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

R=Radius of circular loop. Given that semicircle KNM lies
in the x-z plane while the semicircle KLM lies in the y-z plane.
Both the semicircle have their centres located at the origin.
(a) Charge on the particle released at the origin=q
Velocity of the particle, `vecv=-v_0hati`
Magnetic field at centre O due to current carrying loop
KLM lying in y-z plane, `vecB_2=(mu_0I)/(4R) (hatj)`.
Thus, the two fields at O are mutually perpendicular in
vector form. Total field at O can be expressed as
`vecB=-hatiB_1+hatjB_2=(mu_0I)/(4R)(-hati+hatj)`


Hence, instantaneous force acting on the charged particle.
released at O.


`vecF=q(vecvxxvecB)=q[-v_0hatixx(-hati+hatj)(mu_0I)/(4R)]`
`=(qmu_0Iv_0)/(4R)[hatixx(hati-hatj)]=-(qmu_0Iv_0)/(4R) hatk,`
(b) External uniform magnetic field `vecB_(ext)=Bhatj`
As semicircular wires are placed in uniform magnetic field,
these loops can be reduced to straight wires each of length
2R placed along z-axis (by joining initial point K and final
point M).
`vecF_(KLM)=vecF_(KMN)=vecF_(KM)=BI2Rhati`
Net force due to both the wires:
`vecF=vecF_(KLM)+vecF_(KMN)=4BIRhati`