In the options above, there seems to be L³ , it is not correct. It should be L².
The metallic rod is tied to one end of a rod and the other end is left free. The other end of the string is fixed at the center of a horizontal table. The rod rotates on the horizontal table about an axis perpendicular to it and passing through its center. The angular speed = ω. Assume that there is a metallic frame on which the ends of the rod are travelling. The ends of the rod are always on the metallic points. So a circular loop is established.
Magnetic field B is perpendicular to the metallic rod. Area covered by the rod and the loop: A. To find A, we take the radius of the revolution of the center of the rod, which is 2L + L/2 = 2.5 L. Then the angle of revolution theta = ω * time t. The r * theta gives the length through which the rod moves. Its length is L.
A = [ 2.5 L * (ω t) ] L = 2.5 L² ω t
EMF induced = -dΦ / dt = - d/dt [Φ] = - d/dt [B A]
= - d/dt [ B 2.5 L² ω t ]
= - 2.5 B L² ω