A uniform conducting wire of length $12 \mathrm{a}$ and resistance $'R'$ is wound up as a current carrying coil in the shape of,

$(i)$ an equilateral triangle of side $'a'.$

$(ii)$ a square of side $'a'.$

The magnetic dipole moments of the coil in each case respectively are:

Question

A uniform conducting wire of length $12 \mathrm{a}$ and resistance $'R'$ is wound up as a current carrying coil in the shape of,

$(i)$ an equilateral triangle of side $'a'.$

$(ii)$ a square of side $'a'.$

The magnetic dipole moments of the coil in each case respectively are:

NEET 2021, Medium

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Solution

$x=\sqrt{a^{2}-\frac{a^{2}}{4}}=\sqrt{\frac{3 a^{2}}{4}}=\sqrt{\frac{3 a^{2}}{4}}=\frac{\sqrt{3}}{2} a$

$\mathrm{A}_{1}=\frac{1}{2} \times \mathrm{a} \times \frac{\sqrt{3}}{2} \mathrm{a}$

$\mathrm{A}_{1}=\frac{\sqrt{3}}{4} \mathrm{a}^{2}$

$\mu_{1}=\mathrm{N}_{1} \mathrm{IA}_{1}$

$\mu_{1}=\frac{4 \mathrm{I} \sqrt{3}}{4} \mathrm{a}^{2}$

$\mu_{1}=\sqrt{3} \mathrm{I} a^{2}$

$A_{2}=a^{2}$

$\mu_{2}=N_{2} I A_{2}$

$=3 \times I \times a^{2}$

$\mu_{2}=3 \mathrm{Ia}^{2}$

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