Electric Potential Energy
Kanishka Kislaya - Spring 2026
Learning Goals:
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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 arrangement of the charges, not the path taken to put them there.
A Mathematical Model
The fundamental equation for the electric potential energy between two point charges, $q_1$ and $q_2$, separated by a distance $r$, is:
[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 in the system. For three charges, it looks like this:
[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]
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
Examples
Simple
Two protons ($q = 1.6 \times 10^{-19} \text{ C}$) are held $1 \times 10^{-10} \text{ m}$ apart. Calculate the potential energy of the system.
- 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
Three $1\mu\text{C}$ charges are placed at the corners of an equilateral triangle with side lengths of $0