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Magnetic force
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Asked by swayamagarwal2114 | 17 Jul, 2022, 12:04: PM
answered-by-expert Expert Answer
workdone W = begin mathsize 14px style integral subscript a superscript 2 a end superscript F left parenthesis x right parenthesis d x end style
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
 
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
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
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
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
 

 
Answered by Thiyagarajan K | 17 Jul, 2022, 14:34: PM

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