11.8 Resistor-Capacitor (RC) Circuits

TLDR

RC circuits combine resistors and capacitors, and the capacitor charges or discharges over time instead of instantly. The key number is the time constant $\tau = R_{eq} C_{eq}$: after one time constant a charging capacitor reaches about 63% of its final charge, and a discharging one drops to about 37% of its starting charge. For AP Physics 2, you describe this behavior qualitatively and find equivalent capacitance for series and parallel combinations.

RC circuits combine resistors and capacitors, so they build on your work with capacitors and DC circuit analysis and add a time element.

The Time Constant and How Capacitors Charge

• Capacitors in series: $\frac{1}{C_{eq,s}} = \sum_i \frac{1}{C_i}$, and the result is smaller than the smallest capacitor. Series capacitors share the same magnitude of charge.
• Capacitors in parallel: $C_{eq,p} = \sum_i C_i$, so adding parallel capacitors increases total capacitance.
• Time constant: $\tau = R_{eq} C_{eq}$, measured in seconds when $R$ is in ohms and $C$ is in farads.
• One time constant gives about 63% of final charge when charging, and about 37% of starting charge remaining when discharging.
• At t = 0, an uncharged capacitor acts like a wire (current flows freely). After a long time, a fully charged capacitor acts like an open branch (zero current).
• Voltage, branch current, and stored energy all change with time and approach steady state asymptotically.

Equivalent Capacitance

When multiple capacitors are connected in a circuit, you can analyze them as a single equivalent capacitance ($C_{eq}$). This simplifies the circuit by treating several components as one.

For capacitors connected in series:

• The equivalent capacitance is found using the inverse sum: $\frac{1}{C_{\text{eq}, \text{s}}}=\sum_{i} \frac{1}{C_{i}}$
• The equivalent capacitance is always less than the smallest individual capacitance in the series.
• With two capacitors: $\frac{1}{C_{eq,s}} = \frac{1}{C_1} + \frac{1}{C_2}$

For capacitors connected in parallel:

• The equivalent capacitance is the sum of all individual capacitances: $C_{\text {eq,p }}=\sum_{i} C_{i}$
• With two capacitors: $C_{eq,p} = C_1 + C_2$

Notice the pattern is the reverse of resistors: capacitors in series add like resistors in parallel, and capacitors in parallel add like resistors in series.

RC Circuit Behavior

RC circuits show time-dependent behavior, which is what makes them different from simple resistor circuits where currents settle instantly.

Time Constant

The time constant ($\tau$) measures how quickly a capacitor charges or discharges:

• It equals the product of equivalent resistance and equivalent capacitance: $\tau = R_{eq} C_{eq}$
• It has units of seconds when resistance is in ohms and capacitance in farads.
• It sets the characteristic time scale for the circuit's response.

For a charging capacitor:

• After one time constant, the charge reaches about 63% of its final value.
• After about five time constants, the capacitor is practically fully charged.

For a discharging capacitor:

• After one time constant, the charge drops to about 37% of its initial value.
• After about five time constants, the capacitor is practically fully discharged.

Charging and Discharging

When a capacitor charges in an RC circuit:

1. At first, the uncharged capacitor acts like a wire, so charge flows easily onto the plates.
2. As charge builds up, the potential difference across the capacitor increases.
3. This rising potential difference opposes further current.
4. The branch current gradually decreases.
5. The stored electric potential energy increases as charge accumulates.
6. Voltage, branch current, and stored energy all change with time and approach steady-state values asymptotically.
7. Eventually the capacitor reaches its fully charged state: maximum potential difference and zero current in that branch.

When a capacitor discharges:

1. It starts at maximum charge and potential difference.
2. Current flows in the opposite direction compared to charging.
3. The charge on the plates and the stored energy begin to decrease immediately.
4. The plate charge, potential difference, and branch current all decrease toward zero asymptotically.
5. This continues until steady state is reached.

After times much greater than the time constant, you can model the capacitor and its branch using steady-state conditions. A charging capacitor approaches its maximum potential difference with zero current in that branch. A discharging capacitor approaches zero charge, zero potential difference, and zero current.

