Question 253 of 276

A uniform conducting wire of length is $24a,$ and resistance $R$ is wound up as a current carrying coil in the shape of an equilateral triangle of side $a$ and then in the form of a square of side $a$. The coil is connected to a voltage source ${V}_{0}.$ The ratio of magnetic moment of the coils in case of equilateral triangle to that for square is $1:\sqrt{y}$ where $y$ is ____________.

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Answer: 3

$4a{n}_{1}=24a$
thus ${n}_{1}=6$ turns

$3{an}_{2}=24a\Rightarrow {n}_{2}=8$ turns
$\dfrac{{M}_{1}}{{M}_{2}}=\dfrac{{n}_{1}i{A}_{1}}{{n}_{2}i{A}_{2}}=\dfrac{6\times {a}^{2}}{8\times {a}^{2}\times \dfrac{\sqrt{3}}{4}}=\dfrac{\sqrt{3}}{1}$
$\dfrac{{M}_{2}}{{M}_{1}}=\dfrac{1}{\sqrt{3}}=\dfrac{1}{\sqrt{y}}$
$y=3$