1.

Two parallel resistanceless rails are connected by an inductor or 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 towards the rightward direction. Then select the corrent option(s).

Answer»

velocity of the CONDUCTOR is half of the initial velocity after a displacement of the conductor `d=sqrt((3J^(2)L)/(4B^(2)l^(2)m))`
velocity of the conductor is half of the initial velocity after a displacement of the conductor `d= sqrt((3J^(2)L)/(B^(2)l^(2)m))`
current flowing through the inductor at the instant when velocity of the conductor is half of the initial velocity is `i=sqrt((3J^(2))/(4Lm))`
current flowing through the inductor at the instant when velocity of the conductor is half of the initial velocity is `i=sqrt((3J^(2))/(mL))`

SOLUTION :`L=(di)/(dt)=BVL`
`rArr int di=(Bl)/(L)int vdt rArr t rArr i=(Bl)/(L)x ""`….(i)
`F=ma rArr -iBl=mv (dv)/(DX)`
`rArr -(B^(2)l^(2)x)/(L)=mv(dv)/(dx)rArr - (B^(2)l^(2))/(mL)int_(0)^(d)xdx=int_(v_(0))^(v_(0)//2)vdv`
`rArr -(B^(2)l^(2)d^(2))/(2mL)=(-3v_(0)^(2))/(8)[v_(0)=(J)/(m)] rArr d=sqrt((3J^(2)L)/(4B^(2)l^(2)m))`
Putx = d in (i),`i=(Bl)/(L)sqrt(3J^(2)L)/(4B^(2)l^(2)m)=sqrt((3J^(2))/(4Lm))`


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