# Rotating Rod In Magnetic Field

Consider a horizontal circular loop of radius $$r$$ carrying a current $$I$$. Concentric and coplanar with the loop is a smooth metallic ring of radius $$a$$, containing a metallic rod $$OA$$ of mass $$m$$, as shown. The rod can freely rotate about $$O$$, the center of arrangement. Between $$O$$ and the circumference of ring, a resistor load $$R$$(not shown in the figure) is connected, which doesn't obstruct the motion of rod. The rod is given an angular velocity $$\omega_{0}$$. Find an expression for the maximum angle $$\theta$$ through which the rod rotates, if $$a<<r$$.

Find the value of $$\displaystyle 2 \Bigg|\bigg(\theta - \frac{16}{3} R m \omega_{0} \bigg(\frac{r}{\mu_{0} I a} \bigg)^2\bigg)\Bigg|$$ for the values given below :

Details and assumptions
The resistances of the rod and ring can be neglected.
$$m = 1g$$
$$R = 1 \Omega$$
$$\omega_{0} = 1 rad/s$$
$$r = 1m$$
$$a = 1cm$$
$$I = 1000 A$$
$$\mu_{0} = 4 \pi \times 10^{-7} H/m$$

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