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A uniform magnetic field of 20 T exists on the right side of a boundary in a gravity-free space. A circular arc of radius 2 cm, made of conducting wire of total resistance 4 ohm, is rotated about the point O at a constant angular speed of 2 rad/s, sweeping across the boundary into the field. Find the power required to maintain this constant angular velocity during the interval when the arc is cutting the field lines (t = pi/6 s to t = pi/3 s).

1. 64 micro W
2. 32 micro W
3. 128 micro W
4. 16 micro W

Correct answer: 16 micro W

Solution

As the arc rotates, the effective rate of change of area inside the field is (1/2)*r²*omega, so the induced EMF is e = B*(1/2)*r²*omega = 20 * 0.5 * (0.02)² * 2 = 20*0.5*0.0004*2 = 8e-3 V = 8 mV. Since rotation is at constant angular velocity, the mechanical power supplied equals the electrical power dissipated in the resistance: P = e²/R = (8e-3)² / 4 = 6.4e-5/4 = 1.6e-5 W = 16 micro W.

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