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A uniform conducting

Question:

A uniform conducting wire of length is $24 \mathrm{a}$, 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 $\mathrm{V}_{0}$. The ratio of magnetic moment of the coils in case of equilateral triangle to that for square is $1: \sqrt{\mathrm{y}}$ where $\mathrm{y}$ is

Solution:

In triangle shape $\mathrm{N}_{\mathrm{t}}=\frac{24 \mathrm{a}}{3 \mathrm{a}}=8$

In square $\mathrm{N}_{\mathrm{s}}=\frac{24 \mathrm{a}}{4 \mathrm{a}}=6$

$\frac{M_{t}}{M_{3}}=\frac{N_{t} I_{t}}{N_{s} I A_{s}}$ [I will be same in both]

$=\frac{8 \times \frac{\sqrt{3}}{4} \times \mathrm{a}^{2}}{6 \times \mathrm{a}^{2}}$

$\frac{\mathrm{M}_{\mathrm{t}}}{\mathrm{M}_{\mathrm{s}}}=\frac{1}{\sqrt{3}}$

$y=3$

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