Solution
Torque Analysis:
The limiter opens when the magnetic force lifts the ring off the support P. We consider torques about the hinge O.
- Gravitational Torque: The weight $mg$ acts at the center of mass. For a uniform semi-circle, the horizontal distance of COM from O is $r$. $\tau_g = mg r$ (Clockwise).
- Magnetic Torque: The magnetic force on the semi-circular arc carrying current $I$ in a perpendicular field $B$ is equivalent to the force on the diameter. $F_m = I(2r)B$. This force acts at the midpoint of the diameter (due to symmetry). The lever arm from O is $r$. $\tau_m = (2IrB) r$ (Counter-Clockwise).
Condition for Opening:
The circuit opens when the magnetic torque exceeds the gravitational torque:
$$ \tau_m > \tau_g $$ $$ 2Ir^2B > mg r $$ $$ 2IrB > mg $$ $$ I > \frac{mg}{2rB} $$
$$ I_{min} = \frac{mg}{2Br} $$