A long straight wire is coplanar with a current carrying circular loop of radius R as shown in Fig. Current flowing through wire and the loop is `I_0` and I, respectively. If distance between centre of loop and wire is `r=2R,` calculate force of attraction between the wire and the loop.

Explanation

Consider two equal elemental
arc length MN and M'N' of lengths
`Rd theta`. The distance of each elemental
length from straight wire is
`x=(r-Rcos theta)`
`=R(2-cos theta)…..(i)`
The magnetic induction at these
elemental arc lengths due to current
`I_0` in the wire is by
`B=(mu_0I_0)/(2pix)=(mu_0I_0)/(2piR(2-cos theta)).....(ii)`
The magnitude of the force on each elemental length
`dF'=BI(dl)=BI(Rd theta)`
`=(mu_0I_0IRd theta)/(2piR(2-cos theta))=(mu_0I_0I d theta)/(2pi(2-cos theta))`
The direction of dF' is shown in Fig.

The resultant force dF is given by
`dF=dF'.2cos theta=(mu_0I_0i d theta)/(2pi(2-cos theta))xx2cos theta`
`=(mu_0I_0I)/pi[(cos theta d theta)/(2-cos theta)]`
`F=(mu_0I_0I)/pi int_0^(pi)(cos theta )/(2-cos theta)d theta=mu_0I_0I((2-sqrt3)/(sqrt3))`