Prove that the parallel conductors of infinite length, carry current in the same direction, attravt each other. Deduce the expressions for the force per unit length experienced by each conductor.

Explanation

Expression for the force between two parallel current carying wires :
Consider two long wires AB and CD carrying currents `l _(1) and l _(2)` let r be the sparation between them. The magnetic field produced by current `l _(1)` at any point on wire CD is

`B _(1) = ( mu _(0))/( 4pi) ( 2 I _(1))/( r).`
This fiels acts perpendicular to the wire CD and points in the plane of the paper i.e. inwards. It exerts a force on current carrying wire CD. The forec eacting on length l of the wire CD will be ` F _(2) = I _(2) lB _(1) sin 90^(@) = I _(2) l B _(1) = I _(2) l ( mu _(0))/( 4pi ) ( 2pi)/(r) = ( mu _(0) I _(1)I _(2))/( 2 pi r ) l therefore` Force per unit length, `F = ( F _(2))/( l ) = ( mu _(0) I _(1) I _(2))/( 2 pi r ).`
This force acts at right angles to CD towards AB in the plane of the paper.
Similarly, an equal force is exerted in the wire AB by the field of wire CD. Thus when the current in the two wires are in the same direction, the force between them are attractive i.e., `vec F _(1) = - vec F _(2).` And also parallel currents attracts and antiparallel currents repel.