Question

In an R-C circuit, when the switch is closed, the response _______.

• A. do not vary with time
• B. decays with time
• C. rises with time
• D. first increases and then decreases

Hint:

At steady-state ($t \rightarrow \infty$), a capacitor acts as an open circuit to DC signals. Since it blocks DC at steady state, the current must eventually fall to zero, confirming that the current response decays over time.

Answer 1

The Correct Option is B

Concept: When a switch is activated inside a standard series Resistor-Capacitor (R-C) network connected to a constant DC excitation source, transient currents and voltages are governed by exponential time functions containing the circuit time constant $\tau = RC$. The transient charging current response $i(t)$ as a function of elapsed time is expressed mathematically as: $$i(t) = I_0 e^{-\frac{t}{RC}}$$ Where $I_0 = \frac{V_s}{R}$ represents the initial peak surge current at the instant the switch closes ($t = 0^+$). Similarly, if evaluating a source-free discharging R-C network, both the capacitor voltage $v_c(t)$ and the loop current $i(t)$ decline over time following the relationship: $$v_c(t) = V_0 e^{-\frac{t}{RC}}$$ Step-by-step Evaluation:
• Observe the exponential factor $e^{-\frac{t}{RC}}$ present in these fundamental transient responses.
• As time progresses ($t \rightarrow \infty$), the value of $e^{-t/\tau}$ approaches zero: $$\lim_{t \rightarrow \infty} e^{-\frac{t}{RC}} = 0$$
• This continuous reduction over time represents a classic exponential decay. Therefore, the natural transient loop current response of an R-C circuit decays with time.