A copper wire \( a b \) of length \( l \), resistance \( r \) and mass \( m \) starts sliding at...

A copper wire \( a b \) of length \( l \), resistance \( r \) and mass \( m \) starts sliding at \( t=0 \) down a smooth, vertical, thick pair of connected conducting rails as shown in figure. A uniform magnetic field \( B \) exists in the space in a direction perpendicular to the plane of the rails. Which options are correct?
(a) The magnitude and direction of the induced current in the wire when speed of the wire \( v \) is \( \frac{v B \ell}{r}, a \) to \( b \).
(b) The downward acceleration of the wire at this instant is \( g-\frac{B^{2} \ell^{2}}{m r} v \).
(c) The velocity of the wire as a function of time is \( v_{m}\left(1-e^{-g t / v_{m}}\right) \) where \( v_{m}=\frac{m g r}{B^{2} \ell^{2}} \).
(d) The displacement of the wire as a function of time is \( v_{m} t-\frac{v_{m}^{2}}{g}\left(1-e^{-g t / v_{m}}\right) \) where \( v_{m}=\frac{m g r}{B^{2} \ell^{2}} \).