How do two parallel current-carrying wires attract or repel each other?

The interaction between two parallel current-carrying wires is a fundamental aspect of electromagnetism. Each wire produces its own magnetic field due to the electric current flowing through it, and the force between the wires arises from the interaction of one wire's current with the magnetic field generated by the other.12

According to the right-hand rule, the magnetic field lines circle around a current-carrying wire in a plane perpendicular to the wire. The nature of the force between two parallel wires depends on the relative direction of these currents:1

• Attraction: When currents in two parallel wires flow in the same direction, each wire exerts an attractive force on the other.12
• Repulsion: When currents flow in opposite directions, each wire exerts a repulsive force on the other.12

The magnitude of the force per unit length ($F/l$) between two long, parallel wires separated by a distance $r$ is given by the formula:
$$\frac{F}{l}=\frac{{\mu }_0{I}_1{I}_2}{2\pi r}$$
where $I_1$ and $I_2$ are the currents in the wires and ${\mu }_0$ is the magnetic permeability of free space.1