Q. 404.0( 2 Votes )

A circular loop of radius r carrying a current i is held at the center of another circular loop of radius R (>>r) carrying a current I. The plane of the smaller loop makes and angle of 30° with that of the larger loop. If the smaller loop is held fixed in this position by applying a single force at a point on its periphery, what would be the minimum magnitude of the force?

Answer :

Given:


Current carried by circular loop of radius r = i


Current carried by circular loop of radius R(>>r) = I


Angle between the planes of the two loops = 300


Diagram:



Formula used:


Magnetic field at the center of a circular loop of radius r


M= ,


where


I = current through the loop,


r = radius of the circle,


𝛍0 = 4π x 10-7 kgm s-2 A-2


Magnetic torque acting on a loop of radius r



where


I = current carried by the loop,


A = area of loop,


B = magnetic field,


F = force on periphery


Therefore, magnetic field at the center of the loops due to the bigger loop(B) =


Area of the smaller loop(A) = πr2


Therefore, torque on the smaller loop = i (A X B), since the smaller loop carries current i.


A = area of smaller loop, B = magnetic field due to bigger loop.


Now, we know that A X B = |A||B|sinθ,


Where


θ is the angle between the area vector of the smaller loop and the magnetic field of the larger loop.


|A| = magnitude of area of smaller loop,


|B| = magnitude of magnetic field due to larger loop.


Given:


The angle between the area vector of the smaller loop and the magnetic field of the larger loop is θ = 300.


Hence, torque acting on the smaller loop = i(|A||B|sin 300) =


We know, |A| = πr2, |B| =


Hence,


Torque on smaller loop due to magnetic field,



Torque due to external force on its periphery,



Therefore



Where


F = external force,


r = perpendicular distance between the line of action of force and the point at which the torque is acting.


Here, F and r are perpendicular to each other, since the force is being applied to the periphery.


Here r = radius of smaller loop.


Hence,


which gives us


(Answer)


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