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)` Magnetic filed `(vec(B))` at the origin `=` magnetic field due to other semicirle `KLM +` Magnetic field due to other semicircle `KNM`.
Therefore,
`vec(B)=-(mu_(0)I)/(4R)(-hat(i))+(mu_(0)I)/(4R)impliesvec(B)=-(mu_(0))/(4R)hat(i)+(mu_(0)I)/(4R)(-hat(i)+hat(j))`
`|vec(B)` due to a circular current carrying loop is
`(mu_(0)I)/(2R)` For semicircle it is half `|`
Therefore magnetic force acting on the particle
`vec(F)=q{(-v_(0)hat(i))xx(-hat(i)+hat(j))xx(mu_(0)I)/(4R)}=(-mu_(0)qv_(0)I)/(4R)hat(k)`
`(B) vec(F)_(KLM)=vec(F)_(KNM)=vec(F)_(KM)`
and `vec(F)_(KLM)=BI(2R)hat(i)=2BIRhat(i)`
Total force on the loop `=4BIRhat(i)`