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A uniform magnetic field $\bar{B} = 3 \hat{i} + 4 \hat{j} + \hat{k}$ exists in region of space. A semicircular wire of radius 1 m carrying current 1A, having its centre at is $(2, 2, 0)$ placed in $x - y$ -plane as shown in figure. The force on semi circular wire will be

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$\sqrt{2} (- \hat{i} + \hat{j} + \hat{k})$

$\sqrt{2} (\hat{i} - \hat{j} + \hat{k})$

$\sqrt{2} (\hat{i} + \hat{j} + \hat{k})$

$\sqrt{2} (\hat{i} + \hat{j} - \hat{k})$

answer is B.

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Detailed Solution

$= \bar{A C} = 2 R (\cos 45 \hat{i} + \sin 45 \hat{j}) = \sqrt{2} (\hat{i} + \hat{j})$

$= i (\bar{i} \times \bar{B}) = 1 (\sqrt{2} (\hat{i} + \hat{j}) \times (3 \hat{i} + 4 \hat{j} + \hat{k})]$

$\bar{F} = \sqrt{2} (\hat{i} - \hat{j} + \hat{k})$

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