A 1.0 m long metallic rod is rotated with an angular frequency of $500\, rad\, s^{-1}$ with one end hinged at the center and the other end at the circumference of a circular metallic ring of radius 1 m about an axis passing through the center and perpendicular to the plane of the ring. A constant and uniform magnetic field of 0.6 T parallel to the axis exists everywhere. The emf developed between the center and the ring is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 150 V
$\text{Given: length (radius) } R = 1.0~\text{m},\;\omega = 500~\text{rad/s},\; B = 0.6~\text{T}$
$\text{At a distance } r \text{ from centre, linear speed } v = \omega r$
$\text{Motional emf of a small element } dr: d\mathcal{E} = B\,v\,dr = B\,\omega\,r\,dr$
$\text{Total emf between centre and rim: } \mathcal{E} = \int_{0}^{R} B\omega r\,dr = B\omega \int_{0}^{R} r\,dr = B\omega \left[\frac{r^{2}}{2}\right]_{0}^{R} = \frac{1}{2}B\omega R^{2}$
$\mathcal{E} = \frac{1}{2}\times 0.6 \times 500 \times (1.0)^{2}$
$\mathcal{E} = 0.3 \times 500 = 150~\text{V}$
$\text{Answer: } \mathcal{E} = 150~\text{V}$