Questions: Transient Response in RC Circuits

Question 1

A capacitor in a series RC circuit is initially uncharged (V_i = 0). A 12V source is switched in at t = 0, with R = 10 kΩ and C = 100 μF. What is the capacitor voltage at t = τ?

• A. 12 V — the capacitor fully charges after one time constant
• B. 6 V — the capacitor charges linearly, reaching half-voltage at τ
• C. Approximately 7.58 V — the capacitor covers about 63% of the gap from 0 V to 12 V
• D. 0 V — no significant charging occurs until t = 2τ

Answer: (C)

At t = τ, the capacitor has covered 1 − e⁻¹ ≈ 63.2% of the gap between initial (0 V) and final (12 V): 0.632 × 12 ≈ 7.58 V. Option A is the most common misconception — τ is not the time to full charge. The charging curve is exponential (not linear), asymptotically approaching 12 V. At t = 5τ the capacitor is within ~0.7% of 12 V; it theoretically never reaches it in finite time.

Question 2

In a series RC circuit, both R and C are doubled. How does the time constant τ change?

• A. τ is halved — larger capacitance and larger resistance have opposing effects
• B. τ is doubled — only capacitance affects τ; resistance is irrelevant
• C. τ is quadrupled — since τ = RC and both factors double, the product quadruples
• D. τ is unchanged — the ratio of R to C stays the same

Answer: (C)

τ = RC. Doubling both gives τ_new = (2R)(2C) = 4RC = 4τ. Both factors independently increase τ: resistance limits the current that can flow into the capacitor (slowing charging), and capacitance requires more charge to reach the final voltage (also slowing charging). These effects multiply, not cancel. The unit check confirms it: Ω × F = (V/A)(C/V) = C/A = seconds.

Question 3

A capacitor in an RC circuit reaches its final voltage exactly at t = 5τ seconds.

• T. True
• F. False

Answer: False

The capacitor voltage v_C(t) = V_f + (V_i − V_f)·e^(−t/τ) approaches V_f asymptotically — the term e^(−t/τ) never reaches exactly zero for finite t. At t = 5τ, it is within 1 − e⁻⁵ ≈ 99.3% of V_f, close enough for most engineering purposes, but the exact final value is only reached in the mathematical limit as t → ∞. The common claim 'fully charged after 5τ' is a practical approximation, not an exact statement.

Question 4

To find the complete time-domain voltage response for any first-order RC circuit, it is sufficient to know three quantities: the initial capacitor voltage, the final capacitor voltage, and the Thévenin resistance seen by the capacitor.

• T. True
• F. False

Answer: True

The solution is fully determined by v_C(t) = V_f + (V_i − V_f)·e^(−t/τ), where τ = R_Thévenin × C. V_i comes from energy continuity (capacitor voltage cannot jump at the switching instant). V_f comes from DC steady-state analysis (treat the fully charged capacitor as an open circuit). τ comes from the Thévenin resistance seen by the capacitor after switching. These three numbers, and nothing more, determine the exponential transient completely.

Question 5

Why can't the voltage across a capacitor change instantaneously when a switch opens or closes in a circuit?

Model answer: Because instantaneous voltage change would require infinite current. The capacitor's constitutive relation is i = C·(dv/dt): current equals capacitance times the rate of voltage change. If voltage changed instantaneously, dv/dt would be infinite, which would require infinite current — impossible in a circuit with finite resistance. The energy stored in the capacitor's electric field must be redistributed gradually through the resistive network, producing the exponential transient.

This energy-continuity principle is what fixes the initial condition V_i: at the moment of switching (t = 0⁺), the capacitor voltage equals what it was just before switching (t = 0⁻), even though the circuit topology has changed. The rest of the circuit may change instantaneously, but the capacitor voltage cannot. This initial condition, combined with the new steady-state and the time constant, fully specifies the exponential transient that follows.