An arrangement of three parallel straight wires placed perpendicular to plane of paper carrying same current ‘I’ along the same direction is shown in Fig. Magnitude of force per unit length on the middle wire ‘B’ is given by

Concept:

Whenever current flows through a conductor magnetic field is produced around it. Magnetic field depends on the current in the conductor and distance of the point from the conductor. Direction of magnetic field is determined using right hand thumb rule.

Magnetic field is calculated using the given formula:

\(B=\frac{\mu_0I}{2\pi r}\)

where B and I is the magnitude of magnetic field and current respectively, r is the distance from the wire where the magnetic field is calculated.

Force per unit length

\(F=I(\vec{L}\times\vec{B})\)

where F is the force on the conductor of L length carrying I amount of current when placed in ‘B’ magnetic field.

Then, Force per unit length is 

\(\frac{F\ }{L}=IB\)

Using \(B=\frac{\mu_0I}{2\pi r}\), we get

\(F'=\frac{F\ }{L}=\frac{\mu_0I^2}{2\pi r}\)

Calculation:

Force between BC and AB will be same in magnitude.

\(F_{BC}=F_{BA}=\frac{\mu_0l^2}{2\pi d}\)

F = \(\sqrt2F_{BC}\)

= \(\sqrt2\frac{\mu_0}{2\pi}\frac{l^2}{d}\)

\(\rm F = \frac{\mu_0l^2}{\sqrt2\pi d}\)