---
title: "AP Physics C E&M 11.8: RC Circuits"
description: "Review AP Physics C: E&M 11.8, including resistor-capacitor circuits, time constant formula, equivalent capacitance, capacitors in series and parallel, Kirchhoff's loop rule, charging and discharging behavior, and RC differential equations."
canonical: "https://fiveable.me/ap-physics-c-e-m/unit-11/8-resistor-capacitor-rc-circuits/study-guide/qy6QreLu93jx043L"
type: "study-guide"
subject: "AP Physics C: E&M"
unit: "Unit 11 – Electric Circuits"
lastUpdated: "2026-06-09"
---

# AP Physics C E&M 11.8: RC Circuits

## Summary

Review AP Physics C: E&M 11.8, including resistor-capacitor circuits, time constant formula, equivalent capacitance, capacitors in series and parallel, Kirchhoff's loop rule, charging and discharging behavior, and RC differential equations.

## Guide

RC circuits combine resistors and capacitors so that charge, current, and [voltage](/ap-physics-c-e-m/key-terms/voltage "fv-autolink") change over time instead of staying fixed. The [time constant](/ap-physics-c-e-m/key-terms/time-constant "fv-autolink") $\tau = R_{\mathrm{eq}}C_{\mathrm{eq}}$ controls how fast a capacitor charges or discharges, and capacitor combinations follow rules that are the reverse of resistor combinations.

## Why This Matters for the AP Physics C: E&M Exam

This topic pulls together [Kirchhoff's loop rule](/ap-physics-c-e-m/unit-11/6-kirchhoffs-loop-rule/study-guide/Nxj12AcsMRzP3ejn "fv-autolink"), capacitors, and resistors into one model where quantities depend on time. You will be asked to write and interpret the differential equation from the loop rule, predict short-term and long-term behavior, and use the exponential charging and discharging functions. Because [Unit 11](/ap-physics-c-e-m/unit-11 "fv-autolink") is one of the most heavily weighted units on the exam, you should be ready to handle RC circuits both as quick conceptual checks and as multi-step calculations.

The free-response section includes an Experimental Design and Analysis question, and RC circuits fit that style well. You may need to design a procedure to measure a time constant, linearize exponential data to find a slope, or justify claims using your data and the underlying physics.

## Key Takeaways

- Capacitors in series add as reciprocals, so the [equivalent capacitance](/ap-physics-c-e-m/key-terms/equivalent-capacitance "fv-autolink") is less than the smallest one; capacitors in parallel simply add.
- Series capacitors carry the same magnitude of charge on each plate because of [conservation of charge](/ap-physics-c-e-m/key-terms/conservation-of-charge "fv-autolink").
- Apply Kirchhoff's loop rule to get the governing equation $\varepsilon = \frac{dq}{dt}R + \frac{q}{C}$.
- The time constant is $\tau = R_{\mathrm{eq}} C_{\mathrm{eq}}$: about 63% charged after one $\tau$, about 37% of charge left after one $\tau$ when discharging.
- At $t = 0$ an uncharged capacitor acts like a wire (maximum current); after $t \gg \tau$ a fully charged capacitor acts like an open branch (zero current).
- Charging and discharging follow exponential functions, so charge, voltage, current, and stored energy all approach [steady state](/ap-physics-c-e-m/key-terms/steady-state "fv-autolink") asymptotically.

## Equivalent Capacitance

Capacitors can be arranged in different configurations within a circuit, and these arrangements affect how they store charge and energy.

**[Series Connection](/ap-physics-c-e-m/key-terms/series-connection "fv-autolink").** When capacitors are connected end-to-end, they form a series combination.

- The equivalent capacitance is calculated using:

$$\frac{1}{C_{\mathrm{eq}, \mathrm{s}}}=\sum_{i} \frac{1}{C_{i}}$$
- For two capacitors in series, this simplifies to: $$C_{eq} = \frac{C_1 C_2}{C_1 + C_2}$$
- The equivalent capacitance is always smaller than the smallest individual [capacitance](/ap-physics-c-e-m/key-terms/capacitance "fv-autolink") in the series.
- Each capacitor in series has the same magnitude of charge on its plates due to conservation of charge.

