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A square loop of side $A$ carries a current $i .$ It is placed as shown in Fig.. Find the magnetic moment of the loop.

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$\vec{M} = \frac{3 i l^{2}}{2} (\sqrt{3} \hat{i} - \hat{j})$

$\vec{M} = \frac{i l^{2}}{4} (\sqrt{3} \hat{i} - \hat{j})$

$\vec{M} = i l^{2} (\sqrt{3} \hat{i} - \hat{j})$

$\vec{M} = \frac{i l^{2}}{2} (\sqrt{3} \hat{i} - \hat{j})$

answer is C.

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Detailed Solution

$\vec{M} = i A \hat{n}$

where $M = i l^{2}$

and $\hat{n} = \sin 30 ° \hat{i} + \cos 30 ° \hat{j} = \frac{\hat{i}}{2} - \frac{\sqrt{3} \hat{j}}{2}$

$\vec{M} = \frac{i l^{2}}{2} (\sqrt{3} \hat{i} - \hat{j})$

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