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

`-(mu_(00I))/(4R) q v_(0) hatk` (ii) `F_(1) = 2I R B F_(2) = 2I R`
`B` Net Force ` =F_(1) + F_(2) = 4 I R B hati` .

Explanation

`B_(1)` = field due to `KNM`
`B_(2) =` field due to to `KLM`
field at origin `oversetrarr(B_(0)) = oversetrarr(B_(1))oversetrarr(B_(2))`
` = (mu_(0))/(4pi) (piI)/(R ) hatj + (mu_(0))/(4pi) (-hati) = (mu_(0))/(4) (I)/R (-hati +hatj)`
Force on particle
`oversetrarrF = q (oversetrarrV xx oversetrarrB) = q V_(0) B [-hati xx (-hati +hatj)]`
`gV_(0) B_(0) (-hatk) = (mu_(0)I)/(4R) qV_(0) (-hatk)`
(ii) In a uniiform field the force on the curved loop is `oversetrarrF = i (oversetrarrl xx oversetB)` where `vecl =rarr ` vector from one end of loop to the other end
`:. oversetrarrF_(KLM) = i (KoversetrarrM xx oversetrarrB) i [2R (-hatk)xx hatB hatj] =2BiR hati`
`:.oversetrarrF = oversetrarrF_(KLM) + oversetrarrF_(KLM) =4BiRhati` .
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