A 100 V DC source charges a capacitor. What is the capacitor voltage after one time constant τ from the start of charging?

Introduction:First-order RC charging follows a well-known exponential. This question checks your ability to apply the standard time-constant result to find the capacitor voltage after one τ when charging from a step DC source.

Given Data / Assumptions:

• Step input Vs = 100 V applied at t = 0.
• Capacitor initially uncharged.
• Time of interest t = τ = R * C.

Concept / Approach:The capacitor voltage during charging is Vc(t) = Vs * (1 − e^(−t/τ)). Substituting t = τ yields Vc(τ) = Vs * (1 − e^(−1)). The numerical factor 1 − e^(−1) ≈ 0.632 applies to any RC with a step input.

Step-by-Step Solution:

Verification / Alternative check:The complementary current at t = τ is i(τ) = (Vs/R) * e^(−1) ≈ 0.368 * (Vs/R); this pairs with Vc(τ) ≈ 0.632 * Vs, confirming standard RC behavior.

Why Other Options Are Wrong:

• 37.8 V or 38 V: These correspond to the remaining fraction e^(−1) of the initial step on the resistor, not the capacitor voltage at τ.
• 101 V: Exceeds the source; impossible without an inductor or overshoot mechanism.

Common Pitfalls:Confusing the capacitor's voltage with the resistor's drop at t = τ, or rounding errors that swap 0.632 and 0.368 fractions.

Final Answer:63 V.