Force of Attraction between two Long Parallel Wires

The two-wire demonstration that opened this chapter (Fig. 10.1, Section 10.1) showed QUALITATIVELY that current-carrying wires exert forces on one another -- attractive for parallel currents, repulsive for antiparallel ones -- but left the underlying mechanism and the exact magnitude of the force unexplained. With the tools of the intervening sections now in hand -- the Biot-Savart-law field of a long straight wire (Section 10.10.1) and the force on a current-carrying wire in a magnetic field (Section 10.5.1) -- that force can now be derived precisely.

Consider two long, straight, parallel wires, separated by a perpendicular distance $d$, carrying currents $I_1$ and $I_2$ respectively (Fig. 10.17). The magnetic field produced by wire 1, AT THE LOCATION of wire 2, is, directly from Section 10.10.1,

directed (by the right-hand rule) into the plane of the paper if the currents are drawn in the plane, circling around wire 1. Wire 2, carrying current $I_2$, now sits in this field produced by wire 1, and so experiences a force given directly by the straight-wire force law of Section 10.5.1: for a segment of wire 2 of length $L$,

directed, by applying the Lorentz force law with this specific field's direction, TOWARDS wire 1 -- i.e. attractive, for the case where both currents flow in the same direction (as drawn). Dividing through by $L$ gives the FORCE PER UNIT LENGTH between the two wires,

attractive whenever $I_1$ and $I_2$ flow in the SAME direction (parallel currents), and, by an identical calculation (only the direction of the resulting force flips), repulsive whenever they flow in OPPOSITE directions (antiparallel currents) -- precisely reproducing, now with an exact formula attached, the qualitative Fig. 10.1 demonstration from the very start of the chapter: parallel currents attract, antiparallel currents repel. …