In a circuit shown in the figure, the capacitor $C$ is initially uncharged and the key $K$ is open. In this condition, a current of $1 \mathrm{~A}$ flows through the $1 \Omega$ resistor. The key is closed at time $t=t_0$. Which of the following statement($s$) is(are) correct?

[Given: $\mathrm{e}^{-1}=0.36$ ]

($A$) The value of the resistance $R$ is $3 \Omega$.

($B$) For $t$

($C$) At $t=t_0+7.2 \mu \mathrm{s}$, the current in the capacitor is $0.6 \mathrm{~A}$.

($D$) For $t \rightarrow \infty$, the charge on the capacitor is $12 \mu C$.

Q: In a circuit shown in the figure, the capacitor $C$ is initially uncharged and the key $K$ is open. In this condition, a current of $1 \mathrm{~A}$ flows through the $1 \Omega$ resistor. The key is closed at time $t=t_0$. Which of the following statement($s$) is(are) correct? [Given: $\mathrm{e}^{-1}=0.36$ ] ($A$) The value of the resistance $R$ is $3 \Omega$. ($B$) For $t$ ($C$) At $t=t_0+7.2 \mu \mathrm{s}$, the current in the capacitor is $0.6 \mathrm{~A}$. ($D$) For $t \rightarrow \infty$, the charge on the capacitor is $12 \mu C$.

c (image) By writing voltage drop across $1 \Omega$ $\Rightarrow 0+5+1 \times 1=\mathrm{V}$ $\quad \mathrm{V}=6$ $\Rightarrow \text { Similarly across } \mathrm{R}$ $0+15-\mathrm{I} \times \mathrm{R}=6$ $\mathrm{IR}=9$ $\Rightarrow \text { across } 3 \Omega$ $\quad 6-3 \mathrm{I}_1=0$ $\mathrm{I}_1=2 \mathrm{~A}$ Hence option $(B)$ is correct $\Rightarrow \quad \mathrm{I}=1+2$ $\mathrm{I}=3$ $\mathrm{IR}=9$ $\mathrm{R}=3 \Omega$ Option $(A)$ is correct (image) $\varepsilon=\frac{\frac{15}{3}+\frac{5}{1}+\frac{0}{3}}{\frac{1}{3}+\frac{1}{1}+\frac{1}{3}}=10 \times \frac{3}{5}=6 \mathrm{~V}$ $\mathrm{q}_{\max }=2 \times 6=12 \mu \mathrm{C}$ $\mathrm{i}=\frac{6}{3.6} \mathrm{e}^{-\frac{\mathrm{t}}{\tau}}$ $=\frac{5}{3} \mathrm{e}-\frac{7.2}{7.2}=\frac{5}{3} \mathrm{e}^{-1} \approx 0.6 \mathrm{~A}$

Question

In a circuit shown in the figure, the capacitor $C$ is initially uncharged and the key $K$ is open. In this condition, a current of $1 \mathrm{~A}$ flows through the $1 \Omega$ resistor. The key is closed at time $t=t_0$. Which of the following statement($s$) is(are) correct?

[Given: $\mathrm{e}^{-1}=0.36$ ]

($A$) The value of the resistance $R$ is $3 \Omega$.

($B$) For $t$

($C$) At $t=t_0+7.2 \mu \mathrm{s}$, the current in the capacitor is $0.6 \mathrm{~A}$.

($D$) For $t \rightarrow \infty$, the charge on the capacitor is $12 \mu C$.

A$A,B,C,$
B$A,B,D$
C$A,B,C,D$
D$A,C,D$

IIT 2023, Medium

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