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 A flows through the $1 Ω$ resistor. The key is closed at time $t = t_{0}$ . Which of the following statement(s) is (are) correct [2023]

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

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 A flows through the $1 Ω$ resistor. The key is closed at time $t = t_{0}$ . Which of the following statement(s) is (are) correct [2023]

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

Smart Achievers · Accepted Answer

(1, 2, 3, 4)

From circuit diagram, voltage drop across $1 Ω$ resistance

$V = 5 + 1 \times 1 = 6 V$

$∴$ $15 - I \times R = 6$ $\Rightarrow I R = 9$

$Across 3 Ω resistance,$ $6 - 3 I_{1} = 0$

$∴$ $I_{1} = 2 A$

Therefore for $t < t_{0}$ the value of current $I_{1}$ is 2A.

By Kirchhoff's current law,

$I = 1 + I_{1}$ $\Rightarrow I = 1 + 2 = 3 A$

$∵$ $I R = 9$ $\Rightarrow 3 \times R = 9$ $\Rightarrow R = 3 Ω$

$E_{eq} = \frac{\frac{15}{3} + \frac{5}{1} + \frac{0}{3}}{\frac{1}{3} + \frac{1}{1} + \frac{1}{3}} = 10 \times \frac{3}{5} = 6 V$

$q_{\max} = C V = 2 \times 6 = 12 μ C$

$Current in the circuit$ $i = \frac{6}{3.6} e^{\frac{t}{τ}}$

$= \frac{5}{3} e^{- 7.2 / 7.2} = \frac{5}{3} e^{- 1} \approx 0.6 A$

$∴$ $options (1, 2, 3, 4) are correct.$

Solution

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 A flows through the $1 Ω$ resistor. The key is closed at time $t = t_{0}$ . Which of the following statement(s) is (are) correct [2023]

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

1 The value of the resistance R is $3 Ω$ .

2 For $t < t_{0}$ , the value of current $I_{1}$ is 2 A

3 At $t = t_{0} + 7.2 μ s$ , the current in the capacitor is 0.6 A

4 For $t \to \infty$ , the charge on the capacitor is $12 μ C$ .

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Solution

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 A flows through the 1 Ω resistor. The key is closed at time t=t0. Which of the following statement(s) is (are) correct [2023] [Given: e-1=0.36]

1 The value of the resistance R is 3Ω.

2 For t<t0, the value of current I1 is 2 A

3 At t=t0+7.2 μs, the current in the capacitor is 0.6 A