AP Physics C: E&M Unit 2: Electric Potential Notes

Introduction

Electric potential turns a conservative vector interaction into a scalar energy map: charge configurations have potential energy, potential is energy per charge, and potential differences predict energy transfer.

Begin by declaring the charge system and the zero of potential energy. Pair energy belongs to the configuration, not to either charge alone. Electric potential removes the test-charge factor, so source contributions add as scalars even though the field recovered from the potential is a vector.

A potential difference connects work, field, and motion through $\Delta U=q\Delta V$ and $\Delta V=-\int\vec E\cdot d\vec l$ . Conservation predicts a particle’s speed when all transfers are included, but charge sign and the local field are still needed to determine force direction.

Configuration energy

sum each unique charge pair once

Potential

add scalar source contributions and choose a reference

Field from potential

use the negative spatial gradient

Energy conservation

translate $q\Delta V$ into kinetic or external-work changes

Current exam guidance: Aligned to the current 2026–27 AP Physics C: Electricity and Magnetism framework, with calculus-based field, circuit, and induction models, explicit assumptions, and experimental reasoning.

9.1

Electric Potential Energy

How does a collection of charges store electrostatic interaction energy?

Short answer

Relative to a stated zero, add $k_eq_iq_j/r_{ij}$ once for every distinct pair; the sign records whether assembly required positive or negative external work.

Know firstConservative work · Coulomb force · System energy

Energy belongs to the configuration

For two point charges with zero energy chosen at infinite separation, $U=k_eq_1q_2/r$ . Like-sign pairs have positive $U$ because an external agent must do positive work to assemble them quasistatically from infinity. Unlike-sign pairs have negative $U$ because the electric force releases energy during assembly. A negative value is permitted; it marks a configuration below the chosen zero.

For many point charges, $U=\sum_{i<j}k_eq_iq_j/r_{ij}$ . The restriction $i<j$ counts each pair once. Do not add a charge’s interaction with itself; ideal point-charge self-energy is singular and continuous-distribution assembly needs a different integral treatment. Energy differences obey $\Delta U=-W_E$ , where $W_E$ is work done by the electric force.

Physics model

Electric potential energy belongs to the charge system

Change the separation of +2 μC and −3 μC point charges. With zero energy at infinity, their interaction energy is negative and falls as the charges move closer.

Charge separation2 m

Separation: 2 m
Potential energy: -0.027 J
Force direction: toward smaller separation

positive chargenegative chargepotential-energy curvecurrent state

U_E=k\frac{q_1q_2}{r}

Read this figure without the graphic

Unlike point charges accompany a negative potential-energy curve. A moving state marker links their current separation to the energy measured from zero at infinity.

Current quantity	Value
Separation	2 m
Potential energy	-0.027 J
Force direction	toward smaller separation
Model relation	$U_E=k\frac{q_1q_2}{r}$

Tall Cockcroft–Walton voltage multiplier with repeated capacitor and rectifier stages inside a laboratory. — A voltage multiplier builds a large potential-energy change in stages
Each repeated stage contributes to a much larger total potential difference. A charge moving through that total difference changes electric potential energy by its charge multiplied by the change in potential.
Use this visual to see
Relate electric potential energy to the accumulated potential difference across a multistage device.
What it cannot show
The photograph shows a vertical sequence of electrical stages, but it does not identify the voltage of a stage, total voltage, charge motion, polarity, stored energy, or conversion losses.

Build a three-charge pair ledger

System: Three fixed point charges at the vertices of an equilateral triangle.
Frame: Only pair separations matter; every side has length $a$ .
Assumptions: Zero potential energy is assigned to infinite separation and no self-energy is included.
Givens: The charges are $q$ , $-2q$ , and $3q$ .

List the three unique edges: $(q,-2q)$ , $(q,3q)$ , and $(-2q,3q)$ .
Their dimensionless charge products are $-2q^2$ , $+3q^2$ , and $-6q^2$ .
Because every separation is $a$ , factor out $k_e/a$ .
Add the three pair coefficients: $-2+3-6=-5$ .

Result

U=-\frac{5k_eq^2}{a}

Units:

k_eq^2/a

has units joules.

Boundary cases

$U\to0$ as every pair separation tends to infinity.
Permuting charge labels cannot change the total pair energy.

The triangle has three edges, and adding their signed pair energies once gives a total coefficient of negative five.

Count each pair exactly once

Pair	$q_iq_j/q^2$	Contribution
$q$ with $-2q$	$-2$	$-2k_eq^2/a$
$q$ with $3q$	$+3$	$+3k_eq^2/a$
$-2q$ with $3q$	$-6$	$-6k_eq^2/a$
total	$-5$	$-5k_eq^2/a$

Evaluate a charge configuration

Declare which charges belong to the system.
State the zero of potential energy.
List every unordered pair once.
Use the actual pair separation and signed charge product.
Add scalar contributions and test large-separation or permutation limits.

Keep these three ideas

Potential energy is a property of a chosen charge configuration.
Every unique pair contributes once.
Negative energy is meaningful relative to the chosen zero.

Electric Potential