Question

A current-carrying uniform square frame is suspended from hinged supports as shown in the diagram such that it can freely rotate about its upper side. The length and mass of each side of the frame are $2m$ and $4kg$ respectively. A uniform magnetic field $\overset{ \rightarrow }{\text{B}} = \left(3 \hat{\text{i}} + 4 \hat{\text{j}}\right)$ is applied. When the wireframe is rotated to $45^\circ $ from vertical and released it remains in equilibrium. What is the magnitude of current (in $A$ ) in the wireframe?

Answer 1

$\overset{ \rightarrow }{\mu }$ (Magnetic moment of loop) when it is lifted by $\left(45\right)^{^\circ }=\left(\text{i l}\right)^{2}\left(\frac{\hat{\text{j}} + \hat{\text{k}}}{\sqrt{2}}\right)$
$\therefore \quad \vec{\tau}$ due to magnetic field $=\vec{\mu} \times \vec{B}=\frac{i 1^2}{\sqrt{2}}[(\hat{j}+\hat{k}) \times(3 \hat{i}+4 \hat{j})]$
$\overset{ \rightarrow }{\tau}$ due to $mg$ (about top edge) $=4\text{mg}\frac{l}{2}cos45^\circ \hat{\text{i}}$
$∴ \, \, $ For equilibrium net torque along $x$ -axis = $0$
$∴ \, \, \frac{4 \text{mgl}}{2 \sqrt{2}}=\frac{4 \text{i l}^{2}}{\sqrt{2}}\Rightarrow \text{i}=\frac{\text{mg}}{2 l}=\text{10 A}$