For an RC circuit with a step input Vs at t = 0, what is the expression for the capacitor voltage v_C(t)?

• A. v_C(t) = Vs e^{-t/RC}
• B. v_C(t) = Vs (1 - e^{-t/RC})
• C. v_C(t) = Vs
• D. v_C(t) = t e^{-t/RC}

Answer: (B)

When a step voltage Vs is applied to a series RC with the capacitor initially uncharged, the capacitor voltage rises toward Vs with a time constant RC. The current through the resistor is i = (Vs − vC)/R, and since i = C dvC/dt, you get C dvC/dt = (Vs − vC)/R. Rewriting gives dvC/dt + (1/RC) vC = Vs/(RC). With the initial condition vC(0) = 0, the solution is vC(t) = Vs [1 − e^(−t/(RC))]. This shows an exponential charging toward Vs, starting from 0 and approaching Vs as t → ∞.

This matches the intuitive behavior: at t = 0 the capacitor is uncharged, so it begins charging, and after a long time it equals the supply voltage. The other forms don’t fit the charging scenario: one would imply a decay from Vs to 0, a constant final value with no dynamics, or an inappropriate time dependence that doesn’t satisfy the governing differential equation and initial condition.