Rotating Rod In Magnetic Field

Electricity and Magnetism Level 5

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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