Question

Derive an expression for the force per unit length between two infinitely long parallel straight conductors carrying currents $I_1$ and $I_2$ at separation $d$. Under what condition the force is repulsive.

Answer 1

The expression for the force between two parallel current carrying wires.

Consider two long parallel wires ABAB and CDCD carrying current $I_1$I1 and $I_2$I2. Let rr be the separation between them.
The magnetic field produced by current $I_1$ at any point on wire CDCD is
$B_1=\frac{{\mu }_o{I}_1}{2\pi r}$
This field acts perpendicular to the wire CDCD and points into the plane of the paper. It exerts a force on current carrying wire CDCD. The force acting on length ll of the wire CDCD will be
$F_2=I_{2}l{B}_{1}sin{90}^o=I_{2}l.\frac{{\mu }_o{I}_1}{2\pi r}.l$
Force per unit length,
$f=F_2/l=\frac{{\mu }_o{I}_1{I}_2}{2\pi r}$
When the currents in the two parallel wires flow in opposite directions (anti-parallel), the forces between the two wires are repulsive.
Thus, parallel currents attract and antiparallel currents repel.