A metal rod of length $$L$$ rotates about one end at origin with a uniform angular velocity $$\omega$$. The magnetic field radially falls off as $$B(r) = B_0 e^{-\lambda r}$$. $$\lambda$$ being a positive constant. The EMF induced (neglecting the centripetal force on electrons in the rod) is :

$$B_0\omega\left[\frac{1}{\lambda^2} - e^{-\lambda L}\left(\frac{1}{\lambda^2} + \frac{L}{\lambda}\right)\right]$$

$$B_0\omega\left[\frac{1}{\lambda^2} + e^{-\lambda L}\left(\frac{1}{\lambda^2} + \frac{L}{\lambda}\right)\right]$$

$$B_0\omega\left[\frac{4}{\lambda^2} - e^{-2\lambda L}\left(\frac{1}{\lambda^2} + \frac{2L}{\lambda}\right)\right]$$

$$B_0\omega\left[\frac{3}{\lambda^2} - e^{-3\lambda L}\left(\frac{3}{\lambda^2} + \frac{L}{\lambda}\right)\right]$$