In Fig a rod of length `l` and mass `m` moves with an intial velocity `u` on a fixed frame containing inductor `L` and resistance `R.PQ` and `MN` are smooth conducting wires. There is auniform magnetic field of strength `B`. Initially, there is no current in the inductor. find the total cherge flows through the inductor by the time, velocity of rod becomes `nu_(f)` and the rod has travelled a distance `x`.

Answer

`(-(B^(2)l^(2))/(R ) x - m(nu_(f) - u))/(Bl)`

Explanation

`I = i_(1) + i_(2)`

`(Bl nu)/(R ) = i_(2) rArr Bl nu = L(di_(1))/(dt)`
`m(d nu)/(dt) = - IlB = - (i_(1) + (Bl nu)/(R )) lB`
`rArr m int d nu = - lB int i_(1) dt - (B^(2)l^(2))/(R )int nu dt`
`rArr m(nu_(f) - u) = - lBQ - (B^(2)l^(2))/(R ) x`
`(nu_(f) =` velocity when it has moved a distance `x`)
`rArr Q = ((-B^(2)l^(2))/(R) x - m(nu_(f) - u))/(Bl)`