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A circular loop of radius R is bent alo...

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.

Text Solution

Verified by Experts

The correct Answer is:
A, B, C, D
(a) Magnetic field ` (vec(B))` at the origin ` = Magnetic field due to semicircle KLM + Magnetic field due to other semicircle KNM`.
Therefore, ` vec(B) = (mu_(0)I)/( 4 R ) ( -hat(i) )+ (mu_(0)I)/( 4 R ) (- hat(j))`
rArr ` vec(B) = - (mu_(0)I)/( 4 R) hat(i) + (mu_(0)I)/( 4 R)hat(j) = (mu_(0)I)/( 4 R)(- hat(i) + hat(j))`
(b) ` vec(F)_(KLM) = vec(F)_( KNM) = BI( 2 R ) hat(i) = 2 BIRhat(i)`
`vec(F)_(KM) = BI ( 2 R )hat(i) = 2 BIRhat(i)`
Therefore , `vec(F)_(1) = vec(F)_(2) = 2 BIR hat(i)` or total force on the loop,
` vec(F) = vec(F)_(1) + vec(F)_(2) rArr vec(F) = 4 B I R hat(i)`
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