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A circular loop of wire with radius of 1.0 m lies in the xy plane. A...

A circular loop of wire with radius of 1.0 m lies in the xy plane. A uniform 1.0 T external magnetic field is directed along the z axis. The wire is a conductor, and carries current of 1.0 Amperes. What is the tension (or compression) in the wire due to the external field?

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Answer #1

The loop may be divided into a large number of small length elements. When a current I is passed through the loop placed in the magnetic field such that the plane of the loop is perpendicular to field, then force on each element is

dF=Idl×B=IdlBsin90o=IdlB

Perpendicular to current element Idl as well as magnetic field

B. Hence, the loop opens into a circle.

Consider an element of length dl of circle of radius r, making an angle alpha at centre.

If T is the tension in the wire, then force toward centre:

2Tsin2α​=IBdl

For small angle α,sin2α​=2α​

2T⋅2α​=IBdl

T=αIBdl​=IBr(∴α=rdl​)

=IB(2π1​)(∴l=2πr)

=1×1×(2×3.14×1​)=6.28N

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