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.

\( m = 1g\)

\( R = 1 \Omega \)

\( \omega_{0} = 1 rad/s\)

\( r = 1m\)

\( a = 1cm\)

\( I = 1000 A\)

\( \mu_{0} = 4 \pi \times 10^{-7} H/m\)