A conducting wire bent in the form of a parabola $y^{2} = x$ , is carrying a current I = 1 A in anticlockwise sense as shown. This wire is placed in a uniform magnetic field $\vec{B} = - 2 \hat{k}$ tesla. The unit vector in the direction of force is

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$\frac{\hat{i} + 2 \hat{j}}{\sqrt{5}}$

$\frac{3 \hat{i} + 4 \hat{j}}{5}$

$\frac{\hat{i}}{\sqrt{2}}$

$\frac{\hat{i} + \hat{j}}{\sqrt{2}}$

answer is B.

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Detailed Solution

$a \vec{b} - O \vec{b} - O \vec{a} = 4 \hat{i} - 2 \hat{j} - \hat{i} - \hat{j} = 3 \hat{j} - 3 \hat{j}$
$\vec{F} = i (\vec{l} \times \vec{B}) = 1 (3 \hat{i} - 3 \hat{j}) \times (- 2 \hat{k}) = 6 \hat{j} + 6 \hat{i}$
$\vec{F} = \frac{6 \hat{j} + 6 \hat{i}}{6 \sqrt{2}} = \frac{\hat{i} + \hat{j}}{\sqrt{2}}$