Electric Potential Energy

Main Idea

Electric Potential Energy ([math]\displaystyle{ U_e }[/math]) is the energy stored within a system of charged objects as a result of their relative positions and the electrostatic forces they exert on one another. Much like gravitational potential energy depends on the distance between masses, electric potential energy depends on the distance between charges. However, unlike gravity the electric force can be either attractive or repulsive. This means that the total [math]\displaystyle{ U_e }[/math] can be either positive or negative. The magnitude of this energy is directly proportional to the product of the charges and inversely proportional to the separation distance between them.

Conceptually, potential energy represents the "potential" for the system to do work. If you hold two positive charges close together and release them, the stored electric potential energy is translated into kinetic energy as they fly apart.

A key convention in electrostatics is the "Zero Reference Point." We define the electric potential energy of a system to be zero when the charges are infinitely far apart ([math]\displaystyle{ r = \infty }[/math]). This makes sense intuitively: if two charges are on opposite sides of the universe, they aren't interacting, so they have no stored energy. As they move closer, the work done to move them—either by the field or by an external force—changes that energy value. For attractive opposite charges, the energy becomes more negative as they get closer (an "energy well"). For repulsive like charges, the energy becomes more positive (an "energy hill").

Mathematical Model

General Form for Electric Potential Energy

For two point charges [math]\displaystyle{ q_1 }[/math] and [math]\displaystyle{ q_2 }[/math] separated by a distance [math]\displaystyle{ r }[/math]: [math]\displaystyle{ U_e = k \frac{q_1 q_2}{r} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r} }[/math]

• [math]\displaystyle{ U_e }[/math]: Total electric potential energy, measured in Joules ([math]\displaystyle{ J }[/math]).
• [math]\displaystyle{ q_1, q_2 }[/math]: The values of the two charges in Coulombs ([math]\displaystyle{ C }[/math]). Include the positive or negative signs of the charges in this calculation.
• [math]\displaystyle{ r }[/math]: The distance between the centers of the two charges in meters ([math]\displaystyle{ m }[/math]).
• [math]\displaystyle{ k }[/math]: Coulomb's constant ([math]\displaystyle{ \approx 8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2 }[/math]).

Electric Potential Energy Near a Charged Surface (Uniform Field)

When a charge [math]\displaystyle{ q }[/math] is moving in a uniform electric field [math]\displaystyle{ E }[/math] (like the constant field between two capacitor plates), the formula simplifies: [math]\displaystyle{ U_e = qEd }[/math]

• [math]\displaystyle{ E }[/math]: The electric field strength ([math]\displaystyle{ \text{N/C} }[/math] or [math]\displaystyle{ \text{V/m} }[/math]).
• [math]\displaystyle{ d }[/math]: The displacement of the charge parallel to the field lines ([math]\displaystyle{ m }[/math]).
• This is the electrical equivalent of the gravitational formula [math]\displaystyle{ U_g = mgh }[/math].

Relation to Electric Force (Calculus)

The potential energy is the negative integral of the internal conservative force over a displacement. By integrating Coulomb’s Law from a reference point at infinity to a distance [math]\displaystyle{ r }[/math]: [math]\displaystyle{ U_e = -\int_{\infty}^{r} \vec{F}_e \cdot d\vec{r} = -\int_{\infty}^{r} \frac{k q_1 q_2}{r^2} dr = \frac{k q_1 q_2}{r} }[/math]

Computational Model

To understand [math]\displaystyle{ U_e }[/math] computationally, we often use energy graphs to plot [math]\displaystyle{ U_e }[/math], [math]\displaystyle{ K }[/math] (Kinetic Energy), and [math]\displaystyle{ E_{total} }[/math] vs. distance.

VPython Example

This script models an electron being fired at a fixed negative "target" charge. As it slows down due to repulsion, you can see the [math]\displaystyle{ K }[/math] decrease and [math]\displaystyle{ U_e }[/math] increase on a graph.

# Objects and Constants
k = 9e9
target = sphere(pos=vector(0,0,0), q=-1e-6, color=color.red)
electron = sphere(pos=vector(0.5,0,0), q=-1.6e-19, m=9.1e-31, color=color.blue)
electron.p = vector(-1e-23, 0, 0) # Initial momentum toward target

# Graphs
energy_graph = gdisplay(xtitle="Distance (m)", ytitle="Energy (J)")
U_plot = gcurve(color=color.blue, label="Ue")
K_plot = gcurve(color=color.yellow, label="K")
Total_plot = gcurve(color=color.green, label="Total E")

dt = 1e-15
while electron.pos.x > 0.01:
    rate(1000)
    # Physics Calculations
    r_vec = electron.pos - target.pos
    r_mag = mag(r_vec)
    Force = (k * target.q * electron.q / r_mag**2) * norm(r_vec)
    
    electron.p = electron.p + Force * dt
    electron.pos = electron.pos + (electron.p / electron.m) * dt
    
    # Energy Calculations
    Ue = k * target.q * electron.q / r_mag
    K = mag(electron.p)**2 / (2 * electron.m)
    
    # Plotting
    U_plot.plot(pos=(r_mag, Ue))
    K_plot.plot(pos=(r_mag, K))
    Total_plot.plot(pos=(r_mag, Ue + K))

Guide on Solving Problems

The principle of Conservation of Energy is the most effective way to solve [math]\displaystyle{ U_e }[/math] problems:

[math]\displaystyle{ K_i + U_{ei} + W_{ext} = K_f + U_{ef} }[/math]

1. Draw a Diagram: Label the initial and final positions of all charges.
2. Define the System: Usually, the system includes all interacting charges so that the internal work is accounted for as potential energy.
3. Identify Energy Types: Determine if the charge is moving in a uniform field ([math]\displaystyle{ qEd }[/math]) or toward a point charge ([math]\displaystyle{ kqQ/r }[/math]).
4. Solve for the Unknown: This is usually the final velocity ([math]\displaystyle{ v_f }[/math]) or the distance of closest approach ([math]\displaystyle{ r_{min} }[/math]).

