A metallic ring of radius r with a uniform metallic spoke of negligible mass and length r is rotated about its axis with angular velocity ω in a perpendicular uniform magnetic field B as shown in figure. The central end of the spoke is connected to the rim of the wheel through a resistor R as shown. The resistor does not rotate, its one end is always at the center of the ring and the other end is always in contact with the ring. A force F as shown is needed to maintain constant angular velocity of the wheel. F is equal to (the$$ ring and the spoke has zero $$resistance)

Question

A metallic ring of radius r with a uniform metallic spoke of negligible mass and length r is rotated about its axis with angular velocity ω in a perpendicular uniform magnetic field B as shown in figure. The central end of the spoke is connected to the rim of the wheel through a resistor R as shown. The resistor does not rotate, its one end is always at the center of the ring and the other end is always in contact with the ring. A force F as shown is needed to maintain constant angular velocity of the wheel. F is equal to (the$$ ring and the spoke has zero $$resistance)

Infinity Learn · Accepted Answer

Induced emf in the spoke is shown in figure below.$$e=12B$$ω$$r2$$,$$I=eR=B$$ω$$r22RThere$$ will be no induced emf separately in parts ADC or AEC.$$Fb=IrB=B2$$ω$$r32RBalancing$$ torque about E: $$Fr=Fbr/2$$⇒ $$F=Fb2=B2$$ω$$r34RNote$$: Force due to current s $$I/2$$ will act on circular parts also, but their torque about E will be zero.

Detailed Solution

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