Two infinitely long straight parallel wires, '1' and '2', carrying steady currents I₁ and I₂ in the same direction are separated by a distance d. Obtain the expression for the magnetic field  due to the wire '1' acting on wire '2'. Hence find out, with the help of a suitable diagram, the magnitude and direction of this force per unit length on wire '2' due to wire '1'. How does the nature of this force changes if the currents are in opposite direction? Use this expression to define the S.I. unit of current.

Consider a straight conductor XY lying in the plane of paper. Consider a point P at a perpendicular distance from the straight conductor.

Magnetic field induction (B) at a point P due to current I passing through conductor XY is given by,
 
where,  are the angles made by point X and Y respectively.
At the centre of the infinite long wire, ₁ = ₂ = 90^o

So, magnetic field is given by, 

Magnetic field produced by current I₁ at any point on conductor Rs is given by, 

 

Force acting on length l of the conductor RS will be,

An equal force is exerted into the wire PQ by the field of conductor RS which is given by, 

Thus, the force is attractive when the current is acting along the same direction. When, current flows in opposite direction, the forces between the two conductors are repulsive.

One Ampere is that value of constant current which when flowing through each of the two parallel uniform long conductors placed in free space at a distance of 1m from each other will attract or repel with a force of 2 10^-7 Newton per metre of their length.