In the RC circuit in the figure, a DC voltage E is applied to an uncharged capacitor by closing the switch at t = 0. With the time constant τ = RC, which of the following is closest to the capacitor voltage vC at t = 5τ, as a percentage of the final value E?

Answer: B. About 99.3%

Method 1 (substitute into the formula): the capacitor voltage is vC(t) = E(1 − e^(−t/τ)), so at t = 5τ it is 1 − e^(−5) ≈ 1 − 0.0067 = 0.993, about 99.3%. Method 2 (rule of thumb per time constant): each τ multiplies the remaining difference from the final value by e^(−1) ≈ 0.37, so the voltage reaches 63% at 1τ, 86% at 2τ, 95% at 3τ, 98% at 4τ and 99.3% at 5τ. In theory it never reaches 100%, so in practice the transient response is considered finished after about 5τ.

• A. About 95% is the value at t = 3τ.
• B. Correct. 1 − e^(−5) ≈ 99.3%.
• C. The voltage approaches exponentially, so it never reaches the final value exactly in a finite time.