Two parallel resistanceless rails are connected by an inductor of inductance \( L \) at one end as shown in figure. A magnetic field \( B \) exists in the space which is perpendicular to the plane of the rails. Now a conductor of length \( l \) and mass \( m \) is placed transverse on the rail and given an impulse \( J \) toward the rightward direction. Then choose the correct option(s).
(1) Velocity of the conductor is half of the initial velocity after a displacement of the conductor \( d=\sqrt{\frac{3 J^{2} L}{4 B^{2} l^{2} m}} \)
(2) Current flowing through the inductor at the instant when velocity of the conductor is half of the initial velocity is \( i=\sqrt{\frac{3 J^{2}}{4 L m}} \)
(3) Velocity of the conductor is half of the initial velocity after a displacement of the conductor \( d=\sqrt{\frac{3 J^{2} L}{B^{2} l^{2} m}} \)
(4) Current flowing through the inductor at the instant when velocity of the conductor is half of the initial velocity is \( i=\sqrt{\frac{3 J^{2}}{m L}} \)
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