Electric Potential Energy | AP Physics 2 Unit 2 Study Guide

Getting Started

This chapter explores the energy stored within a system of stationary electric charges. We will investigate the microscopic scale of atoms and subatomic particles, asking a core question: How much work does it take to assemble a specific arrangement of charges, and what does the resulting stored energy tell us about the stability of that arrangement? This concept, known as electric potential energy, is fundamental to understanding everything from chemical bonds to the operation of electronic devices.

What You Should Be Able to Do

After completing this section, you will be able to:

Define electric potential energy as the work done by an external force to configure a system of charges.
Calculate the electric potential energy for a pair of point charges.
Calculate the total electric potential energy for a system of three or more point charges by summing the energy of each unique pair.
Interpret the sign (positive or negative) of the electric potential energy in terms of the interaction (repulsive or attractive) and the work required to assemble the system.

Key Concepts & Mechanisms

This section analyzes electric potential energy through the lens of Interactions & Conservation, focusing on how energy is stored in a system due to the work done against electrostatic forces.

System & Preconditions

System: Our system consists of two or more point charges. A point charge is an idealized charged object of negligible size.
Preconditions & Idealizations: We assume the charges are initially infinitely far apart from each other. At this infinite separation, the force between them is zero, and we define the electric potential energy of the system to be zero. We also assume the charges are brought together from this infinite separation so slowly that they have no kinetic energy at any point (quasi-statically).

Key Steps / Relations

Work and Potential Energy: The electric potential energy ( $U_{E}$ ) of a system of charges is defined as the total work done by an external force to assemble the charges from an infinite separation to their final positions. The unit for energy is the Joule (J). This is an application of the work-energy principle: $W_{e x t er na l} = Δ U_{E}$ . Since we define the initial potential energy at infinite separation as zero ( $U_{E, ini t ia l} = 0$ ), the potential energy of the final configuration is simply equal to the work done: $U_{E, f ina l} = W_{e x t er na l}$ .
Energy of Two Point Charges: To bring a charge $q_{2}$ from infinity to a distance $r$ from a fixed charge $q_{1}$ , an external agent must do work against the electric force between them. This work, and thus the stored potential energy, is given by the equation:
$U_{E} = \frac{1}{4 π ϵ _{0}} \frac{q _{1} q _{2}}{r} = k \frac{q _{1} q _{2}}{r}$
- $q_{1}$ and $q_{2}$ are the magnitudes of the point charges in Coulombs (C).
- $r$ is the final separation distance between the centers of the charges in meters (m).
- $k$ is Coulomb's constant, approximately $8.99 \times 1 0^{9} N \cdot m^{2} / C^{2}$ .
Energy of a Multi-Charge System: Electric potential energy is a scalar quantity, not a vector. To find the total potential energy of a system with three or more charges, we calculate the potential energy for every unique pair of charges and then add the results together. For a system of three charges ( $q_{1}, q_{2}, q_{3}$ ), the total potential energy is:
$U_{E, t o t a l} = U_{12} + U_{13} + U_{23}$
$U_{E, t o t a l} = k \frac{q _{1} q _{2}}{r _{12}} + k \frac{q _{1} q _{3}}{r _{13}} + k \frac{q _{2} q _{3}}{r _{23}}$
Note that we only count the interaction between each pair once (e.g., $U_{12}$ is the same as $U_{21}$ ).

Outputs & Effects

Positive Potential Energy ( $U_{E} > 0$ ): This occurs when two interacting charges have the same sign (both positive or both negative). The electric force between them is repulsive. An external agent must do positive work (push them together) to assemble the system, storing energy within it. A system with positive potential energy will spontaneously fly apart if its components are released, converting the stored potential energy into kinetic energy.
Negative Potential Energy ( $U_{E} < 0$ ): This occurs when two interacting charges have opposite signs. The electric force between them is attractive. The system does work on the external agent as the charges are brought together (i.e., the external agent must do negative work, or hold them back). A system with negative potential energy is bound; energy must be added to the system to separate the charges to infinity.

Regulation & Limits

Domain of Validity: This model is precise for static (non-moving) point charges in a vacuum.
Zero-Point Definition: The value of potential energy is relative. The choice to define $U_{E} = 0$ at $r = \infty$ is a convention, but it is the standard for this formula. Changing this definition would add a constant to all energy calculations.
Scalar Summation: Remember that energy is a scalar. When calculating the total potential energy of a system, you perform simple algebraic addition, including the positive or negative signs of the charges. There are no components or directions to consider, unlike with electric forces or fields.

Key Models & Diagrams

The process for calculating the total electric potential energy of a system can be visualized as a simple workflow.

Step	Description	Mathematical Representation
1. Identify System	List all point charges ( $q_{1}, q_{2}, ..., q_{n}$ ) and the distances between each unique pair ( $r_{12}, r_{13}, r_{23}, ...$ ).	System: ${q_{1}, q_{2}, q_{3}}$ Distances: $r_{12}, r_{13}, r_{23}$
2. Calculate Pairwise Energy	Use the potential energy formula to calculate the energy for every unique pair of charges. Keep track of the sign.	$U_{12} = k \frac{q _{1} q _{2}}{r _{12}}$ $U_{13} = k \frac{q _{1} q _{3}}{r _{13}}$ $U_{23} = k \frac{q _{2} q _{3}}{r _{23}}$
3. Sum the Energies	Add the scalar energy values from all pairs to find the total potential energy of the system.	$U_{E, t o t a l} = U_{12} + U_{13} + U_{23}$
4. Interpret the Result	Analyze the sign of $U_{E, t o t a l}$ . A net positive value suggests the system is unstable and required energy to assemble. A net negative value suggests the system is bound and released energy upon assembly.	If $U_{E, t o t a l} < 0$ , the system is bound. If $U_{E, t o t a l} > 0$ , the system is unbound.

Key Components & Evidence

Electric Potential Energy ( $U_{E}$ ): A scalar quantity representing the energy stored in a system of charges due to their configuration. Measured in Joules (J).
Point Charge ( $q$ ): The fundamental component of the system. Its sign determines the nature of the interaction. Measured in Coulombs (C).
Separation Distance ( $r$ ): The distance between a pair of charges. Potential energy is inversely proportional to this distance. Measured in meters (m).
Coulomb's Constant ( $k$ ): A fundamental constant of proportionality that relates electric charge to electric force and energy.
Work ( $W$ ): The mechanical process of energy transfer that defines potential energy. $W_{e x t} = Δ U_{E}$ . Measured in Joules (J).
Superposition Principle for Energy: The total energy of a system is the scalar sum of the energies of its constituent pairs. This is an observable principle that simplifies complex systems.
Zero-Energy at Infinity: The convention that two charges have zero potential energy when infinitely far apart. This is the baseline from which all potential energy is measured.
Bound System: A system with negative total potential energy (e.g., a proton and an electron in a hydrogen atom). Energy must be supplied to separate the components.
Unbound System: A system with positive total potential energy (e.g., two protons held near each other). The system will release energy by flying apart if not constrained.

Skill Snapshots

Causation

Interaction: Bringing two like charges closer together (decreasing $r$ ) causes the system's electric potential energy to become more positive, as an external force must do work against the repulsion.
Interaction: Allowing two opposite charges to move closer together causes the system's electric potential energy to become more negative, as the attractive electric field does positive work.
Interaction: Adding a third positive charge to a system of two positive charges causes the total potential energy of the system to increase, as two new repulsive interactions are added.

Comparison

Electric Potential Energy vs. Electric Force: Electric potential energy is a scalar property of a system (a single number with no direction), whereas electric force is a vector acting on a single charge (requiring magnitude and direction).
Repulsive vs. Attractive System: A system of two protons has a positive potential energy that decreases toward zero as they are separated, while a system of a proton and an electron has a negative potential energy that increases toward zero as they are separated.
Energy of a Pair vs. System: The potential energy of a single pair of charges ( $U_{12}$ ) depends only on those two charges, while the total potential energy of a system ( $U_{t o t a l}$ ) is the scalar sum of the energies of all unique pairs within it.

Change Over Time

Baseline: A system of two protons is held stationary 1 meter apart. The system has a specific, positive electric potential energy.
Change 1: If an external agent moves one proton so the separation distance becomes 0.5 meters, the potential energy of the system increases (becomes more positive).
Change 2: If the protons are released from their 1-meter separation, they will accelerate away from each other. The system's potential energy will decrease, converting into the kinetic energy of the protons.
Continuity: Throughout any process involving only conservative electric forces, the total energy of the system (the sum of electric potential energy and kinetic energy) is conserved.

Common Misconceptions & Clarifications

Misconception: Electric potential energy and electric force are the same thing.
- Clarification: Force is a vector; it has a magnitude and a direction, and it describes the push or pull on a charge. Potential energy is a scalar; it is a property of the entire system's configuration and has no direction. The force is related to how the energy changes with position ( $F = - d U / d r$ ), but they are distinct concepts.
Misconception: You must use vector addition to find the total potential energy of a system.
- Clarification: Because energy is a scalar, you find the total potential energy by simple algebraic addition. You calculate the energy for each pair (a positive or negative number) and add them up.
Misconception: A negative potential energy means the energy is less than zero or has been lost.
- Clarification: A negative sign for potential energy indicates a bound system. It means that the system is more stable than its constituent parts would be if they were infinitely far apart. Energy must be added to the system to break it apart. For example, the Earth-Sun system has negative gravitational potential energy.
Misconception: When calculating the total energy of a three-charge system, you add the energy of charge 1 with 2, charge 2 with 3, and charge 3 with 1, and that's it.
- Clarification: This is correct! A common mistake is to double-count pairs. The interaction between charge 1 and 2 ( $U_{12}$ ) is the same as the interaction between 2 and 1 ( $U_{21}$ ). For $N$ charges, there are $N (N - 1) /2$ unique pairs to sum.

One-Paragraph Summary

Electric potential energy ( $U_{E}$ ) quantifies the energy stored in a system of point charges based on their relative positions. It is defined as the work required by an external force to assemble the charges from an infinite separation, where $U_{E}$ is conventionally set to zero. For any pair of charges, this energy is calculated by $U_{E} = k (q_{1} q_{2}) / r$ . Because energy is a scalar, the total potential energy of a multi-charge system is the simple algebraic sum of the potential energies of all unique pairs. A positive total energy signifies a repulsive, unbound system that required work to assemble, while a negative total energy signifies an attractive, bound system that released energy upon assembly. This concept is a cornerstone of energy conservation in electrostatic systems.

Electric Potential Energy - AP Physics 2: Algebra-Based Study Guide