RC charging (repaired for solvability): An RC low-pass (first-order) is driven by two identical rectangular pulses from 0 V to 12 V, applied back-to-back with no gap. Each pulse width equals 5 time constants (5τ), so the node sees a total “high” duration of 10τ before returning low. What is the output voltage across the capacitor at the end of the second pulse (assume it started at 0 V and the op-amp/buffer, if present, is ideal)?

Introduction / Context:
Time-response questions with “the given circuit” often reference an RC network responding to pulses. To make the problem solvable without the missing diagram, we specify a standard first-order RC low-pass driven by two consecutive 0→12 V pulses, each lasting 5τ. This lets us compute the capacitor voltage using the classic exponential charging law and determine the voltage at the precise moment the second pulse ends.

Given Data / Assumptions:

• First-order RC, initially uncharged (V_C(0) = 0 V).
• Input = 12 V high for 5τ, immediately followed by another 12 V high for 5τ (total 10τ high), then low.
• Ideal buffer/op-amp if used; no loading on the capacitor; τ = R * C is constant.

Concept / Approach:
For a step from 0 to V_S, the capacitor charges as V_C(t) = V_S * (1 − e^(−t/τ)). After a duration of nτ, the residual error to the final value is e^(−n). At 5τ, the charge has reached ≈ 99.3% of its final value; at 10τ, ≈ 99.995% of final. With V_S = 12 V, the end-of-second-pulse voltage is essentially the final value, i.e., approximately 12 V.

Step-by-Step Solution:

Verification / Alternative check:
If each pulse were only 3τ wide, the end of the second pulse would be V_C(6τ) = 12 * (1 − e^(−6)) ≈ 12 * 0.9975 ≈ 11.97 V—still very close to 12 V. Increasing to 10τ makes it indistinguishable from 12 V at normal resolution.

Why Other Options Are Wrong:

Common Pitfalls:
Forgetting that the RC never reaches the final value mathematically in finite time; confusing a single 5τ pulse with two back-to-back 5τ pulses; neglecting that “virtual ground” buffers prevent loading of the RC node.

Final Answer:
12 V