Forces between currents and torque on loops

One current makes a field; a second current sitting in that field feels a force. So two current-carrying wires push or pull each other.

For long parallel wires a distance d apart, the force per metre of length is:

f = µ₀ I₁I₂ / (2πd)

• Same direction currents attract.
• Opposite direction currents repel.

This is the opposite of charges, where 'like' repel. Example: 5 A and 10 A, 10 cm apart: f = 2 × 10⁻⁷ × 50 / 0.1 = 10⁻⁴ N/m.

This force defines the ampere: 1 A is the current that, flowing in two long parallel wires 1 m apart, gives a force of 2 × 10⁻⁷ N per metre.

A loop in a uniform field: the forces on opposite sides are equal and opposite, so there is no net force, but they twist the loop.

Its magnetic moment is m = NIA, a vector along the loop's normal (by the right-hand rule). The torque is:

τ = m × B, size NIAB sin θ, with θ the angle between the normal and B.

Example: 50 turns of area 0.002 m², carrying 1 A, in 0.5 T, with the plane of the coil along B (θ = 90°): τ = 50 × 1 × 0.002 × 0.5 = 0.05 N m, the largest possible.

The torque turns the loop until m lines up with B: that is the stable position, with energy U = –mB cos θ at its lowest. This turning effect drives electric motors and meters.