Question

A wire carrying a $10\text{A}$ current is bent to pass through various sides of a cube of side $10\text{cm}$ as shown in the figure. A magnetic field $\overrightarrow{B}=(2\hat{i}-3\hat{j}+\hat{k})\text{T}$ is present in the region. Find the net torque on the loop.

• A. $-0.1\hat{i}+0.4\hat{k}\text{Nm}$
• B. $-0.1\hat{i}-0.4\hat{k}\text{Nm}$
• C. $0.2\hat{i}-0.4\hat{k}\text{Nm}$
• D. $0.1\hat{i}-0.4\hat{k}\text{Nm}$

Answer 1

The correct option is C $0.2\hat{i}-0.4\hat{k}\text{Nm}$

Let us consider the given loop to be a super position of three loops as shown in the figure. The area vector of the loops $\left(1\right),\left(2\right)$ and $3)$ are also shown.


From the figure, $\overrightarrow{A_1}=(\frac{1}{2}\times 10\times 10\times {10}^{-4})\hat{j}{\text{Am}}^2$

$\overrightarrow{A_2}=(\frac{1}{2}\times 10\times 10\times {10}^{-4})\hat{k}{\text{Am}}^2$

$\overrightarrow{A_3}=(10\times 10\times {10}^{-4})\hat{i}{\text{Am}}^2$

Therefore, Magnetic moment vectorr, $\overrightarrow{\mu }=10(0.01\hat{i}+0.005\hat{j}+0.005\hat{k}){\text{Am}}^2$

$\Rightarrow \overrightarrow{\mu }=(0.1\hat{i}+0.05\hat{j}+0.05\hat{k}){\text{Am}}^2$

Thus, torque acting on the loop, $\overrightarrow{\tau }=(0.1\hat{i}+0.05\hat{j}+0.05\hat{k})\times (2\hat{i}-3\hat{j}+\hat{k})$

$\overrightarrow{\tau }=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 0.1& 0.05& 0.05\\ 2.0& -3.0& 1.0\end{array}\right|=(0.2\hat{i}-0.4\hat{k})\times {10}^{-4}\text{Nm}$


Hence, option (c) is the correct answer.