Topic: Network Theory
A steady state is achieved in following Network at $t=0$, find the time when $Vc(t)$ will be maximum
- A. $ln(9)$ sec
- B. $ln(9)^{0.125}$ sec
- C. $ln(9)^{0.25}$ sec
- D. 9 sec
Answer: B. $ln(9)^{0.125}$ sec
Explanation: Solving the differential equation for the RLC circuit with the given initial conditions yields an overdamped response for the capacitor voltage. Taking the derivative of $Vc(t)$ and setting it to zero gives an equation of the form $e^{-t} = 9e^{-9t}$, which simplifies to $e^{8t} = 9$. Thus, $t = \frac{1}{8}\ln(9) = 0.125 \ln(9)$, typically represented as $\ln(9)^{0.125}$ seconds in this context.