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a hypothetical magnetic field existing in a region is given by vector B=B.er,where er denotes unit vector along the radial direction.a circular loop of radius a,carrying a current i,is placed with its plane parallel to X-Y plane and the center at (0,0,d).find the magnitude of the magnetic force acting on the loop.

a hypothetical magnetic field existing in a region is given by vector B=B.er,where er denotes unit vector along the radial direction.a
circular loop of radius a,carrying a current i,is placed with its plane parallel to X-Y plane and the center at (0,0,d).find the magnitude of the magnetic force acting on the loop.

Grade:12

1 Answers

askIITianexpert IITDelhi
8 Points
11 years ago

If you could visualize the above problem pictorially then it's quite easy to understand.

The magnetic field is radially outward here & the current loop intersect these magnetic lines in such a way that all these lines together with loop make an inverted right cylindrical cone with vertex at origin and height being 'd' & radius being 'a'.

Now if you could grasp the above then it's cakewalk now.For each infinitesimal length 'dl' of loop(around at each point) the magnetic field lines intersect it at angle 'x' with Z-axis here with "tanx=a/d".

Now Z-component of the field is perpendicular to loop at each point.If you apply right hand rule to this the resulting force will be either radially inward on the loop or outward correspondingly for both possible directions(clockwise & anticlockwise as you see the loop from origin) of current 'i'.

Due to circular symmetry of force here there will be no net force on the loop due to Z-component of magnetic field.

Now consider the other component of field at each point.It is radially outward at each point so perpendicular to 'idl'.Applying right hand rule will give force in the Z-direaction.Value of field is 'Bcosx'.

Now seeing from origin if current is:

                                                               1.)Clockwise then force will be towards origin or in the (-ve) Z-direction .

                                                                2.)Anti-clockwise then force will be away from origin or in the (+ve) Z-direction.

   At each 'dF' will simply add up to give magnitude of the net force F=i(2¶a)Bcosx=2¶Bi[a*d/(a2+d2)1/2]

As i say earlier the 3-D view of the problem here will just show you how trivial it is.But yes it could also be solved analytically using rigorous vector algebra where you have to work in cylindrical co-ordinate system.I would always prefer above method because that is what physics is all about.......visualizing!!!

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