A uniform conducting wire of length is $24 {a}$, 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 ..... .
JEE MAIN 2021, Diffcult
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In triangle shape ${N}_{{t}}=\frac{24 {a}}{3 {a}}=8$
In square ${N}_{{s}}=\frac{24 {a}}{4 {a}}=6$
$\frac{{M}_{{t}}}{{M}_{3}}=\frac{{N}_{{t}} {IA}_{{t}}}{{N}_{{s}} {I}_{{s}}}[{I}$ will be same in both $]$
$=\frac{8 \times \frac{\sqrt{3}}{4} \times a^{2}}{6 \times a^{2}}$
$\frac{{M}_{{t}}}{{M}_{{s}}}=\frac{1}{\sqrt{3}}$
${y}=3$
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