First slide

Force on a current carrying wire placed in a magnetic field

Question

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

Difficult

A

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

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

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

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

Solution

Force on semi-circular wire will be same as the force on straight wire AC.
$\begin{array}{l} \vec{l} = A \vec{C} = 2 R (\cos 45^{0} \hat{i} + \sin 45^{0} \hat{j}) = \sqrt{2} (\hat{i} + \hat{j}) \\ \vec{F} = I \vec{l} \times \vec{B} = 1 \times \sqrt{2} (\hat{i} + \hat{j}) \times (3 \hat{i} + 4 \hat{j} + \hat{k}) \\ = \sqrt{2} [\hat{i} - \hat{j} + \hat{k}] \end{array}$

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