**[Parallel Connection](/ap-physics-c-e-m/key-terms/parallel-connection "fv-autolink").** When capacitors are connected across the same two points, they form a [parallel combination](/ap-physics-c-e-m/key-terms/parallel-combination "fv-autolink").

- The equivalent capacitance is simply the sum of individual capacitances:

$$C_{\mathrm{eq}, p}=\sum_{i} C_{i}$$
- For two capacitors in parallel: $$C_{eq} = C_1 + C_2$$
- Parallel capacitors all experience the same voltage across their terminals.
- The total charge is distributed among the capacitors proportionally to their capacitance values.

Notice the pattern is the opposite of resistors: series resistors add directly, but series capacitors add as reciprocals.

## RC Circuit Behavior

### Fundamental Differential Equation

When a resistor and capacitor are combined in a circuit with a voltage source, their behavior is governed by a differential equation derived from Kirchhoff's loop rule.

$$\mathcal{E}=\frac{d q}{d t} R+\frac{q}{C}$$

This equation describes how the charge on the capacitor changes over time, where:
- $\mathcal{E}$ is the [electromotive force](/ap-physics-c-e-m/key-terms/electromotive-force "fv-autolink") (voltage source)
- $R$ is the resistance
- $C$ is the capacitance
- $q$ is the charge on the capacitor
- $\frac{d q}{d t}$ represents the current flowing through the resistor

Solving this equation gives the time-dependent behavior of the circuit, showing how charge, current, and voltage evolve after a [switch](/ap-physics-c-e-m/key-terms/switch "fv-autolink") is opened or closed.

### Time Constant

The time constant characterizes how quickly an [RC circuit](/ap-physics-c-e-m/key-terms/rc-circuit "fv-autolink") responds to changes.

$$\tau=R_{\mathrm{eq}} C_{\mathrm{eq}}$$

The time constant has several important interpretations:

- It represents the time required for a charging capacitor to reach approximately 63% of its final value.
- During discharge, it is the time needed for the charge to decrease to about 37% of its initial value.
- After about 5 time constants, the circuit is treated as having essentially reached steady state.
- The time constant has units of seconds and measures how fast or slow the circuit responds.

For complex circuits with multiple resistors and capacitors, find the [equivalent resistance](/ap-physics-c-e-m/key-terms/equivalent-resistance "fv-autolink") and equivalent capacitance first, then use them in $\tau$.

### Capacitor Charging

When a capacitor is connected to a voltage source through a resistor, it begins to charge. This process follows an exponential pattern set by the time constant.

- **Initial behavior**: An uncharged capacitor initially acts like a wire, allowing maximum current to flow.
- **Charge accumulation**: As charge builds up on the plates, the voltage across the capacitor increases.
- **Current reduction**: The increasing capacitor voltage opposes the source voltage, gradually reducing the current.
- **Mathematical description**: The charge on the capacitor during charging follows:
  $$q(t) = Q(1 - e^{-t/\tau})$$
  where $$Q = C\mathcal{E}$$ is the maximum charge.
- **Voltage and current**:
  $$V_C(t) = \mathcal{E}(1 - e^{-t/\tau})$$
  $$I(t) = \frac{\mathcal{E}}{R}e^{-t/\tau}$$
- **Energy storage**: As the capacitor charges, the stored [electric potential energy](/ap-physics-c-e-m/unit-9/1-electric-potential-energy/study-guide/InqS68GlgFXEOpRi "fv-autolink") increases according to $$U = \frac{1}{2}CV_C^2 = \frac{q^2}{2C}$$ and approaches a maximum asymptotically.
- **Steady-state behavior**: After a long time ($$t \gg \tau$$), the capacitor is fully charged for modeling purposes. The [potential difference](/ap-physics-c-e-m/key-terms/potential-difference "fv-autolink") across it reaches its maximum value equal to the battery emf, and the current in that branch becomes zero.