Examples

Simple

Question: How much work is required to bring a [math]\displaystyle{ +2 \mu C }[/math] point charge from very far away to a distance of [math]\displaystyle{ 0.1 \text{ m} }[/math] from a fixed [math]\displaystyle{ +4 \mu C }[/math] charge?

• Solution: Since [math]\displaystyle{ W = \Delta U_e }[/math] and [math]\displaystyle{ U_{initial} = 0 }[/math]:

[math]\displaystyle{ U_e = \frac{(9 \times 10^9)(2 \times 10^{-6})(4 \times 10^{-6})}{0.1} }[/math]

[math]\displaystyle{ U_e = 0.72 \text{ Joules} }[/math].

Medium

Question: A proton is released from rest at a distance of [math]\displaystyle{ 1 \times 10^{-10} \text{ m} }[/math] from another fixed proton. What is the speed of the moving proton when it has reached a distance of [math]\displaystyle{ 1 \times 10^{-9} \text{ m} }[/math]?

• Solution: Using [math]\displaystyle{ K_i + U_{ei} = K_f + U_{ef} }[/math]:

[math]\displaystyle{ 0 + \frac{k q^2}{r_i} = \frac{1}{2}m v^2 + \frac{k q^2}{r_f} }[/math]

[math]\displaystyle{ v = \sqrt{\frac{2kq^2}{m} \left( \frac{1}{r_i} - \frac{1}{r_f} \right)} }[/math]

Plugging in [math]\displaystyle{ q = 1.6 \times 10^{-19} \text{ C} }[/math] and [math]\displaystyle{ m = 1.67 \times 10^{-27} \text{ kg} }[/math] gives [math]\displaystyle{ v \approx 1.57 \times 10^5 \text{ m/s} }[/math].

Hard

Question: An alpha particle ([math]\displaystyle{ q=2e, m=6.64 \times 10^{-27} \text{ kg} }[/math]) is fired with a velocity of [math]\displaystyle{ 2 \times 10^7 \text{ m/s} }[/math] directly at a gold nucleus ([math]\displaystyle{ q=79e }[/math]). What is the distance of closest approach?

• Solution: At the closest point, [math]\displaystyle{ v_f = 0 }[/math], so all initial kinetic energy has become potential energy.

[math]\displaystyle{ K_i + U_{ei} = U_{ef} }[/math] (Assume [math]\displaystyle{ U_{ei} \approx 0 }[/math] if starting from far away)

[math]\displaystyle{ \frac{1}{2}m v^2 = \frac{k(2e)(79e)}{r_{min}} }[/math]

[math]\displaystyle{ r_{min} = \frac{2k(2e)(79e)}{mv^2} \approx 2.74 \times 10^{-14} \text{ m} }[/math].

Connectedness

In industrial engineering and chemistry, Electric Potential Energy is the primary driver of reaction kinetics. The "Activation Energy" required for a chemical reaction is often the energy needed to overcome the electric repulsion between electron clouds. In aerospace, [math]\displaystyle{ U_e }[/math] is critical for understanding the "plasma sheath" that forms around re-entry vehicles, which can block radio communications. Furthermore, every battery in your pocket stores energy in the form of electric potential energy, specifically by using chemical reactions to separate charges.

History

The mathematical framework for potential energy was developed in the 18th and 19th centuries. While Charles-Augustin de Coulomb provided the force law in 1785, it was Siméon Denis Poisson who expanded this into potential theory in 1813. Later, James Clerk Maxwell unified these concepts into a set of equations that showed how energy is not just "between" charges, but stored within the electric field itself.

See also

• Electric Potential — Often confused with [math]\displaystyle{ U_e }[/math]; this is energy per unit charge ([math]\displaystyle{ V }[/math]).
• Capacitors — Devices that store large amounts of [math]\displaystyle{ U_e }[/math] for rapid release.
• Equipotential Surfaces — Regions where [math]\displaystyle{ U_e }[/math] remains constant.

Further reading

• Matter and Interactions by Chabay and Sherwood. This textbook provides a deep dive into the computational modeling of electric systems.
• OpenStax University Physics Vol 2 — Free resource for more derivations.

External links

• PhET Simulation: Charges and Fields — Interactive way to visualize potential and field lines.
• Video: Visualizing Voltage with 3D Animations — A high-quality visual guide to how energy is stored in fields.
• Video: Electric Potential Energy Explained — A clear breakdown of the work-energy relationship in electrostatics.
• Video: Voltage, Electric Energy, and Capacitors — Crash Course Physics #27, connecting theory to real-world applications.

References

• Chabay, R., & Sherwood, B. (2015). Matter and Interactions. Wiley.
• Knight, R. D. (2016). Physics for Scientists and Engineers: A Strategic Approach. Pearson.