A current carrying wire of length $L$ is transformed into a coil of $N$ turns. To achieve maximum magnetic moment, the coil
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$M=\mathbf{N I} A$
$\mathrm{M}=\mathrm{N} \times \mathrm{I} \times \mathrm{n} \mathrm{R}^{2}$
$\mathrm{M}=\mathrm{N} \times \mathrm{I} \times \mathrm{N} \times \frac{\mathrm{L}^{2}}{4 \mathrm{n}^{2} \mathrm{N}^{2}}$
$\mathrm{L}=\mathrm{N} \times 2 \mathrm{nR}$
$\Rightarrow \mathrm{R}=\frac{\mathrm{L}}{2 \mathrm{nN}} \quad \mathrm{M} \times \frac{1}{\mathrm{N}}$
For $M_{\max } N=1$ and area of circle is maximum
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