$A$ 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 :

• A
  $B_0 \omega [\frac{1}{\lambda^2} - e^{-\lambda L} (\frac{1}{\lambda^2} + \frac{L}{\lambda})]$
• B
  $B_0 \omega [\frac{1}{\lambda^2} + e^{-\lambda L} (\frac{1}{\lambda^2} + \frac{L}{\lambda})]$
• C
  $B_0 \omega [\frac{4}{\lambda^2} - e^{-2\lambda L} (\frac{1}{\lambda^2} + \frac{2L}{\lambda})]$
• D
  $B_0 \omega [\frac{3}{\lambda^2} - e^{-3\lambda L} (\frac{3}{\lambda^2} + \frac{L}{\lambda})]$