RC Circuit Charging & Discharging

Q: What is the time constant of a circuit with a 1MΩ resistor and a 4μF capacitor?

4.0 seconds \tau = RC = (10^6 \Omega) \times (4 \times 10^{-6} F) = 4.0s.

Q: After one time constant (t = \tau), what percentage of the final charge has a capacitor reached during the charging process?

63.2% 1 - e^{-1} \approx 0.632. After one time constant, the capacitor is roughly 63% charged.

Q: When a capacitor is fully charged in a DC circuit, what is the current flowing through that branch?

Zero. Once fully charged, the potential difference across the capacitor equals the battery voltage, and there is no longer a potential difference across the resistor to drive current. The capacitor behaves like an open switch.

Analyze RC circuits where capacitors charge and discharge through resistors with exponential time dependence. Understand the time constant τ = RC that characterizes how quickly the capacitor charges to 63% of maximum voltage or discharges to 37% of initial voltage. Visualize voltage and current curves using Q(t) = Q₀(1 - e^(-t/τ)) for charging and Q(t) = Q₀e^(-t/τ) for discharging. Apply RC circuits to timing applications, filters, and camera flashes.

WHAT IS AN RC CIRCUIT?

An RC circuit is a circuit containing a resistor (R) and a capacitor (C) connected in series or parallel. In these circuits, the current is not constant but changes over time as the capacitor charges or discharges. The behavior is governed by the time constant, , which determines how quickly the voltage and current change.

CHARGING VS. DISCHARGING

When charging, the capacitor voltage starts at zero and asymptotically approaches the battery voltage. The current starts at its maximum () and exponentially decays to zero. When discharging, both the voltage and current decay exponentially from their initial values toward zero.

HOW TO USE THIS VISUALIZATION

1. **Switch the Circuit**: Toggle between "Charging" (battery connected) and "Discharging" (capacitor connected only to resistor). 2. **Adjust Components**: Change R and C values and watch the time constant () update. 3. **View Graphs**: Observe the and graphs. See how larger makes the curves flatter (slower change). **Try This**: Set R=10kΩ and C=100μF. Calculate . Then set R=20kΩ and C=50μF. Compare the charging times. (They should be the same since is the same).

CORE FORMULAS

Time Constant

Charge during charging process

Voltage during discharging process

AP EXAM CONNECTION

Unit: Unit 4: DC Circuits (Topic 4.3)
Learning Objective: PVFE-3.C

COMMON MISCONCEPTIONS

Thinking the current is constant (it is always changing in an RC circuit).
Confusing the charging and discharging equations (charging has the form).
Believing a capacitor charges fully in exactly one time constant (it takes roughly 5τ to reach 99%).

KEY TAKEAWAYS

Time constant is determined by RC product.
Voltage/current change exponentially.
Charging: Current decreases as voltage increases.
Discharging: Both V and I decrease to zero.
Full charge takes approximately 5 time constants.

PRACTICE QUESTIONS

Q1 (QUANTITATIVE): What is the time constant of a circuit with a 1MΩ resistor and a 4μF capacitor?

Show Answer & Explanation

Answer: 4.0 seconds

Explanation: .

Q2 (QUANTITATIVE): After one time constant (), what percentage of the final charge has a capacitor reached during the charging process?

Show Answer & Explanation

Answer: 63.2%

Explanation: . After one time constant, the capacitor is roughly 63% charged.

Q3 (CONCEPTUAL): When a capacitor is fully charged in a DC circuit, what is the current flowing through that branch?

Show Answer & Explanation

Answer: Zero.

Explanation: Once fully charged, the potential difference across the capacitor equals the battery voltage, and there is no longer a potential difference across the resistor to drive current. The capacitor behaves like an open switch.