Torque on a current loop
Forces on a rectangular loop
Hang a rectangular loop of wire, carrying a current I, in a uniform magnetic field B. Its sides have lengths a and b. What does the field do to it?
First, let the field lie in the plane of the loop, along the two sides of length b. Those two sides are parallel to B, so they feel no force at all.
The two sides of length a are perpendicular to B. Each feels a force I a B. The currents in them run opposite ways, so one force points into the screen and the other out of it.
The two forces are equal and opposite, so the net force on the loop is zero. But they do not act along the same line. They form a couple, which turns the loop.
The arm of the couple is b. So the torque is I a B times b, which is I A B, where A equals a b is the area of the loop.
Now turn the loop so that its normal makes an angle θ with B. The forces on the sides of length b now cancel along the same line, so they add no torque.
The forces on the sides of length a are still I a B, but their arm shrinks to b sin θ. So the torque is τ equals I A B sin θ.
The torque is largest when the plane of the loop lies along B, θ = 90°. It is zero when the plane of the loop is perpendicular to B, θ = 0°.
The loop as a magnetic dipole
Define a vector m, the magnetic moment of the loop: its size is I A, and it points along the normal given by the right-hand thumb rule, fingers along the current. Its unit is ampere metre squared.
Then the torque is simply τ equals m cross B. For a coil of N turns, m equals N I A, and τ equals N I A B sin θ.
This torque always tries to turn m into line with B. At θ = 0°, m along B, there is no torque, and a small push brings the loop back: stable equilibrium.
At θ = 180°, m against B, there is no torque either. But the slightest push and the loop flips right over: unstable equilibrium.
Example: a coil of 100 turns, radius 10 centimetres, carries 3.2 amperes. Its area is π times 0.1 squared, 3.14 × 10⁻² square metre. So m equals 100 times 3.2 times 0.0314: about 10 ampere metre squared.
Place it in a field of 2 tesla with its normal at 90° to the field. The torque is m B, 10 times 2: 20 newton metre. At 30°, it is half of that, 10 newton metre.
Now recall the field of a loop far along its axis: μ₀ I R² over 2x³. Since m equals I πR², this is B equals μ₀ m over 2π x³.
Compare the electric dipole: E equals 2p over 4πε₀ x³. Same shape, with p replaced by m and 1 over ε₀ by μ₀. A current loop is a magnetic dipole.
Its field lines come out of one face, the north pole, and go into the other, the south pole, just like a small bar magnet. But unlike charges, isolated magnetic poles have never been found.
Try it: a loop between the poles
- At θ = 90°, only the two vertical sides feel forces, and they form a couple. Drag the scene to see it from above.
- Move θ towards 0°: the forces on the vertical sides stay the same size, but their arm shrinks, so τ = mB sin θ falls.
- Press “Release the loop”: it swings and settles with m along B.
- Double N or I: m and the torque double. The torque depends only on the area, not on the shape.
Facing the wrong way
The magnetic moment m of a current loop points exactly opposite to a uniform field B (θ = 180°). Which is true?
- The torque is largest, mB
- The torque is zero and the loop is in stable equilibrium
- The torque is zero and the loop is in unstable equilibrium
- The loop feels a net force along B
Answer
C. τ = mB sin θ and sin 180° = 0, so there is no torque. But turn the loop a little and the torque turns it further away, until m lines up with B: the equilibrium is unstable. In a uniform field the net force on any current loop is zero.
Torque on a coil
A circular coil of 100 turns and radius 10 cm carries a current of 3.2 A. It is placed in a uniform field of 2 T with its normal at 30° to the field. What is the torque on it?
- 0.10 N m
- 10 N m
- 17 N m
- 20 N m
Answer
B. m = NIA = 100 × 3.2 × π(0.1)² ≈ 10 A m². Then τ = mB sin θ = 10 × 2 × sin 30° = 10 N m. Using cos 30° gives 17 N m; forgetting sin θ gives 20 N m; forgetting N gives 0.10 N m.
Summary
- Torque on a loop:
τ = m × B, τ = N I A B sin θ - Magnetic moment:
m = N I A (A m²) - Far field on the axis:
B = μ₀ m / 2πx³
- In a uniform field the net force on a current loop is zero, but there can be a torque.
- The torque is largest when the plane of the loop lies along B, and zero when m is along or against B.
- m along B is stable equilibrium; m against B is unstable.
- A current loop behaves as a magnetic dipole, like a small bar magnet.