Question

Switch \(S_1\) is closed at time \(t = 0\) in the network shown. The expression for the voltage across capacitor \(C\) is _______.

• A. \(10 \cdot (1 + e^{-40t})\)
• B. \(10 \cdot (1 - e^{-40t})\)
• C. \(10 \cdot (1 + e^{-25t})\)
• D. \(10 \cdot (1 - e^{-25t})\)

Hint:

For RC circuits, voltage across capacitor = \(V(1 - e^{-t/\tau})\) where \(\tau = RC\).

Answer 1

The Correct Option is D

Step 1: Identify Thevenin resistance
The total resistance in series with the capacitor after closing switch is: \[ R = R_1 \parallel R_2 = \frac{20k \cdot 20k}{20k + 20k} = 10k\ \Omega \] Step 2: Time constant
\[ \tau = RC = 10k \cdot 4\mu F = 40000 \mu s = 0.04s \] \[ \text{So, } \tau = \frac{1}{25} \Rightarrow e^{-t/\tau} = e^{-25t} \] Step 3: Capacitor charging formula: \[ V_C(t) = V_{final}(1 - e^{-t/\tau}) = 10(1 - e^{-25t}) \]