Question

In the circuit shown below, if the key is pressed at time $t=0$ , then which of the following statements is/are true ?

• A. The voltmeter displays $-5\text{V}$ as soon as the key is pressed, and displays $+5\text{V}$ after a long time .
• B. The voltmeter will display $0\text{V}$ at time $t=\ln 2$ seconds
• C. The current in the ammeter becomes $\frac{1}{e}$ of the Initial value after $1\text{second}$
• D. The current in the ammeter becomes zero after a long time

Answer 1

Answer: (A), (B), (C), (D)

The current in the ammeter becomes zero after a long time

At $t=0$ capacitors act as short circuit, so the voltmeter in the given connection displays $-5\text{V}$.

After a long time, (i.e as $t\to \infty$), capacitors act as open circuit, so the voltmeter which will be in series with the two resistors carries close to zero current (Resistance of ideal voltmeter is infinity), Thus, voltmeter displays $+5\text{V}$ .

Option (a) is correct.

Let us consider $C=20\mu \text{F}$ and $R=25\text{k}\Omega$ and also suppose that the current passing through different arms of circuit are as shown in the figure.

From the figure,

$q_x=2CV(1-e^{-t/2CR})$

$\Rightarrow x=\frac{V}{R}\left(e^{-t/2CR}\right)$

$q_y=CV(1-e^{-t/2CR})$

$\Rightarrow y=\frac{V}{2R}\left(e^{-t/2CR}\right)$

From figure, $V_A-V_B+V_B-V_C=V_A-V_C$

$\Rightarrow \frac{q_x}{2C}+\Delta V=y\left(2R\right)$

For $\Delta V=0$ , $y=\frac{q_x}{4CR}$

Substituting this in the above equation we get,

$\frac{q_x}{4CR}=\frac{V}{2R}\left(e^{-t/2CR}\right)$

$\Rightarrow \frac{q_x}{2CV}=e^{-t/2CR}$

$\Rightarrow (1-e^{-t/2CR})=e^{-t/2CR}$

$\Rightarrow 1=2e^{-t/2CR}$

$\Rightarrow t=2CR\ln 2$

Substituting the values of $C$and $R$ gives,

$t=\ln 2\text{sec}$

Option (b) is correct.

Current through the ammeter $I=x+y$

$\Rightarrow I=\frac{3V}{2R}\left(e^{-t/2CR}\right)$

At $t=0\to I=\frac{3V}{2R}=I_o\left(\text{say}\right)$

Thus, at $t=1\text{sec}\to I=\frac{I_0}{e}$

Option (c) is correct.

As $t\to \infty ;I\to 0$ So, option (d) is also correct.

Hence, all the options are correct alternatives for the given question.