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Magnetic force
![question image](http://images.topperlearning.com/topper/new-ate/top_mob165803967382654298088d71b14-10db-412d-8f91-10caf3ad4681.jpg)
Asked by swayamagarwal2114 | 17 Jul, 2022, 12:04: PM
workdone W = ![begin mathsize 14px style integral subscript a superscript 2 a end superscript F left parenthesis x right parenthesis d x end style](https://images.topperlearning.com/topper/tinymce/cache/8b1e4512a76fecb2760c38aa91955894.png)
![begin mathsize 14px style integral subscript a superscript 2 a end superscript F left parenthesis x right parenthesis d x end style](https://images.topperlearning.com/topper/tinymce/cache/8b1e4512a76fecb2760c38aa91955894.png)
where F(x) is magnetic force that is given as
F(x) = i B(x) a
where i is induced current in the loop , B(x) is magnetic field at distance x from long wire and a is side of square loop
Induced current = induced EMF / Resistance of loop = ξ / R
magnitude of Induced EMF ξ = rate of chagne of flux of magnetic field
ξ = dφ/dt
![begin mathsize 14px style fraction numerator d phi over denominator d t space end fraction equals space fraction numerator d over denominator d t end fraction left parenthesis space B space A space right parenthesis space equals space A space fraction numerator d B over denominator d t end fraction space equals space a squared space fraction numerator d B over denominator d t end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/9b6200ef336c342a3b1869e8c3bbc7a6.png)
where B is flux density of magnetic field and A is area of loop
Flux density of magnetic field B at a distance x due to current in long wire is given as
![begin mathsize 14px style B space equals space fraction numerator mu subscript o I over denominator 2 pi x end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/cb75b7fa370a30e8d90bc1fc2cf7e451.png)
where μo is permeability of free space and I is current in long wire
![begin mathsize 14px style fraction numerator d B over denominator d t end fraction space equals space fraction numerator mu subscript o I over denominator 2 pi end fraction cross times 1 over x squared cross times fraction numerator d x over denominator d t end fraction space equals space fraction numerator mu subscript o I over denominator 2 pi end fraction space cross times space 1 over x squared space cross times space v end style](https://images.topperlearning.com/topper/tinymce/cache/aed77682c69d468257cf8564a7aa8946.png)
where dx/dt = v is speed of movement of loop
Hence workdone is given as
![begin mathsize 14px style W space equals space integral subscript a superscript 2 a end superscript B left parenthesis x right parenthesis space i space a space d x space equals space 1 over R open parentheses fraction numerator mu subscript o I over denominator 2 pi end fraction close parentheses squared space v space space integral subscript a superscript 2 a end superscript space fraction numerator d x over denominator x cubed end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/d5fa3d2e14e467e1c56b6c52c036cd8a.png)
![begin mathsize 14px style W space equals space 1 over R open parentheses fraction numerator mu subscript o I over denominator 2 pi end fraction close parentheses squared space v space space cross times fraction numerator 3 over denominator 2 a squared end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/8db0c911ebcd1f8145e244dc04edca3d.png)
Answered by Thiyagarajan K | 17 Jul, 2022, 14:34: PM
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