Force between parallel currents

Wire a makes a field B_a = μ₀I_a/2πd all along a parallel wire b a distance d away. Wire b, carrying I_b, feels a sideways force I_b L B_a on a length L.

So the force per unit length is f = μ₀I_aI_b/2πd, equal in size on both wires and opposite in direction, as Newton's third law needs for steady currents.

Currents in the same direction attract; currents in opposite directions repel. This is the reverse of charges, where like repels like.

Definition of the ampere: the steady current that, flowing in two very long, straight, thin parallel wires 1 m apart in vacuum, makes each feel 2 × 10⁻⁷ N per metre. This definition dates from 1946.

In practice the earth's field and stray fields must be removed, long wires are replaced by coils, and a current balance measures the force. One coulomb is then the charge carried past a point in 1 s by 1 A.

Example 4.9: in a horizontal earth's field of 3.0 × 10⁻⁵ T pointing south to north, a wire with 1 A running east to west feels IB = 3 × 10⁻⁵ N per metre (downwards), far more than 2 × 10⁻⁷ N/m; running south to north it feels nothing.