Electrostatic Potential Energy

Electrostatic potential energy is the energy stored in a system of electric charges due to their positions relative to one another. Just as a compressed spring stores elastic potential energy or a raised mass stores gravitational potential energy, separated or assembled charges store energy because of the electric forces acting between them. This energy is entirely a property of the configuration — the arrangement of the charges — not of any single charge in isolation. The concept is deeply practical. Every time a capacitor charges in your phone, every time an ion moves across a cell membrane, or every time lightning discharges from a cloud, electrostatic potential energy is being converted into other forms. Understanding this energy lets physicists and engineers predict how charges will move, how much work an electric field can do, and how to design devices that store or release electrical energy on demand. Electrostatic potential energy is closely related to the concept of electric potential (voltage). While potential energy is a property of a specific charge or charge system at a location, electric potential is a property of the location itself — a field quantity. Mastering the distinction between these two ideas, and the precise mathematical relationship connecting them, is the central goal of this topic.

Key takeaways
  • Work Done by the Electric Force
  • Potential Energy of Two Point Charges
  • Energy of a Charge in an External Field
  • Energy of Multiple Charge Configurations
  • Energy Conservation in Electrostatics

Work Done by the Electric Force

When a charge moves through an electric field, the field does work on it. If a positive charge q moves from point A to point B and the electric force assists the motion, the field does positive work W. By the work-energy theorem, this work comes at the expense of potential energy: the potential energy decreases by exactly W. Conversely, if you push a charge against the electric force, you must do positive work, which is stored as increased potential energy. This mirrors pushing a ball uphill against gravity.

Potential Energy of Two Point Charges

Consider assembling two point charges q₁ and q₂ from infinite separation to a distance r apart. The work you must do against (or with) the Coulomb force equals the electrostatic potential energy U of the pair. Using Coulomb's law and integrating the force over the path from infinity to r, we obtain U = kq₁q₂/r. If both charges have the same sign, U is positive (you had to push them together, storing energy). If they have opposite signs, U is negative (they attracted each other, and the system released energy as they came together, just as a falling mass releases gravitational potential energy).

Energy of a Charge in an External Field

When a single charge q is placed in an external electric potential V (created by other charges or a source like a battery), its potential energy is simply U = qV. Here V is measured in volts (joules per coulomb), so qV is in joules. A positive charge at high potential has large positive potential energy; a positive charge at low potential has smaller potential energy. The charge will spontaneously move toward lower potential energy — from high to low potential — just as a mass rolls downhill. A negative charge does the opposite: it spontaneously moves toward higher potential.

Energy of Multiple Charge Configurations

For a system of N point charges, the total electrostatic potential energy is the sum of all unique pairwise interactions: U = k Σ(i

Energy Conservation in Electrostatics

Since the electrostatic force is conservative, total mechanical energy (kinetic + electrostatic potential) is conserved when no non-electric forces act. If a proton is released from rest near a fixed positive charge, it accelerates away. Its gain in kinetic energy equals its loss in potential energy: ΔKE = −ΔU. This conservation principle is the most powerful tool for solving problems: no need to track the detailed path, only the start and end positions matter.

The Key Equation

U=kq1q2rU = k\frac{q_1 q_2}{r}
  • U — Electrostatic Potential Energy (Joule (J)): The energy stored in the charge pair due to their interaction; positive for like charges, negative for opposite charges.
  • k — Coulomb's Constant (N·m²/C²): Equal to 8.99 × 10⁹ N·m²/C²; sets the scale of the electric force and potential energy in SI units.
  • q_1,\, q_2 — Point Charges (Coulomb (C)): The magnitudes and signs of the two interacting charges. Sign matters: same-sign product gives positive U, opposite-sign gives negative U.
  • r — Separation Distance (Meter (m)): The center-to-center distance between the two point charges. As r → ∞, U → 0 (the reference level).

Worked Example

Problem. A proton (charge +e = 1.60 × 10⁻¹⁹ C) is held 2.00 × 10⁻¹⁰ m from another proton. (a) Calculate the electrostatic potential energy of the system. (b) If both protons are released from rest, find the speed of each proton when they are very far apart (treat them as point charges of mass 1.67 × 10⁻²⁷ kg).

U=1.15×10−18 J;v≈2.62×104 m/sU = 1.15\times10^{-18}\text{ J};\quad v \approx 2.62\times10^{4}\text{ m/s}

Answer. The initial potential energy is 1.15 × 10⁻¹⁸ J. Each proton reaches a speed of approximately 2.62 × 10⁴ m/s (26,200 m/s) when far apart.

Figure
Diagram
Electrostatic Potential Energy vs. SeparationSeparation rPotential Energy UU=0Like charges (q₁q₂ > 0)U > 0, repulsiveUnlike charges (q₁q₂ < 0)U < 0, attractiver₀2r₀U→∞r→0++r+−
Origins · 1 minCharles-Augustin de Coulomb and how this was discovered
CD
Charles-Augustin de Coulomb
1736–1806 · French

Coulomb systematically measured the forces between electrically charged spheres using a torsion balance of his own invention, establishing the inverse-square law that bears his name. His precise quantification of the electrostatic force is the direct foundation for the formula U = kq₁q₂/r, since potential energy is obtained by integrating that force over distance. Before turning to electricity, Coulomb spent years as a military engineer in Martinique, where he studied soil mechanics and structural failure. His rigorous experimental mindset — honed through building fortifications — is exactly what made his electrical measurements so reliable.

  • Electric Potential (Voltage) — Electric potential V is defined as electrostatic potential energy per unit charge (V = U/q). Understanding U is essential before defining or working with V.
  • Coulomb's Law — The formula U = kq₁q₂/r is derived directly by integrating the Coulomb force, so a solid grasp of Coulomb's Law is a prerequisite.
  • Work-Energy Theorem and Conservative Forces — Potential energy is defined through work done by conservative forces. The same logic used for gravitational PE applies directly to electrostatic PE.
  • The Electric Field — The electric field is the gradient of electric potential, and doing work against the field changes potential energy. The two concepts are inseparable.