Capacitance Math and Describing Circuits Over Time

Problem Solving:

• For equivalent capacitance, remember the formulas are flipped from resistors. Series uses the inverse sum; parallel is a direct sum.
• Series check: your answer must be smaller than the smallest capacitor. If it is not, you used the wrong formula.
• For time constants, plug $R_{eq}$ and $C_{eq}$ into $\tau = R_{eq} C_{eq}$. Watch units: convert microfarads to farads before multiplying.

Free Response:

• Use the two-snapshot strategy: describe the circuit at t = 0 (capacitor acts like a wire) and after a long time (capacitor acts like an open branch with zero current).
• When you justify an answer, name the right quantity. Do not mix up potential difference, current, charge, and capacitance.
• If a question asks how voltage or current changes over time, describe the asymptotic approach to steady state instead of trying to write an exponential equation.

Common Trap:

• A fully charged capacitor in steady state has zero current in its branch, but the voltage across it is at maximum, not zero. Students often flip these.

Practice Problem 1: Equivalent Capacitance

Solution:

(a) For capacitors in series: $\frac{1}{C_{\text{eq}, \text{s}}}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}$

Substituting the values: $\frac{1}{C_{\text{eq}, \text{s}}}=\frac{1}{3 \text{ μF}}+\frac{1}{6 \text{ μF}}+\frac{1}{12 \text{ μF}}$ $\frac{1}{C_{\text{eq}, \text{s}}}=\frac{4}{12 \text{ μF}}+\frac{2}{12 \text{ μF}}+\frac{1}{12 \text{ μF}}=\frac{7}{12 \text{ μF}}$ $C_{\text{eq}, \text{s}}=\frac{12 \text{ μF}}{7}=1.71 \text{ μF}$

This is less than 3 μF, the smallest capacitor, which is the expected check.

(b) For capacitors in parallel: $C_{\text{eq}, \text{p}}=C_1+C_2+C_3$

Substituting the values: $C_{\text{eq}, \text{p}}=3 \text{ μF}+6 \text{ μF}+12 \text{ μF}=21 \text{ μF}$

Practice Problem 2: RC Circuit Time Constant

Solution:

The time constant is: $\tau = R \times C$

Substituting the values: $\tau = 220 \text{ Ω} \times 470 \times 10^{-6} \text{ F}$ $\tau = 103.4 \times 10^{-3} \text{ s}$ $\tau = 0.1034 \text{ s}$

By definition, after one time constant (about 0.1034 seconds), the capacitor reaches approximately 63% of its maximum charge.

Common Misconceptions

• Misconception: Capacitors charge instantly.
  Actually: Capacitors do not charge instantly. The voltage and current change over time, set by the time constant.
• Misconception: Series and parallel formulas for capacitors are the same as for resistors.
  Actually: Series and parallel formulas for capacitors are not the same as for resistors. Capacitors in series use the inverse sum, and parallel capacitors add directly.
• Misconception: An uncharged capacitor at t = 0 acts like an open switch.
  Actually: An uncharged capacitor at t = 0 acts like a wire, not like an open switch. The "open branch" behavior happens only after it is fully charged.
• Misconception: A fully charged capacitor has zero voltage.
  Actually: A fully charged capacitor has zero current in its branch but maximum voltage, not zero voltage.
• Misconception: The time constant is the time to fully charge.
  Actually: The time constant is not the time to fully charge. After one $\tau$ you only reach about 63% (charging) or have about 37% left (discharging). Full charge or discharge takes roughly five time constants.
• Misconception: Series capacitors share equal voltage.
  Actually: Series capacitors share equal charge, but they do not generally share equal voltage unless the capacitances are equal.

Related AP Physics 2 Guides

• 11.2 Simple Circuits
• 11.3 Resistance, Resistivity, and Ohm's Law
• 11.5 Compound Direct Current (DC) Circuits
• 11.1 Electric Current
• 11.4 Electric Power
• 11.6 Kirchhoff's Loop Rule