A capacitor is connected directly across a 20 V DC source through a series resistor and allowed to charge. After five time constants (5τ), what is the approximate voltage across the capacitor?

Introduction:
Engineers often estimate settling times using the 5τ rule for first-order systems. This question asks for the capacitor voltage after five time constants during charging from a fixed DC source.

Given Data / Assumptions:

• Supply voltage Vs = 20 V.
• Charging duration t = 5τ.
• Standard RC charging starting from 0 V.

Concept / Approach:
Use Vc(t) = Vs * (1 - exp(-t / τ)). At t = 5τ, exp(-5) is very small, so the capacitor is essentially at the supply voltage within about 0.7% error.

Step-by-Step Solution:
1) Vc(5τ) = 20 * (1 - exp(-5)).2) exp(-5) ≈ 0.0067.3) 1 - 0.0067 = 0.9933.4) Vc ≈ 20 * 0.9933 ≈ 19.866 V.5) Rounded to typical engineering precision, Vc ≈ 20 V.

Verification / Alternative check:
Compare to the 5τ rule of thumb: about 99.3% settled, which is close enough to treat as the final value in most applications.

Why Other Options Are Wrong:
13.5 V and 12.8 V: correspond to earlier times (between 1τ and 2τ), not 5τ.5.0 V: far too low for 5τ; would be near t much less than τ.

Common Pitfalls:
Reporting exact infinite-time behavior; first-order exponentials approach but do not instantaneously reach the final value. Engineers round at 5τ for practicality.

Final Answer:
20 V