Electric Potential Energy

Kanishka Kislaya - Spring 2026

Learning Goals: Define Electric Potential Energy ($U_e$) as a scalar quantity. Calculate the work required to assemble a system of point charges. Apply the principle of conservation of energy to electrostatic systems. Differentiate between potential energy in point-charge systems and uniform fields.

The Main Idea

Electric Potential Energy ($U_e$) is the energy stored within a system of charges due to their relative positions. It represents the external work required to assemble a specific configuration of charges from an initial state where they are infinitely far apart.

Because the electrostatic force is **conservative**, this energy depends only on the final arrangement of the charges, not the path taken to put them there.

• **Attractive vs. Repulsive:** Unlike gravity, which is always attractive, $U_e$ depends on charge signs.
  • If you pull two opposite charges apart, you do work on the system ($U_e$ increases).
  • If you push two like charges together, you do work on the system ($U_e$ increases).
• **Zero Reference:** By convention, $U_e$ is defined as zero when charges are infinitely far apart ($r = \infty$).

Mathematical Models

Point Charge Interactions

The fundamental equation for the potential energy between two point charges, $q_1$ and $q_2$, separated by a distance $r$: [math]\displaystyle{ U_e = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r} }[/math]

For a system containing multiple charges, the total potential energy is the sum of the interaction energies for every unique pair. For three charges: [math]\displaystyle{ U_{total} = k \left( \frac{q_1 q_2}{r_{12}} + \frac{q_1 q_3}{r_{13}} + \frac{q_2 q_3}{r_{23}} \right) }[/math]

Uniform Electric Fields

Near a large charged plate or inside a capacitor where the field $E$ is constant, the potential energy of a charge $q$ at a distance $d$ (parallel to the field lines) is: [math]\displaystyle{ U_e = qEd }[/math] This is the electrical equivalent of the gravitational formula $U_g = mgh$.

Key Relationships

• Work and Energy: $\Delta U_e = -W_{field}$
• Potential Relationship: $U_e = qV$, where $V$ is the Electric Potential.

A Computational Model

In a computational environment like VPython, we calculate the potential energy by iterating through pairs of charges.

# Example snippet for two charges
k = 9e9
q1 = 1e-6
q2 = 2e-6
r = mag(particle2.pos - particle1.pos)
Ue = k * q1 * q2 / r

Conservation of Energy

In an isolated system, total energy ($E_{tot} = K + U_e$) is conserved. If a charge is released from rest in an electric field, its potential energy is converted into kinetic energy: [math]\displaystyle{ K_i + U_{ei} = K_f + U_{ef} }[/math]

Examples

Simple: Two Protons

Two protons ($q = 1.6 \times 10^{-19} \text{ C}$) are held $1 \times 10^{-10} \text{ m}$ apart. Calculate the potential energy.

• Solution: $U_e = (9 \times 10^9) \frac{(1.6 \times 10^{-19})^2}{1 \times 10^{-10}} = 2.3 \times 10^{-18} \text{ J}$.

Middling: Energy Conversion

An electron is released from rest in a uniform field of $500 \text{ N/C}$. After moving $0.2 \text{ m}$, what is its kinetic energy?

• Solution: $\Delta K = -\Delta U_e = -(qEd)$.
• $K_f = -(-1.6 \times 10^{-19} \text{ C})(500 \text{ N/C})(0.2 \text{ m}) = 1.6 \times 10^{-17} \text{ J}$.

Real-World Applications

• Capacitors: These components store $U_e$ to be released quickly, like in a camera flash.
• Chemical Energy: The energy in food or fuel is actually $U_e$ stored in molecular bonds.
• Neurobiology: Nerve signals rely on the potential energy created by ion concentrations across cell membranes.