An RC circuit has R = 10 kΩ and C = 100 μF. The capacitor starts uncharged and is connected to a 10 V source. Approximately how long does it take for the capacitor voltage to reach 6.3 V?
A1 ms
B100 ms
C1 s — one time constant τ = RC
D10 s
τ = RC = (10 × 10³ Ω)(100 × 10⁻⁶ F) = 1 s. After one time constant, v_C = V(1 − e⁻¹) ≈ 10 × 0.632 = 6.32 V ≈ 6.3 V. The time constant is the product of resistance and capacitance — R controls how much current flows for a given voltage difference, and C controls how much charge is needed for a given voltage rise. Both a larger R and a larger C independently slow the transient.
Question 2 Multiple Choice
Why does a capacitor's voltage increase exponentially rather than linearly when connected to a constant voltage source through a resistor?
ABecause the capacitor's capacitance decreases as it charges, reducing its ability to store more charge
BBecause as v_C rises, the voltage difference (V − v_C) across the resistor decreases, reducing current flow and progressively slowing the charging rate
CBecause the resistance increases as current heats the resistor, limiting the charging rate
DBecause KVL only applies during steady state, not during transient charging
The charging current is i = (V − v_C)/R. At t = 0, v_C = 0 so i = V/R (maximum). As the capacitor charges and v_C rises, the voltage available to drive current through R shrinks. Less current means slower charging, which means the rate of voltage rise decreases over time. This self-limiting feedback produces the exponential shape: the closer v_C gets to V, the more slowly it approaches. The process is asymptotic — theoretically never quite reaching V, though for practical purposes it is 'done' after 5τ.
Question 3 True / False
After one time constant τ = RC, a charging RC circuit has reached approximately 63% of its final voltage, regardless of the specific values of R, C, or the source voltage.
TTrue
FFalse
Answer: True
v_C(τ) = V(1 − e⁻¹) = V × 0.6321... ≈ 63.2% of V, always. The universality comes from the normalized form of the solution — the time constant τ = RC is the natural unit of time for any RC circuit, and after exactly one such unit, the exponent is −1 regardless of what the actual second-count is. This is why τ is so useful: a large RC circuit and a small RC circuit both reach 63% of their final voltage after exactly one of their respective time constants.
Question 4 True / False
Increasing primarily the capacitance in an RC circuit speeds up the transient response, since a larger capacitor stores more energy and charges faster.
TTrue
FFalse
Answer: False
Increasing C slows the transient — τ = RC increases proportionally. A larger capacitor requires more charge to reach the same voltage (Q = CV), and since the charging current is limited by R, it takes longer. Thinking of C as 'storing more energy' is not wrong, but it leads to the wrong intuition here: more storage capacity at the same charging rate means it takes longer to fill, not shorter. To speed up an RC circuit, you must decrease R, decrease C, or both.
Question 5 Short Answer
Explain in physical terms why an RC circuit charges exponentially rather than linearly. What causes the charging rate to decrease over time?
Think about your answer, then reveal below.
Model answer: Charging rate is proportional to the current flowing into the capacitor, which by Ohm's law equals (V_source − v_C)/R. As the capacitor charges, its voltage v_C rises, reducing the voltage difference available to drive current through R. Less current means charge accumulates more slowly, which means voltage rises more slowly — which in turn reduces the driving current further. This positive-feedback-in-reverse (a self-limiting process) produces the characteristic exponential decay of the charging rate and exponential rise of the voltage.
The governing differential equation RC(dv_C/dt) + v_C = V has the form 'rate of change is proportional to the remaining gap' — dv_C/dt = (V − v_C)/RC. Whenever the rate of change of a quantity is proportional to how far it is from its final value, the solution is exponential. Linear charging would require constant current, which would require a constant voltage across R, which would require v_C to stay constant — a contradiction. The exponential is not an approximation; it is the exact solution to the physics.