A circular loop of radius `R` is bent along a diameter and given a shapes as shown in the figure. One of the semicircles `(KNM)` lies in the ` x-z` plane with their centres and the other one `(KLM)` in the `y-z` plane with their centres at the origin. current `I` is flowing through each of the semi circles as shown in figure.
(a) A particle of charge `q` is released at the origin with a velocity `vec(v) = -v_(0)hat(i)`. Find the instantaneous force `vec(F)` on the particle . Assume that space is gravity free.
(b) If an external uniform magnetic field `B_(0) hat(j) ` is applied , determine the force `vec(F)_(1) and vec(F)_(2)` on the semicircles `KLM and KNM` due to the field and the net force ` vec(F)` on the loop.

Explanation

(a) Requried force, `vec(F) = q [-V_(0) hat(i)) xx (mu I)/(4R) (hat(j) - hat(i))] = (mu_(0) Iv_(0)q)/(4R) [-hat(k)]`
(b) force one semicircular loop KLM due to external B `hat(j)` is same as the force experienced by straight current carrying conductor along KM
`= - I (2 R hat(k)) xx B hat(j) = 2I BR hat(i)`
Similarly force on the semicircular loop KNM = `2IBR hat(i)`
`:.` Net force on the loop `= vec(F) = 4IBR hat(i)`