Physics · Electromagnetism · Electrostatics
Electric Potential Energy Calculator
Calculates the electric potential energy between two point charges separated by a given distance using Coulomb's law.
Calculator
Formula
U is the electric potential energy in joules (J). k is Coulomb's constant (8.9875 × 10⁹ N·m²/C²). q₁ and q₂ are the magnitudes (and signs) of the two point charges in coulombs (C). r is the separation distance between the charges in meters (m). A positive U indicates a repulsive interaction (same-sign charges), while a negative U indicates an attractive interaction (opposite-sign charges).
Source: Griffiths, D. J. Introduction to Electrodynamics, 4th ed. Cambridge University Press, 2017. §2.4.
How it works
Electric potential energy arises from the Coulomb interaction between charged particles. Whenever two charges are brought from an infinite distance apart to a finite separation r, work is done by or against the electrostatic force. This work is stored as potential energy in the field between the charges — energy that can be recovered if the charges are released. The sign of this energy carries physical meaning: a negative value means the system is bound and energy must be supplied to separate the charges, while a positive value means the configuration is unstable and the charges will repel each other spontaneously.
The formula governing this relationship is U = kq₁q₂/r, where k = 8.9875 × 10⁹ N·m²/C² is Coulomb's constant (equal to 1/(4πε₀)), q₁ and q₂ are the signed charges in coulombs, and r is the center-to-center separation in meters. The formula is derived directly from integrating the Coulomb force over the displacement from infinity to r. Importantly, this calculator accepts charges in microcoulombs (μC) for practical convenience, since nanocoulombs and microcoulombs are the most common scales encountered in laboratory and engineering contexts. The result is returned in joules (J).
Practical applications of this formula span an enormous range of scales. At the atomic level, it underpins the binding energy of ionic crystals like sodium chloride, where alternating positive and negative ions lower their mutual potential energy by forming a lattice. In electronics, it governs the energy stored in the electric field of a capacitor. In particle physics, the potential energy between protons in a nucleus must be overcome by the strong nuclear force. Even Van der Waals interactions between molecules can be traced back to fluctuating electric potential energies at the quantum level.
Worked example
Consider two point charges separated in free space. Charge 1 is q₁ = +2 μC, Charge 2 is q₂ = −3 μC, and their separation is r = 0.05 m (5 cm).
Step 1 — Convert units: q₁ = 2 × 10⁻⁶ C, q₂ = −3 × 10⁻⁶ C.
Step 2 — Apply the formula:
U = kq₁q₂ / r
U = (8.9875 × 10⁹) × (2 × 10⁻⁶) × (−3 × 10⁻⁶) / 0.05
Step 3 — Compute the numerator:
(8.9875 × 10⁹) × (−6 × 10⁻¹²) = −0.053925 J·m
Step 4 — Divide by r:
U = −0.053925 / 0.05 = −1.0785 J
The negative sign confirms that this is an attractive interaction — the opposite-sign charges are in a lower energy state at 5 cm separation than they would be at infinite separation. To pull them apart to infinity, you would need to supply approximately 1.08 joules of work. The Coulomb force magnitude between them is |F| = |U|/r = 1.0785/0.05 ≈ 21.57 N, which can also be verified directly by F = k|q₁||q₂|/r².
Limitations & notes
This calculator assumes both charges are ideal point charges in a vacuum (or free space), using the permittivity of free space ε₀ in Coulomb's constant. In a real dielectric medium, k must be replaced by k/εᵣ, where εᵣ is the relative permittivity of the material — for example, εᵣ ≈ 80 for water, which dramatically reduces the potential energy. The formula also breaks down at extremely small separations (sub-nanometer scale) where quantum mechanical effects dominate and the classical point-charge model is no longer valid. For systems of more than two charges, the total potential energy is the sum of all pairwise interactions and cannot be computed with a single application of this formula. The calculator does not account for any kinetic energy, induced polarization of nearby conductors or dielectrics, or relativistic effects relevant at very high charge velocities. Ensure that the separation distance r is strictly greater than zero to avoid a physically meaningless singularity.
Frequently asked questions
What is the difference between electric potential energy and electric potential?
Electric potential energy (U, measured in joules) is a property of a system of charges and depends on the configuration of all charges involved. Electric potential (V, measured in volts = J/C) is a property of a single point in space, defined as the potential energy per unit positive test charge placed at that point. They are related by U = qV, where q is the charge of interest.
Why can electric potential energy be negative?
Electric potential energy is negative when opposite-sign charges interact. The reference point is defined as U = 0 when the charges are infinitely far apart. Bringing opposite charges closer together requires no external work — the attractive Coulomb force does positive work — so the system's energy drops below the zero reference, becoming negative. This negative energy represents the binding energy of the system.
How do I use this calculator if my charges are in nanocoulombs (nC)?
Simply convert your charges to microcoulombs before entering them. Since 1 μC = 1000 nC, divide your nanocoulomb value by 1000. For example, 500 nC = 0.5 μC. Enter 0.5 in the charge field. Alternatively, if working in coulombs directly, enter the value scaled appropriately — 1 C = 1,000,000 μC.
Does the separation distance r measure edge-to-edge or center-to-center?
The formula U = kq₁q₂/r uses the center-to-center distance between the two point charges. For ideal point charges, this distinction is trivial, but when approximating charged spheres as point charges (which is valid by the shell theorem for uniform spherical charge distributions), r must be measured between the centers of the spheres, not their surfaces.
What value of Coulomb's constant k does this calculator use?
This calculator uses k = 8.9875 × 10⁹ N·m²/C², which is the standard NIST value for Coulomb's constant in vacuum, defined exactly as k = 1/(4πε₀) where ε₀ = 8.854187817 × 10⁻¹² C²/(N·m²) is the permittivity of free space. This value is appropriate for charges in air or vacuum and may differ in other media.
Last updated: 2025-01-15 · Formula verified against primary sources.