Question

Given the initial voltage in a capacitor C is $V_C(0) = 5\text{V}$, $R=10\text{ K}\Omega$ and $C=1\ \mu\text{F}$. For the network shown in the figure, the value of $V_C(t)$ at time $t=\infty$ is:

• A. $0.13\text{ V}$
• B. $0\text{ V}$
• C. $2.5\text{ V}$
• D. $5\text{ V}$

Hint:

In any passive circuit without external independent DC/AC sources, all transient voltages and currents must eventually decay to zero as $t \rightarrow \infty$ because of energy dissipation in the resistors.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The problem presents a source-free parallel RC circuit where the capacitor has an initial charged voltage of $5\text{ V}$ and discharges through a resistor of $10\text{ K}\Omega$.
We need to find the final capacitor voltage $V_C(t)$ as $t \rightarrow \infty$.

Step 2: Key Formula or Approach:
The voltage across a discharging capacitor in a source-free RC circuit is:
\[ V_C(t) = V_C(0) e^{-t/\tau} \]
where $\tau = RC$.

Step 3: Detailed Explanation:

• Given parameters:
Initial voltage, $V_C(0) = 5\text{ V}$.
Resistance, $R = 10\text{ K}\Omega = 10^4\ \Omega$.
Capacitance, $C = 1\ \mu\text{F} = 10^{-6}\text{ F}$.

• Time constant calculation:
\[ \tau = RC = 10^4 \times 10^{-6} = 0.01\text{ s} \]

• Finding the voltage expression:
\[ V_C(t) = 5 e^{-100t}\text{ V} \]

• At $t = \infty$:
\[ V_C(\infty) = \lim_{t \to \infty} 5 e^{-100t} = 5 \times 0 = 0\text{ V} \]

• All the initially stored energy ($E = \frac{1}{2} C V^2$) is completely converted into heat in the resistor.

Step 4: Final Answer:
The final voltage $V_C(\infty)$ is $0\text{ V}$, which corresponds to Option (B).