Compound Direct Current | AP Physics 2 Unit 3 Study Guide

Getting Started

Most electronic devices are far more complex than a single lightbulb connected to a battery. They contain intricate networks of components, and understanding how they function requires analyzing these compound circuits. This chapter explores how to predict the behavior of direct current (DC) circuits that combine multiple resistors, and how to account for the non-ideal nature of real-world power sources and measurement tools.

What You Should Be Able to Do

After completing this section, you will be able to:

Analyze a circuit diagram to identify resistors connected in series and in parallel.
Calculate the single equivalent resistance for a complex network of resistors.
Model the effect of a battery's internal resistance on the potential difference it supplies to a circuit.
Describe how ammeters and voltmeters are correctly connected to a circuit to measure current and potential difference.
Use Ohm's Law and the rules for equivalent resistance to solve for unknown currents and potential differences in a compound circuit.

Key Concepts & Mechanisms

We can understand compound circuits by comparing the fundamental ways components can be connected and by contrasting idealized models with their real-world counterparts.

Model Comparison: Resistor Connections

Feature	Series Connection	Parallel Connection	Why It Matters
Path for Charge	A single, undivided path. Charge that flows through the first resistor must flow through all subsequent resistors in the series.	Multiple, distinct paths (branches). Charge flow splits, with some flowing through each branch before recombining.	The path determines how current and potential difference are distributed. A single path constrains the current to be the same everywhere in that path.
Current (I)	The current is the same through every resistor in the series. $I_{t o t a l} = I_{1} = I_{2} = \dots$	The total current from the source splits among the parallel branches. The sum of the branch currents equals the total current. $I_{t o t a l} = I_{1} + I_{2} + \dots$	In series, all components experience the same rate of charge flow. In parallel, components can draw different amounts of current depending on their resistance.
Potential Difference (V)	The total potential difference (voltage) supplied by the source is divided among the resistors. The sum of the individual voltage drops equals the total voltage. $V_{t o t a l} = V_{1} + V_{2} + \dots$	The potential difference (voltage) is the same across every branch. $V_{t o t a l} = V_{1} = V_{2} = \dots$	In series, higher-resistance components have a larger voltage drop. In parallel, all components are connected across the same two points, so they must have the same potential difference.
Equivalent Resistance ( $R_{e q}$ )	The equivalent resistance is the sum of the individual resistances. It is always greater than the largest individual resistance. $R_{e q, s} = \sum_{i} R_{i}$	The inverse of the equivalent resistance is the sum of the inverses of the individual resistances. It is always less than the smallest individual resistance. $\frac{1}{R _{e q, p}} = \sum_{i} \frac{1}{R _{i}}$	Adding a resistor in series increases the total opposition to current flow. Adding a resistor in parallel decreases the total opposition by providing an additional path for current.

Model Comparison: Power Sources & Meters

Feature	Ideal Model	Real-World Model	Why It Matters
Battery	An ideal battery is a source of constant potential difference, called electromotive force (EMF or $E$ ). It has no internal resistance.	A real battery has internal resistance (r), which can be modeled as a small resistor in series with the ideal EMF. This causes the terminal voltage (the actual potential difference supplied to the external circuit) to be less than the EMF when current is flowing. $V_{t er mina l} = E - I r$	The internal resistance causes the battery's output voltage to "sag" under a heavy load (i.e., when a large current is drawn). This explains why car headlights dim when you start the engine.
Ammeter	An ideal ammeter has zero resistance so it can be placed in series without affecting the current it is meant to measure.	A real ammeter has a very small, but non-zero, internal resistance.	Because it is connected in series, its small resistance slightly increases the circuit's total resistance, causing the measured current to be slightly lower than the true current.
Voltmeter	An ideal voltmeter has infinite resistance so that when connected in parallel, no current flows through it, leaving the circuit undisturbed.	A real voltmeter has a very large, but finite, internal resistance.	Because it is connected in parallel, it provides an alternate path for current. A tiny amount of current flows through the voltmeter, slightly altering the potential difference it is meant to measure.

Key Models & Diagrams

Analyzing a compound circuit is a systematic process of simplification and reconstruction. The goal is to reduce the entire resistor network to a single equivalent resistance, solve for the total current, and then work backward to find the values for each component.

Flowchart for Compound Circuit Analysis

Identify Series/Parallel Groups
- Examine the circuit diagram.
- Find any set of resistors that are connected purely in series (one after another on a single path) or purely in parallel (connected across the same two points).
$\to$
Calculate Equivalent Resistance ( $R_{e q}$ ) of a Group
- For a series group: $R_{e q, s} = R_{1} + R_{2} + \dots$
- For a parallel group: $R_{e q, p} = (\frac{1}{R _{1}} + \frac{1}{R _{2}} + \dots)^{- 1}$
$\to$
Redraw the Simplified Circuit
- Replace the group you just analyzed with a single resistor labeled with the calculated $R_{e q}$ .
- Repeat steps 1-3 until only one equivalent resistor remains for the entire circuit.
$\to$
Solve for Total Current
- Apply Ohm's Law to the fully simplified circuit.
- $I_{t o t a l} = \frac{V _{so u rce}}{R _{e q, t o t a l}}$
$\to$
Work Backward
- Use the total current and potential difference to find the values in the last simplified step.
- Apply series rules (same I, V adds up) and parallel rules (same V, I adds up) to find the current and voltage for each individual resistor in the original diagram.

