Electric Potential Energy

The Main Idea

Electric Potential Energy is the energy result of the arrangement of charges in an electric field. Just as a mass in a gravitational field has the "potential" to fall and gain kinetic energy, a charge in an electric field has the "potential" to move when released.

• Differences Between Gravitational and Electric Potential Energy

• Attractive vs. Repulsive: Unlike gravity, which is always attractive, Electric Potential Energy depends on charge signs.
  • If you pull two opposite charges apart, you do work on the system (Electric Potential Energy increases).
  • If you push two like charges together, you do work on the system (Electric Potential Energy increases).

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.