Question

A uniform magnetic field $\overrightarrow{B}=B_o\hat{k}$ exists in a region . A current carrying wire is placed in $x-y$ plane as shown in the figure. The magnetic force acting on the part $\text{AB}$ of wire is:

• A. $2B_{o}ia\hat{i}$
• B. $-3B_{o}ia\hat{j}$
• C. $-3B_{o}ia\hat{k}$
• D. $-B_{o}ia\hat{j}$

Answer 1

The correct option is B $-3B_{o}ia\hat{j}$

The wire $AB$ consists of five straight sections i.e $AC,CD,DE,EF$ and $FB$ as shown in the figure.

However, if we look at the wire $AB$ and the overall direction of curent, the effective length vector is ${\overrightarrow{L}}_{AB}$ as shown below


Thus, ${\overrightarrow{L}}_{AB}=3a\hat{i}$
Now,
$\overrightarrow{F}=i({\overrightarrow{L}}_{AB}\times \overrightarrow{B})$

$\overrightarrow{F}=i(3a\hat{i}\times B_o\hat{k})(\because \hat{i}\times \hat{k}=-\hat{j})$

$\overrightarrow{F}=-3B_{o}ai\hat{j}$

Hence, option $\left(B\right)$ is the correct answer.