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 `omega` in a perpendicular uniform magnetic field `B` as shown in Fig. 3.163. 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 as shown is needed to maintain constant angular velocity of the wheel. `F` is equal to (the ring and the spoke has zero resistance)

Options

A. `(B^(2)omega r^(2))/(8R)`
B. `(B^(2)omega r^(2))/(2R)`
C. `(B^(2) omega r^(3))/(2R)`
D. `(B^(2) omega r^(3))/(4R)`

Answer

D

Explanation

`F_(b) = BIL`
Induced current
`I = (Bomega r^(2)//2)/R`

`therefore F_(b) = B(Bomega r^(2))/(2R) r = (B^(2) omega r^(3))/(2R)`
To maintain constant angular velocity:
`F( r) = F_(B) (r/2) rArr F = F_(B)/2 = (B^(2)omega r^(3))/(4R)`