Question

A metallic ring of mass $2kg$ and radius $1m$ with a uniform metallic spoke of same mass $2kg$ and length $1m$ is rotated about its axis with angular velocity $1rev/sec$. in a perpendicular uniform magnetic field $B$ of magnitude $10T$ as shown in figure. The central end of the spoke is connected to the rim of the wheel through a resistor $R$ of magnitude $\pi \Omega$ as shown. The resistor does not rotate, its one end is always at the center of the ring and other end is always in contact with the ring. A force $F$ as shown is needed to maintain constant angular velocity of the spoke then, $F$ is equal to (The ring and the spoke has zero resistance)

• A. $10N$
• B. $20N$
• C. $50N$
• D. None of these

Answer 1

The correct option is C $50N$

$F_b=BIL$
Induced current $I=\frac{(B\omega r^2/2}{R}$
$\therefore F_b=B\left(\frac{B^2\omega r^3}{2R}\right)r=\frac{B^2\omega r^3}{2R}$

To maintain constant angular velocity :
$F\left(r\right)=F_B\left(r/2\right)$
$\Rightarrow F=\frac{F_B}{2}=\frac{B^2\omega r^3}{4R}=\frac{(10{)}^2\times 2\pi \times (1{)}^3}{4\times \pi }=50N$