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.

Answer

(i)`-(mu_(0)I)/(4R)qv_(0)hatk` (ii) `4lRBhati`

Explanation

(a)Magnetic field `(vecB)` at the origion =Magnetic field due to semicircle `KLM+` Magnetic field due to other semicircle `KNM`
`therefore vecB=(mu_(0)I)/(4R)(-hati)+(mu_(0)I)/(4R)(-hatj)`
`vecB=-(mu_(0)I)/(4R)hati+(mu_(0)I)/(4R)hatj=(mu_(0)I)/(4R)(-hati+hatj)`
`therefore` Magnetic force acting on the particle-
`vecF=q(vecvxxvecB)=q(-V_(0)hati)xx(-hati+hatj)}(mu_(0)l)/(4R)`
`vecF=-(mu_(0)qV_(0)l)/(4R)hatk`
(b)`vecF_(KLM)=vecF_(KNM)=vecF_(KM)`
and `vecF_(KM)=BI(2R)hati`
`=2BIRhatj`
`vecF_(1)=vecF_(2)=2BIRhati`
or Total force on the loop `=vecF=vecF_(1)+vecF_(2)`
or `vecF=4 BIRhatj`
If a current carrying wire `ADC` (of any shape) is placed in a uniform magnetic field
`vecB`.Then `vecF_(ADC)=vecF_(AC) or |vecF_(ADC)|=i (AC) B`.
From this we can conclude that net force on a current carrying loop in uniform magnetic field is zero.In the question, segmets `KLM` and `KNM` also form a loop and they are also placed in a uniform magnetic field bu in this case net force on the loop will not be zero.It would had been zero it the current in any of the segments was in opposite direction.