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Mon to Sat - 10 AM to 7 PM Asked by Prashant DIGHE 6th December 2019, 10:56 PM Left side figure shows the current loop made up of circular arc and chord of the same circle. Angle subtended by chord and arc is θ.

To get magnetic induction at P due to current i passing through loop as shown in figure,
let us calculate the magnetic induction separately for arc and chord.

Figure given in the middle helps us to get magentic induction B1 at P.

Let us consider small current element idl which is at a distance l from the foot of perpendicular drawn from P to chord.

Using Biot-Severt's law, we get, magnetic induction dB due to this current element as

dB = ( μo / 4π ) idl sinφ / ( l2 + d2 )  = ( μo / 4π ) idl d / ( l2 + d2 )3/2 ...................(1)

let us use the substitution l = d tanα ( see figure )

Hence dl = d sec2α dα

Using these substitutions, eqn.(1) becomes,  dB = ( μo i / 4π d) cosα dα  ................(2)

We get magnetic induction B1 by integrating eqn.(2) with limits from -θ/2 to +θ/2

B1 = ( μo i / 2π d) sin(θ/2) .....................(3)

since d = R cos(θ/2) , eqn,(3) becomes,  B1 = ( μo i / 2π R) tan(θ/2) ...................(4)1
Direction of this magnetic induction is inward ( please check using right hand rule )
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Right side figure helps us to get magnetic induction at P due to circular arc.

Let us consider current element idl in the circular arc. Magnetic induction dB at P is given by

dB = ( μo / 4π ) ( idl / R2 )

Magnetic induction B2 due to full arc will be sum of above dB over arc length Rθ

Hence magnetic induction B2 = ( μo i / 2πR ) ( θ/2 )
Direction of this magnetic induction is outward
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net magnetic induction due to loop = B1 - B2 = ( μo i / 2π R) [  tan(θ/2) - ( θ/2 ) ]

when θ < π ,  [  tan(θ/2) - ( θ/2 ) ] > 0  , hence direction of net magnetic induction due to loop is inward

when θ > π ,  [  tan(θ/2) - ( θ/2 ) ] < 0  , hence direction of net magnetic induction due to loop is outwar
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Answer to the question :- option (3) inward as long as θ < π
Answered by Expert 8th December 2019, 8:49 PM
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