Question 708: Electrostatics & Capacitance

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

Accepted Answer

At $\mathbf{t} \mathbf{t}_0$
$\begin{aligned} \frac{\varepsilon}{r} & =\frac{15}{3}+\frac{5}{1}+\frac{0}{3} \\ \frac{1}{r} & =\frac{1}{3}+\frac{1}{1}+\frac{1}{3}=\frac{5}{3} \\ \varepsilon & =10 	imes \frac{3}{5} \Rightarrow 6 \mathrm{~V} \\ 	ext { At } t ightarrow \infty & \\ q & =\mathrm{C} \varepsilon \\ & =2 	imes 10^{-6} 	imes 6=12 \mu \mathrm{C}\end{aligned}$
$\begin{aligned} & 	ext { Time constant } 	au=R C \\ & =(r+3) 2 \mu \mathrm{C} \\ & =\left(\frac{3}{5}+3ight) 	imes 12=7.2 \mu \mathrm{s} \\ & i=i . e^{-t / 	au} \\ & i_0=\frac{\varepsilon}{r+3}=\frac{6}{\left(3+\frac{3}{5}ight)}\end{aligned}$
$$
\begin{aligned}
\Rightarrow \quad i & =\frac{6}{\frac{18}{5}} 	imes e^{-7.2 / 1.2} \\
& =\frac{5}{3} 	imes 0.36=0.6 \mathrm{~A}
\end{aligned}
$$
All options are correct.

Answer

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

Answer

For $t < t_0 $, the value of current $I_1$ is $2 \mathrm{~A}$.

Answer

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

Answer

For $t \rightarrow \infty$, the charge on the capacitor is $12 \mu \mathrm{C}$.

Electrostatics & Capacitance

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: e^{-1}=0.36 ]

Choose the correct answer: