Compound Direct Current | AP Physics C: E&M Unit 4 Study Guide

Getting Started

We now move from simple circuits with a single resistor to more complex networks containing multiple resistive elements. These compound circuits, powered by direct current (DC) sources, form the basis of most electronic devices. The core question is how to predict the steady-state flow of charge (current) and the electric potential difference across any part of a circuit, especially when considering non-ideal components like real-world batteries.

What You Should Be Able to Do

Upon completing this section, you will be able to:

Systematically reduce a complex network of series and parallel resistors to a single equivalent resistance.
Model a non-ideal battery using an ideal electromotive force in series with an internal resistance, and calculate the battery's true terminal voltage.
Predict the readings of ideal ammeters and voltmeters placed within a compound circuit and explain how their ideal properties prevent them from altering the circuit's behavior.
Apply the principles of conservation of energy (the Loop Rule) and conservation of charge (the Junction Rule) to analyze circuits that cannot be simplified by series/parallel rules alone.

Key Concepts & Mechanisms

Our analysis of DC circuits is framed by a Dynamics/Fields as Cause perspective. A source of electromotive force establishes an electric field within the circuit's conductors, driving a steady flow of charge against the opposition presented by resistors. The structure of the circuit dictates the resulting currents and potential differences.

System & Preconditions

The system is a direct current (DC) circuit. We operate under the lumped-element model, where components (resistors, batteries) are treated as discrete points connected by ideal conductors. Key idealizations and preconditions include:

Ideal Wires: Connecting wires have zero resistance, meaning there is no change in electric potential along their length.
Quasi-Static Conditions: The circuit is in a steady state. This means the currents are constant in time, and the electric fields that drive them are stable.
Non-Ideal Source: We will model a real battery as an ideal source of electromotive force in series with a small internal resistance.

Key Steps / Relations

The Driver: Electromotive Force ( $E$ ): The primary cause of current is the electromotive force (EMF), symbolized by $E$ and measured in volts (V). It is not a force in the Newtonian sense, but rather the work done per unit charge by a non-electrostatic process (e.g., a chemical reaction in a battery) to move charge from a low potential to a high potential. An ideal battery maintains a constant potential difference equal to its EMF across its terminals, regardless of the current.
The Opposition: Resistance ( $R$ ): As charge carriers move through a resistor, they collide with the lattice of the material, dissipating energy. This opposition to flow is resistance, $R$ , measured in ohms ( $Ω$ ). The electric field does work on the charges, resulting in a potential drop across the resistor given by Ohm's Law, $Δ V = I R$ . In the direction of current $I$ , the potential decreases by $I R$ .
System Simplification: Equivalent Resistance ( $R_{e q}$ ): To analyze the circuit as a whole, we simplify networks of resistors into a single equivalent resistance, which would draw the same total current from the source.
- Resistors in Series: Components are connected end-to-end, providing only one path for the current. The current $I$ is the same through each resistor. The total potential drop is the sum of individual drops. This is a direct consequence of the conservation of energy.
  $R_{e q, s} = R_{1} + R_{2} + \dots = \sum_{i} R_{i}$
- Resistors in Parallel: Components are connected across the same two points, providing multiple paths for the current. The potential difference $Δ V$ is the same across each resistor. The total current is the sum of the currents in each branch, a consequence of the conservation of charge.
  $\frac{1}{R _{e q, p}} = \frac{1}{R _{1}} + \frac{1}{R _{2}} + \dots = \sum_{i} \frac{1}{R _{i}}$
Real-World Sources: Internal Resistance ( $r$ ): A real battery has internal resistance, $r$ , due to the resistance of its chemical components. We model this as an ideal EMF source $E$ connected in series with this resistor $r$ . When the battery supplies a current $I$ to an external circuit, a potential drop $I r$ occurs inside the battery itself.

Outputs & Effects

Current ( $I$ ): The net result of the EMF driving charge through the total equivalent resistance of the circuit is a steady current, $I = E / R_{t o t a l}$ .
Terminal Voltage ( $Δ V_{t er mina l}$ ): For a non-ideal battery, the potential difference measured across its external terminals is less than its EMF. The internal potential drop reduces the available voltage for the external circuit.
$Δ V_{t er mina l} = E - I r$
This equation shows that the terminal voltage depends on the current being drawn. A battery under a heavy load (large $I$ ) will have a lower terminal voltage.

Regulation & Limits

The models we use are valid for steady-state DC circuits. The concept of ideal meters is a crucial approximation. A real ammeter has a small, non-zero resistance, and a real voltmeter has a large, but finite, resistance. In sensitive circuits, these non-ideal properties can slightly alter the very quantities they are intended to measure.

Key Models & Diagrams

The process of analyzing a compound circuit can be visualized as a systematic reduction and subsequent expansion.

Flowchart for Circuit Analysis

Step	Action	Governing Principle / Equation
1. Identify	Locate all simple series or parallel combinations of resistors in the circuit diagram.	Visual inspection of the circuit topology.
2. Simplify	Choose one combination and replace it with its single equivalent resistor.	$R_{e q, s} = \sum R_{i}$ or $\frac{1}{R _{e q, p}} = \sum \frac{1}{R _{i}}$
3. Redraw	Redraw the entire circuit diagram with the new equivalent resistance in place.	Maintain connectivity of the remaining components.
4. Iterate	Repeat steps 1-3 until the entire circuit is reduced to a single source and a single total equivalent resistance, $R_{t o t a l}$ .	Systematic reduction.
5. Solve (Total)	Calculate the total current leaving the source using Ohm's Law.	$I_{t o t a l} = \frac{E}{R _{t o t a l}}$ (where $R_{t o t a l}$ includes any internal resistance).
6. Expand	Work backward through your redrawn diagrams, using the known total current or branch voltages to find the current and voltage for each individual component.	Ohm's Law ( $V = I R$ ), Conservation of Charge (currents at a junction), Conservation of Energy (voltages in a loop).

