In the circuit shown below, the key K is closed at t =0. The current through the battery is

Options

A. `(V(R_(1)+R_(2)))/(R_(1)R_(2))` at `t=0` and `(V)/(R_(2))` at `t=oo`
B. `(V(R_(1)+R_(2)))/(sqrt(R_(1)^(2)R_(2)^(2)))` at `t=0` and `(V)/(R_(2))` at `t=oo`
C. `(V)/(R_(2))` at `t=0` and `(V(R_(1)+R_(2)))/(R_(1)R_(2))` at `t=oo`
D. `(V)/(R_(2))` at `t=0` and `(V(R_(1)+R_(2)))/(R_(1)R_(2))` at `t=oo`

Answer

C

Explanation

Current through battery
`i=i_(1)+i_(2)=(V)/(R_(1))(1-e^(-(R_(2)t)/(L)))+(V)/(R_(2))`
`t=0` , `i=0+(V)/(R_(2))=(V)/(R_(2))`
`t=prop` , `i=(V)/(R_(1))+(V)/(R_(2))=(V(R_(1)+R_(2)))/(R_(1)R_(2))`