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Physics-

General

Easy

Question

A small square loop wire of side $l$ is placed inside a large squre loop of side $L open parentheses L greater than greater than l close parentheses.$ If the loops are coplanar and their centres coincide, the mutual induction of the system is directly proportional to

$\frac{L}{l}$
$\frac{l}{L}$
$\frac{L^{2}}{l}$
$\frac{l^{2}}{L}$

The correct answer is: $fraction numerator l to the power of 2 end exponent over denominator L end fraction$

Considering the larger loop to be made up of four rods each of length $L$ , the field at the centre,
$i. e. comma$ at a distance $open parentheses fraction numerator L over denominator 2 end fraction close parentheses$ from each rod, will be

$B equals 4 cross times fraction numerator mu subscript 0 end subscript over denominator 4 pi end fraction fraction numerator l over denominator d end fraction open square brackets sin invisible function application alpha plus sin invisible function application beta close square brackets$
$i e comma blank B equals 4 cross times fraction numerator mu subscript 0 end subscript over denominator 4 pi end fraction fraction numerator I over denominator open parentheses fraction numerator L over denominator 2 end fraction close parentheses end fraction cross times 2 sin invisible function application 45 degree$
$i e comma blank B subscript 1 end subscript equals fraction numerator mu subscript 0 end subscript over denominator 4 pi end fraction fraction numerator 8 square root of 2 over denominator L end fraction I$
So, the flux with smaller loop
$ϕ equals B subscript 1 end subscript S subscript 2 end subscript equals fraction numerator mu subscript 0 end subscript over denominator 4 pi end fraction fraction numerator 8 square root of 2 over denominator L end fraction l to the power of 2 end exponent I$
and hence, $M equals fraction numerator ϕ subscript 2 end subscript over denominator I end fraction equals 2 square root of 2 fraction numerator mu subscript 0 end subscript over denominator pi end fraction fraction numerator l to the power of 2 end exponent over denominator L end fraction$
or $M proportional to fraction numerator l to the power of 2 end exponent over denominator L end fraction$

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