### Capacitor Discharging

When a charged capacitor is connected across a resistor with no voltage source, it discharges through the resistor.

- **Initial behavior**: At the moment of connection, maximum current flows because of the full voltage across the capacitor.
- **Immediate change in stored quantities**: As soon as discharging begins, the charge on the capacitor and the energy stored in its [electric field](/ap-physics-c-e-m/unit-10/1-electrostatics-with-conductors/study-guide/4Vb5LzwBQm2HSChq "fv-autolink") both start to decrease.
- **[Exponential decay](/ap-physics-c-e-m/key-terms/exponential-decay "fv-autolink")**: The charge, voltage, and current all decrease exponentially with time.
- **Mathematical description**: The charge during discharging follows:
  $$q(t) = Q e^{-t/\tau}$$
  where $$Q$$ is the initial charge.
- **Voltage and current**:
  $$V_C(t) = V_0 e^{-t/\tau}$$
  $$I(t) = \frac{V_0}{R}e^{-t/\tau}$$
  where $$V_0$$ is the initial voltage.
- **Energy change**: As the capacitor discharges, the stored electric potential energy decreases with time and is dissipated as thermal energy in the resistor. The stored energy approaches zero asymptotically.
- **Steady-state behavior**: After a long time ($$t \gg \tau$$), the capacitor is effectively discharged, with charge, capacitor voltage, and current all approximately zero.

## How to Use This on the AP Physics C: E&M Exam

### Problem Solving

- Reduce capacitor networks to a single $C_{\mathrm{eq}}$ and resistor networks to a single $R_{\mathrm{eq}}$ before finding $\tau$.
- For series capacitors, remember the charge is shared equally, so start from the common charge to find each voltage.
- Always check the two limiting cases. At $t = 0$, treat an uncharged capacitor as a wire and find the initial current from the resistors. At $t \gg \tau$, treat the capacitor as an open branch with zero current.

### Free Response

- If asked to set up the model, start from Kirchhoff's loop rule and write $\varepsilon = \frac{dq}{dt}R + \frac{q}{C}$, then state initial conditions like $q(0)$.
- When solving for behavior over time, use the charging or discharging exponential form and identify $\tau$, the maximum charge, and the initial values clearly.
- To find a time, set the exponential expression equal to the target value and take a natural log, as shown in the practice problems below.

### Experimental Design and Analysis

- A common approach is to measure capacitor voltage or current versus time, then linearize. Taking the natural log of an exponential turns it into a straight line whose slope is related to $1/\tau$.
- From the slope and a known $R$ or $C$, you can solve for the unknown circuit value, and you can discuss sources of error such as nonideal meters or resistor heating.

## Common Misconceptions

- Capacitors do not follow the same combination rules as resistors. Series capacitors add as reciprocals (smaller result), and parallel capacitors add directly.
- A capacitor does not "block" current immediately. When uncharged, it acts like a wire at $t = 0$ and allows the largest current; it blocks current only after it is fully charged.
- The capacitor does not reach its final charge in one time constant. One $\tau$ gives about 63% charged, not 100%.
- Current does not keep flowing through a fully charged capacitor branch. At steady state that branch carries zero current, so any series resistor in that branch has zero voltage drop.
- The time constant uses equivalent values. Use $R_{\mathrm{eq}}$ and $C_{\mathrm{eq}}$ for the relevant branch, not just one component, when the circuit has multiple elements.

