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A uniform magnetic field of $3000 \mathrm{G}$ is established along the positive z-direction. A rectangular loop of sides $10 \mathrm{~cm}$ and $5 \mathrm{~cm}$ carries a current of $12 \mathbf{A} .$ What is the torque on the loop in the different cases shown in the figure? What is the force on each case? Which case corresponds to stable equilibrium?

(a) $B$ is along the z-axis and $A$ is normal to the $x-z$ plane.

$\vec{A}=-50 \times 10^{-4} \hat{j}$

$\vec{B}=0.3 \hat{k}$

$\vec{\tau}=12 \times\left(-50 \times 10^{-4}\right) \hat{j} \times 0.3 \hat{k}$

$=-1.8 \times 10^{-2} \hat{i} N m$

The net force is zero and the torque is $1.8 \times 10^{-2} \mathrm{~N} \mathrm{~m}$ along the negative $x$-direction.

(b) Torque, $\vec{\tau}=I \vec{A} \times \vec{B}$

It is clear from the diagram that is perpendicular to the coil. Because the coil forms a $-$ $30^{\circ}$ angle with the y axis, $A$ forms a $30^{\circ}$ angle with the positive x-axis in the negative y-direction, and $B$ is oriented along the z-axis. The angle formed by $A$ and B is $\theta=90^{\circ} .$. Therefore,

Magnitude of torque will be,

$\mathrm{T}=12 \times 50 \times 10^{-4} \times 0.3$

$=1.8 \times 10^{-2} \mathrm{~N} \mathrm{~m}$

The torque direction is $\left(90^{\circ}+30^{\circ}\right)$ from the negative x-axis or $360^{\circ}-120^{\circ}=240^{\circ}$ from the positive x-axis. The loop’s net force is zero.