Source: PHYS 212 Final Examination, University of Illinois at Urbana-Champaign
Tags: RC circuit, time constant, capacitor charging, capacitor energy, exponential decay, DC steady state, inductor behaviour, EMF, resistor, tau, transient response
Difficulty: Intermediate | Prerequisites: Ohm's law, Kirchhoff's voltage law, basic capacitor and inductor definitions.
RC circuits are the simplest example of a time-dependent circuit, and they show up everywhere: camera flashes, heartbeat pacemakers, touchscreens, and digital signal filters. The core idea is that a capacitor does not charge or discharge instantly; it follows an exponential curve whose speed is set by the time constant τ = RC. This topic also covers what capacitors and inductors look like in the long run (t → ∞) when connected to a DC source, which is a favourite exam question because it forces you to think about limiting behaviour rather than formulas. If you are not comfortable with Kirchhoff's voltage law and the definitions of capacitance and inductance, review those before diving in.
When a battery, resistor, and uncharged capacitor are connected in series, the current starts at ε/R and decays exponentially to zero, while the voltage across the capacitor rises from zero to ε. The time constant τ = RC sets the pace. After a very long time the capacitor stores energy ½Cε². In DC steady state, capacitors act as open circuits (no current) and inductors act as ideal wires (no voltage drop).
RC circuit
A circuit containing a resistor and a capacitor in series, often with a battery or switch. It exhibits exponential charging or discharging behaviour. Think of it as the electrical equivalent of filling a tank through a narrow pipe: the flow (current) is fastest at the start and slows as the tank (capacitor) fills.
Time constant (τ)
For an RC circuit, τ = RC, measured in seconds. It is the time for the capacitor voltage to reach about 63% of its final value during charging, or to drop to about 37% during discharging. In simple terms, τ tells you "how long does it take for things to mostly happen."
EMF (ε)
Electromotive force. The voltage provided by the battery, measured in volts. Despite the name, it is not a force; it is the energy per unit charge that the battery supplies.
Capacitor energy
The energy stored in a charged capacitor: U = ½CV². When the capacitor is fully charged to voltage ε, this becomes U = ½Cε². Think of it as the energy you could recover if you let the capacitor discharge through a circuit.
DC steady state
The condition a circuit reaches after all transients have died out (t → ∞) with a constant (DC) source. Currents and voltages are no longer changing with time.
At t = 0 the capacitor is uncharged, so the voltage across it is zero.
Kirchhoff's voltage law around the loop: ε - IR - V_C = 0. With V_C = 0, this gives I = ε / R.
The initial current is determined entirely by the battery and the resistor. The capacitor, being uncharged, behaves like a short circuit (a wire) at this instant.
Current: I(t) = (ε / R) · e^(-t/RC)
Capacitor voltage: V_C(t) = ε · (1 - e^(-t/RC))
Capacitor charge: Q(t) = Cε · (1 - e^(-t/RC))
The current decays from ε/R towards zero. The capacitor voltage rises from zero towards ε. Both processes follow the same exponential time scale, τ = RC.
As t → ∞, the capacitor voltage approaches ε and the current approaches zero. The capacitor is fully charged.
Final charge: Q_max = Cε
Final voltage: V_C = ε
Energy stored: U = ½CV² = ½Cε²
Note that ½Cε² is the correct expression. The answer U = ½RI² does not apply here (that is instantaneous power dissipated in the resistor, not stored energy). U = 0 is wrong because the capacitor is fully charged. U = ε²C lacks the factor of ½.
τ = RC is defined as the product of the resistance (in ohms) and the capacitance (in farads).
After one time constant (t = τ), the capacitor has charged to about 63.2% of its final voltage.
After two time constants, about 86.5%.
After five time constants, the circuit is effectively in steady state (over 99%).
The physical significance: τ governs how quickly the circuit responds. A large resistance slows the current, and a large capacitance means more charge must be moved, so both increase the charging time.
This is about what capacitors and inductors look like long after a DC source has been connected and all transients have settled.
Capacitor in DC steady state: acts like an open circuit. No current flows through it. The voltage across it equals whatever Kirchhoff's laws demand, but the current is zero.
Inductor in DC steady state: acts like an ideal wire (short circuit). It carries current with no voltage drop across it, because V_L = L · dI/dt and dI/dt = 0 in steady state.
