Torque on a Current Loop
Loops twist to line up with the field.
Lesson at a glanceShow the key points
- Γ⃗ = μ⃗ × B⃗ (34.7), μ⃗ = niA⃗ (34.8); valid for any flat loop.
- A⃗ by the right-hand rule; torque turns μ⃗ toward B⃗.
- Orbiting electron: μ = evr/2, μ/L = e/2m.
A current loop in a magnetic field feels no net force, but it does feel a twist. That twist is what spins a motor and swings a meter needle.
A rectangular loop
A rectangular loop PQRS (sides l and b) carries current i in a uniform field B⃗, with the loop’s plane perpendicular to B⃗. The forces on opposite sides are equal, opposite and along the same line: no net force and no torque.
Now rotate the loop by θ about the line joining the midpoints of QR and SP. Sides PQ and RS (length l) still feel ilB each, unchanged in direction, but their lines of action separate: each now has moment (b/2) ilB sin θ about the axis. The forces on the other two sides stay equal, opposite and collinear. Total torque:
These results hold for flat loops of any shape, not just rectangles.
A circular coil of 50 turns and radius 3.0 cm carries 2.0 A. Find its magnetic moment, and the torque on it in a 0.40 T field at 30° to its axis.
A 200-turn coil 2.0 cm × 5.0 cm carries 0.50 A in a 0.80 T field. Its area vector makes 30° with the field. Find the torque.
An orbiting electron is a magnet
An electron circling at speed v on a radius r passes any point v/2πr times a second, so it is a current i = ev/2πr. Its magnetic moment:
The ratio of magnetic moment to angular momentum is the same for every orbit: a link between atoms and magnetism that later chapters build on.
- 1Torque on a loop: Γ⃗ = μ⃗ × B⃗, μ⃗ = niA⃗ (34.7–34.8); no net force in a uniform field.
- 2A⃗ by the right-hand rule; torque turns μ⃗ toward B⃗; zero when aligned.
- 3An orbiting electron has μ = evr/2 and μ/L = e/2m.
Check yourself
Keep exploring
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