## Practice Problem 1: Equivalent Capacitance

> A circuit contains three capacitors with the following values: $$C_1 = 4.0 \mu F$$, $$C_2 = 6.0 \mu F$$, and $$C_3 = 12.0 \mu F$$. Calculate the equivalent capacitance if:
> a) All three capacitors are connected in series
> b) All three capacitors are connected in parallel

**Solution**

a) For capacitors in series, we use the formula:
$$\frac{1}{C_{\mathrm{eq}, \mathrm{s}}}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}$$

Substituting the values:
$$\frac{1}{C_{\mathrm{eq}, \mathrm{s}}}=\frac{1}{4.0 \mu F}+\frac{1}{6.0 \mu F}+\frac{1}{12.0 \mu F}$$
$$\frac{1}{C_{\mathrm{eq}, \mathrm{s}}}=\frac{3}{12.0 \mu F}+\frac{2}{12.0 \mu F}+\frac{1}{12.0 \mu F}=\frac{6}{12.0 \mu F}=\frac{1}{2.0 \mu F}$$

Therefore, $$C_{\mathrm{eq}, \mathrm{s}} = 2.0 \mu F$$

b) For capacitors in parallel, we use the formula:
$$C_{\mathrm{eq}, p}=C_1+C_2+C_3$$

Substituting the values:
$$C_{\mathrm{eq}, p}=4.0 \mu F+6.0 \mu F+12.0 \mu F=22.0 \mu F$$

## Practice Problem 2: RC Circuit Time Constant

> An RC circuit consists of a 100 Ω resistor and a 470 μF capacitor connected in series with a 12 V battery.
> a) Calculate the time constant of the circuit.
> b) How long will it take for the capacitor to charge to 99% of its maximum value?

**Solution**

a) The time constant of an RC circuit is given by:
$$\tau = RC$$

Substituting the values:
$$\tau = (100 \Omega)(470 \times 10^{-6} F) = 0.047 \text{ seconds}$$

b) To find the time needed to reach 99% of the maximum charge, we use the charging equation:
$$q(t) = Q(1 - e^{-t/\tau})$$

We need to find t when $$q(t) = 0.99Q$$:
$$0.99Q = Q(1 - e^{-t/\tau})$$

$$0.99 = 1 - e^{-t/\tau}$$

$$e^{-t/\tau} = 0.01$$
$$-t/\tau = \ln(0.01)$$
$$t = -\tau \ln(0.01)$$
$$t = 0.047 \text{ s} \times 4.605 = 0.216 \text{ seconds}$$

Therefore, it will take approximately 0.216 seconds for the capacitor to charge to 99% of its maximum value.

## Practice Problem 3: Capacitor Discharge

> A 25 μF capacitor is charged to 50 V and then connected across a 10 kΩ resistor.
> a) What is the initial current through the resistor?
> b) How much time will it take for the voltage across the capacitor to drop to 5 V?

**Solution**

a) The initial current can be calculated using [Ohm's law](/ap-physics-c-e-m/unit-11/3-resistance-resistivity-and-ohms-law/study-guide/TnRPkql9C75GQe0d "fv-autolink"):
$$I_0 = \frac{V_0}{R} = \frac{50 \text{ V}}{10 \text{ k}\Omega} = 5 \times 10^{-3} \text{ A} = 5 \text{ mA}$$

b) For a discharging capacitor, the voltage follows:
$$V(t) = V_0 e^{-t/\tau}$$

First, we need to find the time constant:
$$\tau = RC = (10 \times 10^3 \Omega)(25 \times 10^{-6} \text{ F}) = 0.25 \text{ seconds}$$

Now, we need to find t when $$V(t) = 5 \text{ V}$$:
$$5 \text{ V} = 50 \text{ V} \times e^{-t/0.25}$$
$$\frac{5}{50} = e^{-t/0.25}$$
$$0.1 = e^{-t/0.25}$$
$$\ln(0.1) = -t/0.25$$
$$t = -0.25 \times \ln(0.1) = 0.25 \times 2.303 = 0.576 \text{ seconds}$$

Therefore, it will take approximately 0.576 seconds for the voltage to drop to 5 V.

