Torque in AP Physics C: Electricity and Magnetism

In AP Physics C E&M, torque is the rotational effect of a force about an axis (τ = r × F). It is tested mostly as the torque on a current loop in a magnetic field, τ = μ × B with μ = NIA, which can twist a loop even when the net magnetic force on it is zero.

What is Torque?

Torque measures how well a force makes something rotate around an axis. The math is τ = r × F, where r runs from the axis to the point where the force acts. Only the part of the force perpendicular to r does any twisting, so the magnitude is τ = rF sinθ. Push on a door right at the hinge and nothing happens. Push at the handle and it swings. Same force, different torque.

In E&M, torque shows up whenever magnetic forces act on different parts of a current-carrying object. The classic case is a current loop in a uniform magnetic field. The forces on opposite sides of the loop are equal and opposite, so the net force is zero. But those forces don't act along the same line, so they form a couple that twists the loop. All of this packs into one clean formula, τ = μ × B. Here μ = NIA is the loop's magnetic dipole moment, and it points perpendicular to the loop's plane (use the right-hand rule with the current). The magnitude is τ = NIAB sinθ, where θ is the angle between μ and B. Think of it this way: a current loop acts like a tiny compass needle, and the field keeps twisting it until μ lines up with B.

Why Torque matters in AP® Physics C: E&M

Torque lives in Unit 12: Magnetic Fields and Electromagnetism, alongside topic 12.3 (Magnetic Fields of Current-Carrying Wires and the Biot-Savart Law). Once you know how to find magnetic fields and the forces they exert on wires, torque is the next question. Will the wire or loop rotate? This idea turns a page of force vectors into one compact result, τ = μ × B. It's also the working principle of every electric motor.

It matters for the exam because it tests whether you can think about forces at two levels at once. Translational equilibrium (net force) and rotational equilibrium (net torque) are separate conditions. AP questions love objects that satisfy one but not the other. Torque also links E&M back to the rotational dynamics you learned in mechanics, since a torque on a loop with mass produces an angular acceleration through τ = Iα.

How Torque connects across the course

Magnetic Force on Current-Carrying Wires (Unit 12)

Torque is built out of magnetic forces. You find the force on each segment with F = IL × B, then ask where each force acts relative to the axis. For a loop in a uniform field, the forces cancel as a sum but still twist the loop, because they're applied at different points.

Moment Arm and Lever Arm (Unit 12)

The moment arm is the perpendicular distance from the axis to the force's line of action, and torque equals force times that distance. When a loop's plane is parallel to B, the side forces have their biggest moment arm, so torque is at its maximum. When μ lines up with B, the moment arm shrinks to zero and so does the torque.

Angular Acceleration and Rotational Dynamics (Unit 12)

A magnetic torque obeys the same Newton's second law for rotation as any mechanical torque, τ = Iα. Find the torque with NIAB sinθ, find the loop's moment of inertia from its mass and shape, and divide. That's how an E&M problem turns into a rotation problem.

Electric Dipoles in Electric Fields (Unit 8)

An electric dipole in a uniform electric field feels zero net force but a torque τ = p × E, which is the exact twin of τ = μ × B. If you understand one, you understand both. A dipole always gets twisted toward alignment with the field.

Is Torque on the AP® Physics C: E&M exam?

Expect torque in MCQs and as one part of a multi-step FRQ about a loop in a magnetic field. Here's what you'll actually do with it:

• Separate net force from net torque. A common stem puts a rectangular loop in a uniform field with its plane parallel to the field lines and asks you to describe the forces. The right answer is that the net force is zero but there's a nonzero (actually maximum) torque. Questions with perpendicular current-carrying wires ask the same thing a different way. Will the second wire translate, rotate, or both?
• Get the angle right. In τ = NIAB sinθ, θ is measured between the dipole moment μ (the normal to the loop) and B, not between the loop's plane and B. If a problem says the plane makes 60° with the field, the angle you need is 30°. This is the most common trap.
• Chain into rotation. Some problems give you the loop's mass and dimensions and ask for its initial angular acceleration. Compute torque, find the moment of inertia, then use α = τ/I.
• Reason about nonuniform fields. A small loop near a long straight wire sits in a field that changes with distance (topic 12.3). There you have to think about both the torque that aligns μ with B and any net force from the field gradient.

No released FRQ has used the word "torque" as a headline term. Still, loop-in-a-field setups are a natural FRQ scenario, and a question about whether the loop rotates is answered with this idea.

Torque vs Magnetic Force

Magnetic force (F = IL × B or F = qv × B) tells you whether an object gets pushed in some direction. Torque (τ = μ × B for a loop) tells you whether it gets twisted about an axis. They can be totally independent. A current loop in a uniform field has zero net force but can have a large torque. So "no net force" never means "nothing happens." Always check both.

Key things to remember about Torque

• Torque is the rotational effect of a force, calculated as τ = r × F, with magnitude rF sinθ.

• A current loop in a magnetic field feels a torque τ = μ × B, where μ = NIA points perpendicular to the loop by the right-hand rule.

• In a uniform magnetic field, a current loop has zero net force but can still have a nonzero torque, so it rotates without moving its center.

• The torque on a loop is maximum when the loop's plane is parallel to B, and it drops to zero when μ is aligned with B.

• The angle in τ = NIAB sinθ is measured between the normal to the loop and the field, not between the loop's plane and the field.

• A magnetic torque on a loop with mass causes angular acceleration according to τ = Iα, just like a mechanical torque.

Frequently asked questions about Torque

What is torque in AP Physics C E&M?

Torque is the twisting effect of a force about an axis, τ = r × F. In E&M it mostly appears as the torque on a current loop in a magnetic field, τ = μ × B, where μ = NIA. This is the principle that makes electric motors spin.

If the net magnetic force on a current loop is zero, is the torque also zero?

No. In a uniform field the forces on opposite sides of a loop cancel, but they act along different lines, so they create a torque. A loop with its plane parallel to B actually feels the maximum torque, NIAB, even though the net force is zero.

How is torque different from magnetic force?

Magnetic force decides whether an object accelerates in a direction, and torque decides whether it rotates about an axis. You check them separately. A loop in a uniform field shows the difference perfectly, with zero net force but nonzero torque.

What angle do you use in τ = NIAB sinθ?

θ is the angle between the magnetic dipole moment μ (perpendicular to the loop's plane) and the magnetic field. If the loop's plane makes 60° with the field, μ makes 30° with it, so you use sin 30°. Mixing these up is one of the most common mistakes on loop questions.

How do you find the angular acceleration of a current loop in a magnetic field?

First find the torque with τ = NIAB sinθ, then find the loop's moment of inertia about the rotation axis from its mass and shape. Divide to get α = τ/I. For example, a 5.0 cm square loop carrying 1.5 A in a 0.40 T field, with its plane at 60° to the field, feels τ = (1.5)(0.0025)(0.40)(sin 30°) = 7.5 × 10⁻⁴ N·m.