In the figure, a conducting rod of length `l=1` meter and mass `m = 1 kg` moves with initial velocity `u = 5 m//s`. On a fixed horizontal frame containing inductor `L = 2 H` and resistance `R = 1 W`. `PQ` and `MN` are smooth, conducting wires. There is a uniform magnetic field of strength `B = 1T`. Initial there is no current in the inductor. Find the total charge in coulomb, flown through the inductor by the time velocity of rod becomes `v_(f) = 1 m//s` and the rod has travelled a distance `x = 3` meter.

Answer

A, B, C

Explanation

Let `i_(1)` and `i_(2)` be the current through `L` and `R` at any time `t`
`\\ i = i_(2) + i_(2) " "Ϸ (Blv)/(R ) = i_(2)" "` and `" "Bl v = L(di_(1))/(dt)`
Force on condcuting rod `= m(dv)/(dt) = -i l B`
`= -(i_(1) + (Blv)/(R ))l B`

`Ϸ " "m d v = -l B i_(1) dt - (B^(2) l^(2))/(R )v dt`
`Ϸ" " m int dv = -l B int i_(1) dt - (B^(2)l^(2))/(R ) int vdt`
`Ϸ " " m(v_(f) - u) = -l B Q - (B^(2) l^(2))/(R ) x`
(`v_(f) =` velocity, when it has moved a distance `'x'`)
`Ϸ " " Q = (-(B^(2)l^(2))/(R ) x - m(v_(f)-u))/(Bl) = 1C`.