## Related AP Physics C: E&M Guides

- [11.1 Electric Current](/ap-physics-c-e-m/unit-11/1-electric-current/study-guide/9YRMrkv1PVy23BzH)
- [11.2 Electric Circuits](/ap-physics-c-e-m/unit-11/2-electric-circuits/study-guide/17WyJIXaesWwOEX8)
- [11.3 Resistance, Resistivity, and Ohm's Law](/ap-physics-c-e-m/unit-11/3-resistance-resistivity-and-ohms-law/study-guide/TnRPkql9C75GQe0d)
- [11.5 Compound Direct Current Circuits](/ap-physics-c-e-m/unit-11/5-compound-direct-current-circuits/study-guide/lvbJLaPd4EqBAUf6)
- [11.4 Electric Power](/ap-physics-c-e-m/unit-11/4-electric-power/study-guide/u2cRqQTlthIAJtwp)
- [11.6 Kirchhoff's Loop Rule](/ap-physics-c-e-m/unit-11/6-kirchhoffs-loop-rule/study-guide/Nxj12AcsMRzP3ejn)

## Vocabulary

- **Kirchhoff's loop rule**: A principle stating that the sum of potential differences across all circuit elements in a single closed loop must equal zero, based on conservation of energy.
- **RC circuit**: A circuit containing a resistor and capacitor in combination, where the charge and current change over time as the capacitor charges or discharges.
- **asymptotic approach**: The behavior of a quantity that approaches a final value over time but never quite reaches it, as seen in RC circuit charging and discharging.
- **capacitor in parallel**: Capacitors connected with their plates connected together, where each capacitor experiences the same voltage.
- **capacitor in series**: Capacitors connected end-to-end in a single path, where the same charge accumulates on each capacitor plate.
- **charging capacitor**: A capacitor in a circuit that is accumulating charge, with its charge increasing from zero toward a maximum value over time.
- **conservation of electric charge**: The principle that the total electric charge in an isolated system remains constant over time.
- **differential equation**: A mathematical equation that relates a function to its derivatives, used to describe how quantities change over time.
- **discharging capacitor**: A capacitor in a circuit that is losing charge, with its charge decreasing from a maximum value toward zero over time.
- **electric potential energy stored in the capacitor**: The energy stored in the electric field between the capacitor plates, which changes as the capacitor charges or discharges.
- **equivalent capacitance**: The single capacitance value that can replace a combination of capacitors in a circuit while maintaining the same electrical behavior.
- **potential difference across a capacitor**: The voltage between the plates of a capacitor, which changes over time during charging and discharging and reaches a constant value at steady state.
- **steady state**: A condition reached after a long time interval where circuit quantities no longer change with time.
- **time constant**: A characteristic parameter that measures how quickly a circuit reaches steady state, calculated differently for RC and LR circuits.

## FAQs

### What is the time constant formula for an RC circuit?

The time constant is tau = R_eq C_eq. It measures how quickly the capacitor charges or discharges: after one time constant, charging reaches about 63 percent of the final value and discharging falls to about 37 percent of the initial value.

### How do capacitors combine in series?

Capacitors in series add as reciprocals: 1/C_eq = sum 1/C_i. The equivalent capacitance is smaller than the smallest individual capacitance, and series capacitors carry the same magnitude of charge.

### How do capacitors combine in parallel?

Capacitors in parallel add directly: C_eq = sum C_i. Capacitors in parallel share the same potential difference, while total charge is distributed according to capacitance.

### What differential equation describes a charging RC circuit?

A standard loop-rule form is epsilon = R dq/dt + q/C. It comes from Kirchhoff’s loop rule and connects the source voltage, resistor voltage, and capacitor voltage as charge changes over time.

### What happens to an uncharged capacitor at t = 0?

Immediately after an uncharged capacitor is connected in an RC circuit, it acts like a wire because charge can flow easily onto the plates. Current is initially largest, then decreases as the capacitor charges.

### What happens after a long time in an RC circuit?

After a time much greater than the time constant, a charging capacitor is effectively fully charged and the current in that branch is zero. The circuit can then be treated with steady-state conditions.

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