An uncharged capacitor `C` and a variable resistance `R` are connected to an ideal source and two resistors with the help of a switch at `t=0` as shown in the figure. Initially capacitor is uncharged and switch `S` is closed at `t=0 sec`. The graph between `V_(A)-V_(B)` for variable Resistor `R` for its three different values `R_(1)`, `R_(2)`, and `R_(3)` versus time is shown in the figure `II`. Choose the correct statement

Options

A. `R_(1) lt R_(2) lt R_(3)`
B. `R_(1) gt R_(2) gt R_(3)`
C. `|a| lt |b|`
D. `|a| gt |b|`

Answer

b

Explanation

Consider circuit at any time `t` after closing the switch :
`I_(1)=(V)/(2r)`, and `I_(2)=(V)/(R )e^(-(1)/(RC))`
`rArr V_(A)-V_(B)=RI_(2)-rI_(1)=Rxx(V)/(R )e^(-(1)/(RC))-rxx(V)/(2r)`
`rArr V_(A)-V_(B)=(-V)/(2)(1-2e^(-(1)/(RC)))`
`rArr a=(-V)/(2)`, and `v=(+V)/(2)`
When `V_(A)-V_(B)=0rArrt_(0)=RCln(2)`, Hence greater the `R`, greater the `t_(0)`