Key Components & Evidence

Electromotive Force ( $E$ ): The work per unit charge provided by a source, establishing the potential difference that drives current. Unit: volt (V).
Potential Difference ( $Δ V$ ): The change in electric potential energy per unit charge between two points in a circuit. Unit: volt (V).
Current ( $I$ ): The rate of flow of electric charge, $I = d Q / d t$ . Unit: ampere (A).
Resistance ( $R$ ): A measure of a component's opposition to the flow of current. Defined by $R = Δ V / I$ . Unit: ohm ( $Ω$ ).
Equivalent Resistance ( $R_{e q}$ ): The resistance of a single resistor that could replace a more complex network while drawing the same total current. Unit: ohm ( $Ω$ ).
Internal Resistance ( $r$ ): The intrinsic resistance within a source of EMF, like a battery, which causes a drop in terminal voltage when current flows. Unit: ohm ( $Ω$ ).
Terminal Voltage ( $Δ V_{t er mina l}$ ): The actual potential difference measured across the output terminals of a source when it is part of a complete circuit. Unit: volt (V).
Ideal Ammeter: A device to measure current. It is placed in series with a component and has zero resistance ( $R_{A} = 0$ ) so it does not create an additional potential drop.
Ideal Voltmeter: A device to measure potential difference. It is placed in parallel with a component and has infinite resistance ( $R_{V} = \infty$ ) so that no current flows through it.

Skill Snapshots

Causation

Driver → Change: The battery's EMF ( $E$ ) is the driver that causes a steady current ( $I$ ) to flow through the circuit.
Driver → Change: The resistance ( $R$ ) of a component causes a potential drop ( $Δ V = - I R$ ) across it when a current flows.
Driver → Change: The flow of current ( $I$ ) through a battery's internal resistance ( $r$ ) causes the terminal voltage to be less than the EMF ( $Δ V_{t er mina l} < E$ ).

Comparison

Series vs. Parallel Resistors: In a series connection, the current is identical through all components, while potential differences add. In a parallel connection, the potential difference is identical across all components, while currents add.
Ideal vs. Real Battery: An ideal battery provides a constant terminal voltage equal to its EMF ( $Δ V = E$ ). A real battery's terminal voltage decreases as the current it supplies increases ( $Δ V = E - I r$ ).
Ammeter vs. Voltmeter Placement: An ammeter is a low-resistance device placed in series to measure the flow through a path. A voltmeter is a high-resistance device placed in parallel to measure the potential difference across two points.

Change and Continuity

Baseline: A battery with EMF $E$ is connected to a single resistor $R_{1}$ , resulting in current $I_{1} = E / R_{1}$ .
Change 1: A second resistor, $R_{2}$ , is added in series. The total resistance increases to $R_{1} + R_{2}$ , causing the total current drawn from the battery to decrease.
Change 2: Instead, $R_{2}$ is added in parallel to $R_{1}$ . The total resistance decreases, causing the total current drawn from the battery to increase.
Continuity: Throughout these changes, the EMF ( $E$ ) of the ideal battery remains constant. It is a property of the source, independent of the circuit connected to it.

Common Misconceptions & Clarifications

Misconception: Batteries are sources of constant current.
Clarification: Ideal batteries are sources of constant potential difference (EMF). The current they supply is determined by the total resistance of the circuit connected to them, according to Ohm's Law ( $I = E / R_{e q}$ ).
Misconception: Adding more resistors to a circuit always increases the total resistance.
Clarification: This is only true if the resistors are added in series. Adding a resistor in parallel provides an additional path for the current, which decreases the total equivalent resistance. Think of opening another lane on a highway; the overall flow (current) increases for a given motivation (voltage).
Misconception: The voltage is the same everywhere in a parallel branch.
Clarification: The potential difference across each parallel branch is the same. However, if a branch contains multiple resistors in series, the potential will drop across each of those resistors.
Misconception: An ammeter should be connected in parallel, and a voltmeter in series.
Clarification: Connecting a zero-resistance ammeter in parallel would create a short circuit, causing a very large current to bypass the intended component. Connecting an infinite-resistance voltmeter in series would effectively create an open circuit, stopping the flow of current almost entirely.

One-Paragraph Summary

Compound DC circuits are analyzed by systematically simplifying networks of resistors and modeling the non-ideal behavior of power sources. The fundamental rules for calculating the equivalent resistance of series and parallel combinations are direct consequences of the conservation of energy and charge, respectively. By reducing a complex circuit to a single equivalent resistance, we can determine the total current supplied by the source. A real battery's performance is modeled by including an internal resistance, which explains why its terminal voltage drops as the current it delivers increases. Finally, the function of ideal ammeters and voltmeters is understood through their limiting resistances—zero for an ammeter to not impede current, and infinite for a voltmeter to not divert it—allowing for accurate measurement without disturbing the circuit's steady state.

Compound Direct Current Circuits - AP Physics C: Electricity and Magnetism Study Guide