Both statements are correct simultaneously: after a long time with a DC source, the inductor is a wire and the capacitor is a gap.
Initial current: I(0) = ε / R
Current during charging: I(t) = (ε / R) · e^(-t/RC)
Capacitor voltage during charging: V_C(t) = ε · (1 - e^(-t/RC))
Time constant: τ = RC
Energy stored in capacitor: U = ½CV²
Fully charged energy: U = ½Cε²
The RC time constant determines how quickly a camera flash recharges (a large capacitor and small resistor for a quick recharge), how long a delay circuit holds before triggering, and how touchscreen sensors detect your finger (the capacitance changes when you touch the screen, altering the time constant). Defibrillators use a large capacitor charged over several seconds and discharged in milliseconds to deliver a controlled pulse of energy to the heart.
Students often say the initial current is zero because "the capacitor blocks current." At t = 0 an uncharged capacitor has zero voltage across it and behaves like a wire, not an open circuit. It only blocks current after it has charged up.
Confusing energy stored (½Cε²) with energy dissipated. Exactly half the energy supplied by the battery ends up stored in the capacitor; the other half is dissipated as heat in the resistor. This 50/50 split is independent of R.
Thinking inductors block DC current. They oppose changes in current (via V_L = L dI/dt), but once the current is steady, dI/dt = 0 and the inductor has zero voltage drop, acting as a plain wire.
Forgetting that τ has units of seconds. Ohms times farads: (V/A)(C/V) = C/A = seconds. Worth checking if a numerical answer makes sense.
⚠️ Questions 2A and 2B directly test the initial current and the long-time energy. Be ready to identify I(0) = ε/R and U = ½Cε².
⚠️ Question 11 (multi-select) asks which statements about capacitors and inductors in DC steady state are correct. The correct pair is: inductor acts like an ideal wire, capacitor acts like an open circuit.
⚠️ Question 16 (short answer) asks you to define τ and explain its physical significance. Write: τ = RC; it is the time for the capacitor to charge to ~63% of its final voltage (or for the current to decay to ~37% of its initial value); it sets the overall time scale of the transient response.
Fill in the blank: At t = 0, the current in a charging RC circuit is ______.
True or False: After a very long time, the energy stored in the capacitor is ε²C.
Fill in the blank: In DC steady state, a capacitor behaves like a(n) ______.
True or False: An inductor in DC steady state has a large voltage drop across it.
Fill in the blank: The time constant τ = ______ and has units of ______.
Answers: 1. ε/R. 2. False (it is ½Cε²; the factor of ½ is essential). 3. Open circuit. 4. False (V_L = 0 because dI/dt = 0). 5. RC; seconds.
Q: At the instant the switch is closed in a series RC circuit with EMF ε, resistance R, and an uncharged capacitor C, what is the current?
A: I = ε/R. The uncharged capacitor has zero voltage across it, so the full EMF appears across the resistor.
Q: How much energy is stored in the capacitor after a very long time (t → ∞)?
A: U = ½Cε². The capacitor charges to voltage ε, and the energy stored in a capacitor at voltage V is ½CV².
Q: Define the time constant τ for an RC circuit and explain its physical significance.
A: τ = RC. It is the characteristic time scale of the exponential charging (or discharging) process. After one time constant, the capacitor has reached approximately 63% of its final voltage. Equivalently, the current has decayed to about 37% of its initial value. A larger τ means a slower response.
Q: Which statements are correct about DC steady-state behaviour? (a) Inductor = ideal wire, (b) Inductor = open circuit, (c) Capacitor = open circuit, (d) Capacitor = ideal wire.
A: (a) and (c). In DC steady state, dI/dt = 0 so the inductor has no voltage drop (ideal wire), and the capacitor is fully charged so no current flows through it (open circuit).
RC circuits connect to RLC circuits (the next step up, adding an inductor) where oscillation replaces simple exponential decay. The concept of time constants reappears in RL circuits (τ = L/R) and in the damping of RLC oscillations. The energy expression ½CV² comes back whenever you compute energy in capacitor-based systems, including LC oscillations where energy shuttles between capacitor and inductor.
RC circuit, time constant, tau, capacitor charging, exponential decay, ε/R initial current, capacitor energy, ½CV², DC steady state, inductor ideal wire, capacitor open circuit, transient response, Kirchhoff's voltage law, PHYS 212 final review, UIUC electricity and magnetism