A thin stiff insulated metal wire is bent into a circular loop with its two ends extending tangentially from the same point of the loop. The wire loop has mass $$m$$ and radius $$r$$ and it is in a uniform vertical magnetic field $$B_0$$, as shown in the figure. Initially, it hangs vertically downwards, because of acceleration due to gravity $$g$$, on two conducting supports at $$P$$ and $$Q$$. When a current $$I$$ is passed through the loop, the loop turns about the line $$PQ$$ by an angle $$\theta$$ given by

Using magnetic moment of a circular loop: $$M = I A = I(\pi r^2)$$

Using torque due to uniform vertical magnetic field $$B_0$$ when loop tilts by angle $$\theta$$:

$$\tau_m = \vert{}\vec{M} \times \vec{B}_0\vert{} = M B_0 \cos\theta = \pi r^2 I B_0 \cos\theta$$

Using restoring torque due to gravity acting at the center of mass:

$$\tau_g = m g (r \sin\theta) = m g r \sin\theta$$

Equating torques at rotational equilibrium about the axis $$PQ$$:

$$\tau_m = \tau_g \implies \pi r^2 I B_0 \cos\theta = m g r \sin\theta$$

$$\frac{\sin\theta}{\cos\theta} = \frac{\pi r^2 I B_0}{m g r} \implies \tan\theta = \frac{\pi r I B_0}{m g}$$