mathematical operation between two vectors that produces a third vector perpendicular to both, with magnitude equal to the product of their magnitudes and the sine of the angle between them.
- **magnetic field**: A vector field that determines the magnetic force exerted on moving electric charges, electric currents, or magnetic materials.
- **moving charged object**: An object possessing electric charge that is in motion, producing a magnetic field in the surrounding space.
- **perpendicular**: At a 90-degree angle; the magnetic field direction is perpendicular to both the velocity vector and position vector of a moving charged object.
- **position vector**: A vector drawn from a moving charged object to a point in space, used to determine the direction and magnitude of the magnetic field at that point.
- **right-hand rule**: A method for determining the direction of the magnetic field produced by a moving charged object by pointing the thumb in the direction of velocity and curling fingers to show the field direction.
- **velocity**: The rate and direction of motion of an object.

## FAQs

### How does a moving charge create a magnetic field?

A moving charged object produces a magnetic field whose direction is perpendicular to both the velocity and the position vector from the charge to the point.

### What is the magnetic force on a moving charge?

The magnetic force is F_B = q(v x B). Its magnitude depends on charge, speed, magnetic field strength, and the sine of the angle between v and B.

### Does a magnetic force do work on a charge?

No. Magnetic force is always perpendicular to velocity, so it changes the direction of motion but not the speed or kinetic energy.

### How do you use the right-hand rule for magnetic force?

For a positive charge, point your fingers along v, curl toward B, and your thumb gives v x B. For a negative charge, reverse the direction.

### What is a velocity selector?

A velocity selector uses crossed electric and magnetic fields so only particles with speed v = E/B pass through undeflected.

### What is the Hall effect?

The Hall effect is the potential difference that forms across a conductor when moving charge carriers are deflected by a perpendicular magnetic field.

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