First slide

Force on a current carrying wire placed in a magnetic field

Question

one end A of a current carrying wire AB, carrying a current of 2A, lies at the origin and the other end B lies at the point (3, 4) m. A uniform magnetic field $\vec{B} = 2 (\hat{i} + \hat{j} + \hat{k}) T$ exists in that region of space. Then force experienced by the wire AB is

Moderate

A

$8 N$
B

$4 \sqrt{26} N$
C

$12 \sqrt{2} N$
D

$24 N$

Solution

$\begin{matrix} \vec{F} = i \vec{l} \times \vec{B} \\ Here \vec{F} = 2 (3 \hat{i} + 4 \hat{j}) \times 2 (\hat{i} + \hat{j} + \hat{k}) N \\ = 4 |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 4 & 0 \\ 1 & 1 & 1 \end{matrix}| N = 4 (4 \hat{i} - 3 \hat{j} - \hat{k}) N \\ \Rightarrow F = 4 \sqrt{4^{2} + 3^{2} + 1^{2}} N = 4 \sqrt{26} N \end{matrix}$

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