Charged particle trajectory in AP Physics 2

A charged particle trajectory is the curved (often circular) path a charged particle follows in a magnetic field, because the magnetic force F_B = qvB sin θ always acts perpendicular to velocity. The radius of the circle, r = mv/(qB), depends on the particle's mass, speed, charge, and the field strength.

What is Charged particle trajectory?

When a charged particle moves through a magnetic field, the field exerts a force on it given by F_B = qvB sin θ, and that force always points perpendicular to the particle's velocity (CED 12.2.B). A perpendicular force can't speed the particle up or slow it down. It can only change the direction of motion. So instead of accelerating in a straight line like a charge in an electric field would, the particle bends. If the velocity is exactly perpendicular to the field, the bending is perfectly steady and the particle traces a circle.

Here the magnetic force is doing the job of a centripetal force. Set qvB equal to mv²/r and you get the workhorse equation r = mv/(qB). Fast or heavy particles make big circles. Strong fields or large charges make tight circles. The direction of the curve comes from the right-hand rule (and flips for negative charges). One more thing worth remembering: because the force is always perpendicular to velocity, the magnetic force does zero work on the particle, so its speed and kinetic energy never change while it spirals around.

Why Charged particle trajectory matters in AP® Physics 2

This term lives in Topic 12.2 (Magnetism and Moving Charges) in Unit 12 and directly supports learning objective 12.2.B, describing the force a magnetic field exerts on a moving charge. It's also where two big AP Physics ideas collide. The magnetic force from Unit 12 becomes the centripetal force from your mechanics toolkit, and the exam loves questions that make you stitch those together. If you can derive r = mv/(qB) from F_B = qvB and F_c = mv²/r, you've basically unlocked the whole family of problems: mass spectrometers, velocity selectors, and charge-to-mass ratio measurements all start from this one trajectory idea.

How Charged particle trajectory connects across the course

Circular motion and centripetal acceleration (Unit 12 application)

A charged particle trajectory is just uniform circular motion with the magnetic force playing the role of the centripetal force. Setting qvB = mv²/r is the single most common derivation in this topic.

F_B = qvB sin θ (Unit 12)

This equation is the engine behind the trajectory. The sin θ term tells you when curving happens at all. If the particle moves parallel to the field (θ = 0), there's no force and the path is a straight line.

Velocity selector (Unit 12)

A velocity selector crosses an electric field with a magnetic field so only particles with one specific speed travel straight. Particles at any other speed curve off course, which is the trajectory idea used as a filter.

Charge-to-mass ratio (Unit 12)

Rearrange r = mv/(qB) and you get q/m = v/(rB). Measuring the radius of a particle's curved path is exactly how experiments determine charge-to-mass ratios, a classic lab-style question setup.

Is Charged particle trajectory on the AP® Physics 2 exam?

No released FRQ uses the phrase "charged particle trajectory" verbatim, but the physics behind it shows up constantly in Unit 12 questions. Multiple-choice stems typically show a particle entering a field region (often drawn with dots or ×'s for field direction) and ask you to predict the direction of the curve, compare radii of two different particles, or identify what happens when the charge sign flips. Free-response questions tend to ask for the derivation. You combine F_B = qvB with F_c = mv²/r, solve for r, then reason about how changing m, v, q, or B changes the path. Be ready to justify, in words, why the speed stays constant (the force is perpendicular to velocity, so it does no work).

Charged particle trajectory vs Trajectory of a charge in an electric field

An electric field pushes a charge along (or against) the field lines, so the particle speeds up, slows down, or follows a parabolic path like a projectile, and the field does work on it. A magnetic field pushes perpendicular to velocity, so the particle curves into a circle at constant speed and the field does zero work. Electric fields change kinetic energy. Magnetic fields only change direction.

Key things to remember about Charged particle trajectory

• A charged particle moving perpendicular to a uniform magnetic field travels in a circle because the magnetic force always points perpendicular to its velocity.

• The radius of the circular path is r = mv/(qB), which you derive by setting the magnetic force qvB equal to the centripetal force mv²/r.

• The magnetic force does zero work on the particle, so its speed and kinetic energy stay constant even though its direction constantly changes.

• Use the right-hand rule with v and B to find the force direction, then flip the answer if the charge is negative.

• If the particle's velocity is parallel to the field, sin θ = 0, the force vanishes, and the particle moves in a straight line.

• Heavier or faster particles curve in bigger circles, while stronger fields or larger charges curve them more tightly.

Frequently asked questions about Charged particle trajectory

What is a charged particle trajectory in AP Physics 2?

It's the curved path a charged particle follows in a magnetic field. Because the force F_B = qvB sin θ is always perpendicular to the velocity, a particle moving perpendicular to the field traces a circle with radius r = mv/(qB).

Does a magnetic field speed up a charged particle?

No. The magnetic force is always perpendicular to the velocity, so it does zero work and never changes the particle's speed or kinetic energy. It only changes the direction of motion.

How is a particle's path in a magnetic field different from its path in an electric field?

In a magnetic field the path is circular at constant speed, since the force stays perpendicular to velocity. In an electric field the force is along the field lines, so the particle accelerates and follows a straight or parabolic path while gaining or losing kinetic energy.

How do you find the radius of a charged particle's circular path?

Set the magnetic force equal to the centripetal force, qvB = mv²/r, and solve to get r = mv/(qB). The AP exam frequently asks you to do this derivation and then reason about how changing mass, speed, charge, or field strength affects the radius.

What happens if the charged particle moves parallel to the magnetic field?

Nothing curves. When velocity and field are parallel, θ = 0 in F_B = qvB sin θ, so the force is zero and the particle continues in a straight line at constant velocity.