A capacitor initially given a charge `Q_(0)` is connected across a resistor `R` at `t=0`. The separation between the plates changes according to `d=(d_(0))/((1+t))(0 le t lt1)`. A small bulb is connected across the plates of the capacitor which lights when potential difference across the plates of the capacitor reaches `V_(0)`. Find
(`a`) the variation of charge with time
(`b`) time when the bulb will light.

Explanation

Capacitance at `t=0`, `C_(0)=(epsilon_(0)A)/(d_(0))`, `C=C_(0)(1+t)`
Using Kirchoff's law
`(q)/(C )-Ri=0`
(`i`) `(q)/(C_(0)(1+t))+R(dq)/(dt)=0`
`(dq)/(q)=-(1)/(RC_(0))(dt)/((1+t))`
`lnq:|_(Q_(0))^(q)``=-(1)/(RC_(0))ln(1+t):|_(0)^(t)`
`ln(q//Q_(0))=-(1)/(RC_(0))ln(1+t)`
`ln(q//Q_(0))=ln(1+t)^((1)/(RC_(0)))`
`q=Q_(0)(1+t)^((1)/(RC_(0)))`
`V=(q)/(C )=(Q_(0)(1+t)^(-(1)/(RC)))/(C_(0)(1+t))=(Q_(0))/(C_(0))(1+t)^(-((1)/(RC)+1))`
which gives `t={((Q_(0))/(C_(0)V))^(((RC)/(RC+1)))-1}`