Key Components & Evidence

Potential Difference (V or ΔV): The work done per unit charge between two points in a circuit; also called voltage. It is the "push" that drives current. Measured in volts (V).
Current (I): The rate of flow of electric charge. Measured in amperes (A), or "amps."
Resistance (R): A measure of a component's opposition to the flow of current. Measured in ohms (Ω).
Series Connection: A circuit configuration where components are connected end-to-end, providing only one path for current.
Parallel Connection: A circuit configuration where components are connected across the same two points, providing multiple paths for current.
Equivalent Resistance ( $R_{e q}$ ): The resistance of a single, hypothetical resistor that would have the same effect on the total current as a more complex network of resistors.
Internal Resistance (r): The inherent resistance within a real power source (like a battery) that causes a drop in its output voltage when current flows.
Terminal Voltage ( $V_{t er mina l}$ ): The actual potential difference measured across the terminals of a real battery when it is part of a complete circuit.
Ammeter: A device used to measure current. It must be connected in series with the component through which the current is being measured.
Voltmeter: A device used to measure potential difference. It must be connected in parallel with the component across which the voltage is being measured.

Skill Snapshots

Causation:
1. Adding a resistor in series increases the total equivalent resistance, which causes the total current from the battery to decrease (assuming constant voltage).
2. Adding a resistor in parallel decreases the total equivalent resistance by providing a new path, which causes the total current from the battery to increase.
3. Drawing more current from a real battery causes a larger voltage drop across its internal resistance ( $I r$ ), which results in a lower terminal voltage supplied to the external circuit.
Comparison:
1. An ammeter is a low-resistance device connected in series to measure current, whereas a voltmeter is a high-resistance device connected in parallel to measure potential difference.
2. In a series circuit, current is constant through all resistors, but voltage is divided among them. In contrast, in a parallel circuit, voltage is constant across all branches, but current is divided among them.
3. The equivalent resistance of a series circuit is always greater than the largest individual resistor, while the equivalent resistance of a parallel circuit is always less than the smallest individual resistor.
Change Over Time (Circuit Modification):
- Baseline: A battery is connected to a single resistor, $R_{1}$ . A specific current, $I_{1}$ , flows.
- Change 1: A second resistor, $R_{2}$ , is added in series with $R_{1}$ . The total resistance increases to $R_{1} + R_{2}$ , and the total current drawn from the battery decreases.
- Change 2: Instead, $R_{2}$ is added in parallel with $R_{1}$ . The total resistance decreases, and the total current drawn from the battery increases.
- Continuity: Throughout these modifications, the electromotive force (EMF) of the ideal battery remains constant.

Common Misconceptions & Clarifications

Misconception: Adding more resistors to a circuit always increases the total resistance.
- Clarification: This is only true for resistors added in series. Adding a resistor in parallel provides an additional pathway for current, which decreases the overall equivalent resistance of the circuit.
Misconception: The formulas for equivalent resistance are interchangeable.
- Clarification: The formulas are distinct and depend on the connection type. For series, you sum the resistances: $R_{s} = R_{1} + R_{2} + \dots$ . For parallel, you sum the reciprocals of the resistances: $1/ R_{p} = 1/ R_{1} + 1/ R_{2} + \dots$ . A common mistake is to forget the final reciprocal step for parallel circuits (i.e., calculating $1/ R_{p}$ but not solving for $R_{p}$ ).
Misconception: A battery always supplies the voltage printed on its label.
- Clarification: The labeled voltage is the electromotive force (EMF), which is the battery's potential difference under no-load conditions (zero current). As soon as you connect it to a circuit and current flows, its internal resistance causes a voltage drop, making the actual terminal voltage lower than the EMF.
Misconception: Ammeters and voltmeters can be connected anywhere.
- Clarification: Their connection is critical and non-negotiable. Ammeters must be in-line (series) with the part of the circuit you are measuring to "count" the charge passing through. Voltmeters must be connected across two points (parallel) to measure the potential difference between them. Incorrect placement will lead to incorrect readings and can even damage the meter or the circuit.

One-Paragraph Summary

Compound DC circuits are networks of resistors arranged in combinations of series and parallel connections. To analyze them, we systematically simplify the network into a single equivalent resistance using specific rules: summing resistances for series and summing reciprocals for parallel. This allows us to calculate the total current using Ohm's Law. We then work backward to determine the individual current and voltage for each component. This idealized model is refined by considering that real batteries possess internal resistance, which causes their terminal voltage to drop as current increases. Proper circuit analysis also requires understanding that ammeters must be placed in series to measure current, while voltmeters must be placed in parallel to measure potential difference.

Compound Direct Current (DC) Circuits - AP Physics 2: Algebra-Based Study Guide