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An infinitely long current carrying wire and a small current carrying loop are in the plane of the paper as shown. The radius of the loop is a and distance of its centre from the wire is d (d>>a). If the loop applies a force F on the wire then : 

(1) $F\propto\left({a^2\over d^3}\right)$

(2) $F\propto\left({a\over d}\right)$

(3) $F\propto\left({a\over d}\right)^2$

(4) $F = 0$

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Ans: (3) $F\propto\left({a\over d}\right)^2$

Sol:

 

Eqvilent dipole of given loop 

$F=m.{dB\over dr}$

Now ${dB\over dx}={d\over dx}\left({\mu_0I\over2\pi x}\right)$

$\propto{1\over x^2}$

$\implies \text{So }F\propto{M\over x^2}[\because M=NIA]$

$\therefore F\propto{a^2\over d^2}$

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