F_B = q(v × B) in AP Physics C: Electricity and Magnetism

In AP Physics C: E&M, F_B = q(v × B) is the magnetic force on a charge q moving with velocity v through a magnetic field B. Its magnitude is |q|vB sinθ, and its direction is perpendicular to both v and B, set by the right-hand rule (flipped for negative charges).

What is F_B = q(v × B)?

F_B = q(v × B) is the equation for the magnetic force on a moving charged particle. F_B is the magnetic force, q is the charge (with its sign), v is the particle's velocity, and B is the magnetic field. The force comes from a cross product, so two things follow. The magnitude is F = |q|vB sinθ, where θ is the angle between v and B. And the direction is always perpendicular to both v and B.

That cross product carries the whole personality of this force. A charge sitting still feels nothing (v = 0). A charge moving parallel to the field feels nothing (sin 0° = 0). The force is biggest when the charge cuts straight across the field lines (θ = 90°). Because the force is always sideways to the motion, it can bend a particle's path but never speeds it up or slows it down. In other words, the magnetic force does zero work on a moving charge. Think of it as a steering wheel with no gas pedal.

Why F_B = q(v × B) matters in AP® Physics C: E&M

This equation is the foundation of Topic 12.2 Magnetism and Moving Charges in Unit 12: Magnetic Fields and Electromagnetism. Almost everything else in Unit 12 builds on it. The force on a current-carrying wire is this same force added up over all the moving charges in the wire. The circular paths of particles in a field come straight from this force acting as a centripetal force. It also sets up Unit 13, where magnetic forces on charges inside a moving conductor produce motional emf. On the exam, it's where vector reasoning (the cross product and right-hand rule) meets Newton's second law, which is exactly the kind of calculus-and-vectors thinking Physics C rewards.

How F_B = q(v × B) connects across the course

Right-Hand Rule and the Cross Product (Unit 12)

The right-hand rule is just a physical way to evaluate v × B. Point your fingers along v, curl them toward B, and your thumb gives the force direction for a positive charge. For an electron, the force points the opposite way, because q is negative and flips the vector.

Lorentz Force and the Electric Force (Units 8, 12)

Combine Unit 8's electric force with this one and you get the Lorentz force, F = q(E + v × B). When E and B are crossed so the two forces cancel, only particles with speed v = E/B go straight through. That's a velocity selector, and it's a classic problem that blends both units.

Kinematics of a Charged Particle in a Field (Unit 12)

Because the magnetic force stays perpendicular to velocity, it acts as a centripetal force. Setting |q|vB = mv²/r gives a circular path with r = mv/(|q|B). The period T = 2πm/(|q|B) doesn't depend on speed at all, which surprises a lot of people.

Motional EMF and Induction (Unit 13)

When a conducting rod moves through a magnetic field, every charge inside it moves with the rod. So each charge feels q(v × B) along the rod's length. That push separates charge and creates the motional emf ε = BLv, the bridge from magnetic force to electromagnetic induction.

Is F_B = q(v × B) on the AP® Physics C: E&M exam?

Multiple-choice questions usually test this equation conceptually. Fiveable practice questions show the pattern well. One asks which charged object would feel a magnetic force when moving perpendicular to a field. Another asks you to describe the force on an electron moving horizontally through a perpendicular field. To answer these, you need to:

• Check the conditions for any force at all. The object must be charged, moving, and not moving parallel to B.
• Find the direction with the right-hand rule, then flip it for negative charges like electrons.
• Recognize that the force is perpendicular to velocity, so it changes direction but not speed, and does no work.

No released FRQ has used this equation verbatim. Still, free-response problems about charged-particle motion, velocity selectors, and forces on current-carrying wires all depend on it. In those problems, expect to derive radius or period from Newton's second law, justify a path's curvature direction, or explain why kinetic energy stays constant.

F_B = q(v × B) vs Electric force (F = qE)

The electric force acts on any charge, moving or not, and points along E (or opposite it for negative charges), so it can do work and change speed. The magnetic force acts only on moving charges, points perpendicular to both v and B, and never does work. A quick test is to ask whether the force would still exist if the charge stopped. If it would, it's electric. If it wouldn't, it's magnetic.

Key things to remember about F_B = q(v × B)

• The magnetic force on a moving charge is F_B = q(v × B), with magnitude |q|vB sinθ, where θ is the angle between velocity and field.

• A charge that is at rest or moving parallel to the magnetic field feels no magnetic force at all.

• The direction of the force comes from the right-hand rule for positive charges and is reversed for negative charges such as electrons.

• Because the magnetic force is always perpendicular to velocity, it does zero work and cannot change a particle's speed or kinetic energy.

• A charge moving perpendicular to a uniform field travels in a circle of radius r = mv/(|q|B), with a period that does not depend on its speed.

• This same force explains the force on current-carrying wires in Unit 12 and motional emf in Unit 13.

Frequently asked questions about F_B = q(v × B)

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

It's the equation for the magnetic force on a charge q moving with velocity v through a magnetic field B. The force has magnitude |q|vB sinθ and points perpendicular to both v and B, which makes it central to Topic 12.2 in Unit 12.

Does the magnetic force do work on a moving charge?

No. Since F_B is always perpendicular to v, the dot product F · v is zero, so the force does no work. It can bend the path, but the particle's speed and kinetic energy stay constant.

Does a charge moving parallel to a magnetic field feel a force?

No. When v is parallel or antiparallel to B, θ is 0° or 180°, and sinθ = 0, so the magnetic force is zero. The particle just keeps moving in a straight line (unless some other force acts on it).

How is the magnetic force different from the electric force F = qE?

The electric force acts on charges whether they move or not and points along the field, so it can do work. The magnetic force needs motion, points perpendicular to both v and B, and does no work. Together they make up the Lorentz force F = q(E + v × B).

Which way does the magnetic force point on an electron?

Use the right-hand rule for v × B, then reverse the result, because an electron's charge is negative. If the right-hand rule gives +y for a proton, an electron with the same velocity in the same field feels a force in the −y direction.