The Force Between Two Parallel Wires

Picture two straight wires running parallel to each other, each carrying a steady current. Wire 1 makes a magnetic field that circles around it, following the right-hand rule. At the location of wire 2, that field points in some direction perpendicular to both wires. Wire 2's current is a stream of moving charge sitting inside that field, so it feels a force. Flip the roles and wire 2's field does the same thing to wire 1. The two wires end up pushing or pulling on each other through this back-and-forth magnetic interaction, not through anything electric.

To get the force, first find wire 1's field at the distance d where wire 2 sits: B = mu0 I1 / (2 π d). Then treat a length L of wire 2 as carrying current I2 through that field, using F = I2 L B. Substituting gives F/L = mu0 I1 I2 / (2 π d). Work out the direction with the right-hand rule: point fingers along I1, curl toward wire 2 to find B there, then cross I2 L into B. Currents flowing the same way pull the wires together; currents flowing opposite ways push them apart. Neither wire has any excess charge, so there is no electric field doing this work — it is purely a magnetic effect between two moving charge distributions.

This result is the one place where the two halves of the unit meet directly: Ampere's law gives you the field from one wire, and the moving-charge force law turns that field into a force on the other. It also sets up the formal definition of the ampere itself, since this force per unit length between known currents at a known separation is exactly what a lab can measure to calibrate current.