A loop, carring a current i, lying in the plane of the paper, is in the field of a long straight wire with current `i_0` (inward) as shown in Fig. Find the torque acting on the loop.

Explanation

The field due to current carrying wire is tangential to every point on the circular portion of the loop and hence the forces acting on these segments are zero.

Now consider two small elements of length dr at a distance r from the axis symmetrically as shown in fig.. The magnitude of the force experienced by each element is `dF = B i dr = ( (mu_0)/(2pi) (i_0)/(r) ) idr`
On element 1 it is into the page and on 2 it out of the page, `d tau = dF xx 2r sin theta `
` = ( (mu_0i_0 i)/(2pi r) dr) xx 2r sin theta`
Now total torque
` tau = (mu_0 i_0 i sin theta)/(pi) int_(a)^(b) dr = ( mu_0 i_0 i)/(pi) sin theta (b-a)`