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A thin semicircular conducting ring of mass m = 3.0 g and radius R = 1.0 cm can rotate freely in a vertical plane about one of its endpoints O. The system is placed in a uniform horizontal magnetic field of strength B = 1.5 T perpendicular to the plane of the ring. When current I flows through the ring, it tends to rotate and break the circuit. Find the maximum current I_max (in A) at which the ring just disconnects the circuit.
- 0.5
- 1.0
- 1.5
- 2.0
Correct answer: 1.0
Solution
Integrating the magnetic force on each current element of the semicircle about pivot O gives a net torque of 2IR²B (directed to lift the ring). The gravitational torque about O is mgR (centre of mass of a semicircular wire is at 2R/pi from the arc centre, but the horizontal distance from O to the CM is R, giving torque mgR). Setting 2IR²B = mgR: I_max = mg/(2RB) = (3*10⁻³ * 10)/(2 * 0.01 * 1.5) = 0.03/0.03 = 1.0 A.
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