In the figure shown a conducting rod of length `l`, resistnace `R` and mass `m` is moved with a constant velocity `v`.The magnetic field `B` varies with time `t` as `B=5 t`, where `t` is time in second.At `t=0` the area of the loop containing capacitor and the rod is zero and the capacitor is uncharged.The rod started moving at `t=0` on the fixed smooth conducting rails which have negligible resistance.Find (i)The current in the circuit as a function of time `t`.
(ii)If the above system is kept in vertical plane such that the rod can move vertically downward due of gravity and other parts are kept fixed and `B`=constant =`B_(0)`.then find the maximum current in the circuit.

Answer

A, B, C

Explanation

(i)`phi`=flux at time `t`
=`BA=(5t)(lvt)=5lvt^(2)`
`therefore epsilon =|(dphi)/(dt)|=10 l vt`
`q/c=epsilon-iR rArr q/c=10l vt -iR`
Differentiating both sides w.r.t `t`
`10lv-(di)/(dt)R=i/c`
`i=10 lVc(1-e^(-t//cR))`
(ii)`q//c=epsilon-iR=BVl-IR`,
for maximum value of `i,di//dt=0`
`i//c=Bal-di//dtxxR`
`l//c=B((mg-Bil)/m)l-0`
`therefore i=i_(max)=(mg Bl c)/(m+B^(2)l^(2)c)`