---
title: "F_B = q(v × B) — AP Physics C: E&M Definition & Guide"
description: "F_B = q(v × B) gives the magnetic force on a moving charge: a cross product of velocity and field. Core to Topic 12.2 and the Lorentz force on the exam."
canonical: "https://fiveable.me/ap-physics-c-e-m/key-terms/f-b-q-v-b"
type: "key-term"
subject: "AP Physics C: E&M"
unit: "Unit 12"
---

# F_B = q(v × B) — AP Physics C: E&M Definition & Guide

## Definition

F_B = q(v × B) is the magnetic force on a charged particle, equal to the charge times the cross product of its velocity and the magnetic field; the force is perpendicular to both v and B, has magnitude qvB·sinθ, and is zero when the charge is at rest or moving parallel to the field.

## What It Is

F_B = q(v × B) tells you the force a [magnetic field](/ap-physics-c-e-m/unit-12/3-magnetic-fields-of-current-carrying-wires-and-the-biot-savart-law/study-guide/9L5jxtAFTJoI6u5v "fv-autolink") exerts on a single moving charge. Here q is the charge (sign included), v is the particle's velocity vector, and B is the magnetic field vector. Because it's a [cross product](/ap-physics-c-e-m/key-terms/cross-product "fv-autolink"), the force is always perpendicular to both the velocity and the field, and its magnitude is |q|vB·sinθ, where θ is the angle between v and B.

Two consequences fall right out of the math. First, a stationary charge feels no magnetic force at all, since v = 0 kills the cross product. Same deal if the charge moves parallel or antiparallel to B (sinθ = 0). Second, because F_B is always perpendicular to v, the magnetic force can never speed a particle up or slow it down. It only changes direction, which means it does zero work. That's why a charge fired perpendicular to a uniform field traces a circle, with qvB acting as the centripetal force. Direction comes from the right-hand rule, and you flip the answer if q is negative (an [electron](/ap-physics-c-e-m/unit-11/1-electric-current/study-guide/9YRMrkv1PVy23BzH "fv-autolink") pushes the opposite way your fingers say).

## Why It Matters

This equation is the foundation of Topic 12.2, Magnetism and Moving Charges, in [AP Physics C: E&M](/ap-physics-c-e-m "fv-autolink"). Everything else in the magnetism unit builds on it. The force on a current-carrying wire (F = IL × B) is just this equation summed over all the charges drifting in the wire, the Hall effect is this force shoving charges to one side of a conductor, and circular motion of charges in mass spectrometers and cyclotrons comes from setting qvB equal to mv²/r. It's also one half of the full Lorentz force, F = qE + q(v × B), so it links the magnetism unit back to everything you learned about electric fields. If you can take a cross product and use the right-hand rule confidently, you've unlocked most of [Unit 12](/ap-physics-c-e-m/unit-12 "fv-autolink")'s problem types.

## Connections

### [Lorentz force (Unit 12)](/ap-physics-c-e-m/key-terms/lorentz-force)

F_B = q(v × B) is the magnetic half of the full [Lorentz force](/ap-physics-c-e-m/key-terms/lorentz-force "fv-autolink"), F = qE + q(v × B). Velocity selector problems put the two halves in opposition, where qE balances qvB and only charges with v = E/B pass through undeflected.

### [Cross product (Unit 12 math toolkit)](/ap-physics-c-e-m/key-terms/cross-product)

The physics of this equation lives entirely in the cross product. Perpendicularity, the sinθ factor, and the [right-hand rule](/ap-physics-c-e-m/key-terms/right-hand-rule "fv-autolink") aren't extra facts to memorize; they're just what v × B means.

### Kinematics of charged particles (Units 1 and 12)

Since F_B is always [perpendicular](/ap-physics-c-e-m/unit-12/2-magnetism-and-moving-charges/study-guide/aujVCr641dSEbfts "fv-autolink") to v, it acts exactly like a centripetal force from mechanics. Set qvB = mv²/r and you get the radius of circular motion, the bread-and-butter calculation for charges in uniform fields.

### [Hall effect (Unit 12)](/ap-physics-c-e-m/key-terms/hall-effect)

Run a current through a slab in a magnetic field and q(v × B) pushes the moving charges sideways until they pile up on one edge. The resulting voltage even tells you the sign of the charge carriers, which is the Hall effect's claim to fame.

## On the AP Exam

Multiple-choice questions love conceptual traps built on this equation. Expect stems like an electron moving horizontally through a perpendicular field, where you must apply the right-hand rule and then reverse the direction for the negative charge. Other MCQs test whether you know that only moving charges feel a magnetic force, that the force vanishes when v is parallel to B, and that F_B does no work so kinetic energy stays constant. On free-response questions, this equation typically launches a multi-step problem. You might derive the radius or period of circular motion, analyze a velocity selector by balancing qE against qvB, or combine it with kinematics for a charge entering a field region. Always state directions explicitly using the right-hand rule, and watch the sign of the charge. That's the single most common point lost on these problems.

## F_B = q(v × B) vs Lorentz force, F = qE + q(v × B)

F_B = q(v × B) is only the magnetic piece. The full Lorentz force adds the electric force qE, which acts on any charge whether or not it's moving and which CAN do work and change kinetic energy. The magnetic piece requires motion and never does work. If a problem has both E and B fields (like a velocity selector), you need the full Lorentz force, not just the magnetic term.

## Key Takeaways

- The magnetic force on a moving charge is F_B = q(v × B), with magnitude |q|vB·sinθ and direction given by the right-hand rule (reversed for negative charges).
- A charge at rest, or one moving parallel to the magnetic field, experiences zero magnetic force.
- Because F_B is always perpendicular to velocity, the magnetic force does no work and never changes a particle's speed or kinetic energy, only its direction.
- A charge moving perpendicular to a uniform field travels in a circle, and setting qvB = mv²/r gives you the radius r = mv/(qB).
- This equation is the magnetic half of the Lorentz force F = qE + q(v × B), and it scales up to F = IL × B for current-carrying wires.
- The Hall effect, velocity selectors, and mass spectrometers are all direct applications of q(v × B), so mastering this one equation covers a huge slice of Unit 12.

## FAQs

### What is F_B = q(v × B) in AP Physics C: E&M?

It's the equation for the magnetic force on a moving charged particle, where the force equals the charge times the cross product of velocity and magnetic field. The force has magnitude |q|vB·sinθ and points perpendicular to both v and B, per the right-hand rule.

### Can a magnetic force do work on a charged particle?

No. Since q(v × B) is always perpendicular to the velocity, the magnetic force does zero work and cannot change a particle's speed or kinetic energy. It only bends the path, which is why charges in uniform fields move in circles or helices at constant speed.

### How is F_B = q(v × B) different from the Lorentz force?

F_B = q(v × B) is just the magnetic part. The full Lorentz force is F = qE + q(v × B), which adds the electric force qE. The electric part acts on stationary charges and can do work; the magnetic part requires motion and never does work.

### Does a charge at rest feel a magnetic force?

No. With v = 0 the cross product is zero, so a stationary charge feels no magnetic force no matter how strong B is. This is a classic MCQ trap, since the same charge would still feel an electric force from any E field present.

### Which way does the force point for an electron in a magnetic field?

Use the right-hand rule with v and B, then reverse the answer because the electron's charge is negative. An electron moving horizontally through a perpendicular field gets deflected opposite to the direction your right hand predicts for a positive charge.

## Related Study Guides

- [12.2 Magnetism and Moving Charges](/ap-physics-c-e-m/unit-12/2-magnetism-and-moving-charges/study-guide/aujVCr641dSEbfts)

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