In the circuit shown below, the key is pressed at time $t=0$. Which of the following statement(s) is(are) true?

($A$) The voltmeter displays $-5 \mathrm{~V}$ as soon as the key is pressed, znd displays $+5 \mathrm{~V}$ after a long time

($B$) The voltmeter will display $0 \mathrm{~V}$ at time $t=\ln 2$ seconds

($C$) The current in the ammeter becomes $1 / e$ of the initial value after $1$ second

($D$) The current in the ammeter becomes zero after a long time

Question

In the circuit shown below, the key is pressed at time $t=0$. Which of the following statement(s) is(are) true?

($A$) The voltmeter displays $-5 \mathrm{~V}$ as soon as the key is pressed, znd displays $+5 \mathrm{~V}$ after a long time

($B$) The voltmeter will display $0 \mathrm{~V}$ at time $t=\ln 2$ seconds

($C$) The current in the ammeter becomes $1 / e$ of the initial value after $1$ second

($D$) The current in the ammeter becomes zero after a long time

IIT 2016, Advanced

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Solution

Writing instantaneous equation, with $q_1$ being charge on upper and $q_2$ being that on lower capacitor,

$q _1=\left(200 \times 10^{-3}\right)\left(1- e ^{- t }\right)$

$q _2=\left(100 \times 10^{-3}\right)\left(1- e ^{- t }\right)$

$\frac{ q _1}{ C }=\left(50 \times 10^3\right) \frac{ dq }{ dt }$

$\frac{200 \times 10^{-3}\left(1- e ^{- t }\right)}{40 \times 10^{-6}}=\left(50 \times 10^3\right)\left(100 \times 10^{-3}\right) e ^{- t }$

$e ^{- t }=0.5$

$t =\ln (2)$

$I = I _1+ I _2=\left(200 \times 10^{-3}\right)\left( e ^{- t }\right)+\left(100 \times 10^{-3}\right) e ^{- t }$

$=100 \times 10^{-3}\left[2 e ^{- t }+ e ^{- t }\right]$

$=0.3 e ^{- t }$

Att $=\infty, \quad I = o$

Answer is $A,B,C$ and $D.$

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