Coulomb's Law Calculator — Electrostatic Force

Coulomb's Law Guide

What do I need to know about Coulomb's Law?

F = kq₁q₂/r². k = 8.99×10⁹ N·m²/C² (Coulomb's constant = 1/(4πε₀)). q₁, q₂ = charges in coulombs (C). r = separation in metres. The force is repulsive if charges have the same sign, attractive if opposite. 1 microcoulomb (µC) = 10⁻⁶ C. Example: q₁ = +2µC, q₂ = −3µC, r = 10cm. F = 8.99×10⁹ × 2×10⁻⁶ × 3×10⁻⁶ / (0.10)² = 8.99×10⁹ × 6×10⁻¹² / 0.01 = 5.39N. Attractive force (opposite charges). Inverse square law: like gravity, force decreases with the square of distance.

What do I need to know about Electric Field and Potential?

Electric field E = F/q₀ = kq/r². Units: N/C or V/m (equivalent). The field points away from positive charges, toward negative charges. Electric potential V = kq/r (scalar, not vector). Potential energy U = kq₁q₂/r = q₂V. Relationship: E = −dV/dr (field is negative gradient of potential). Equipotential surfaces: spheres concentric with a point charge — no work done moving a charge along an equipotential.

What's the key thing to understand about Comparison with Gravity?

Coulomb's law: F = kq₁q₂/r². Newton's gravitational law: F = Gm₁m₂/r². Both are inverse-square laws. The ratio k/G = 8.99×10⁹ / 6.67×10⁻¹¹ ≈ 10²⁰. Electrostatic force is approximately 10³⁹ times stronger than gravity for a proton-electron pair: F_electric = 8.99×10⁹ × (1.6×10⁻¹⁹)² / (5.3×10⁻¹¹)² = 8.2×10⁻⁸ N. F_gravity = 6.67×10⁻¹¹ × 1.67×10⁻²⁷ × 9.11×10⁻³¹ / (5.3×10⁻¹¹)² = 3.6×10⁻⁴⁷ N. Ratio: 2.3×10³⁹. Why gravity dominates cosmologically: matter is neutral overall, so electric forces cancel out. Gravity, unlike the electric force, is always attractive and never cancels, so it accumulates and dominates at the scale of planets, stars, and galaxies despite being enormously weaker at the level of individual particles.

What should I know about Superposition Principle?

The total electric force on a charge = vector sum of individual forces from all other charges. The total electric field at a point = vector sum of fields from all charges. This is the superposition principle — electric and gravitational fields both obey it. Dipole: two equal and opposite charges (+q and −q) separated by distance d. Electric dipole moment p = qd. Dipole fields decrease more rapidly with distance (1/r³ instead of 1/r²) than a monopole. Biological significance: many molecules are electric dipoles, including water, which is why water is such an effective solvent for other charged or polar substances.

What should I know about Coulomb's Law Formula and Units?

Coulomb's law states F = k×q₁×q₂/r², where k = 8.99×10⁹ N·m²/C² (Coulomb's constant), q₁ and q₂ are charges in coulombs, and r is separation in metres. For two charges of 1μC (1×10⁻⁶ C) separated by 0.1m: F = 8.99×10⁹ × (1×10⁻⁶)² / (0.1)² = 8.99×10⁹ × 1×10⁻¹² / 0.01 = 0.899 N — a surprisingly large force for such tiny charges, illustrating just how strong the electric force is compared to gravity at everyday scales. Like charges repel (positive force value, pushing apart) and opposite charges attract (the force is still calculated with the same formula, but conventionally treated as a negative/attractive direction along the line joining the charges). The inverse-square relationship means doubling the distance reduces the force to one quarter, identical in form to Newton's gravitational law.

What do I need to know about Electric Fields from Point Charges?

The electric field from a point charge is E = k×q/r², which is simply Coulomb's law with the second charge removed — the field represents the force per unit charge that would be felt by a test charge placed at that point, independent of what that test charge's actual value is. For a charge of 5μC at a distance of 0.2m: E = 8.99×10⁹ × 5×10⁻⁶ / (0.2)² = 8.99×10⁹ × 5×10⁻⁶ / 0.04 ≈ 1.12×10⁶ N/C. Multiplying this field by any test charge q placed there (F=qE) recovers Coulomb's law exactly, which is why the field concept is so useful — it separates 'what the source charge creates in the space around it' from 'what happens to any particular charge that enters that space', letting you calculate the effect on multiple different charges without repeating the full Coulomb's law calculation each time.

What do I need to know about Electric Potential Energy Between Charges?

Two point charges separated by distance r store electric potential energy U = kQq/r, directly analogous in form to gravitational potential energy but with an important difference: because charge can be positive or negative, U can itself be positive (repulsive configuration, like charges) or negative (attractive configuration, opposite charges). A negative U means work must be done to pull the charges apart to infinity, while a positive U means they'll fly apart on their own, releasing kinetic energy — exactly the mechanism behind the energy released when similarly charged nuclear fragments separate after fission.

What should I know about Permittivity and the Role of the Medium?

Coulomb's law is usually written F = kq₁q₂/r², with the constant k ≈ 8.99 × 10⁹ N·m²/C² assuming the charges sit in a vacuum (or, to good approximation, air). This constant is actually k = 1/(4πε₀), where ε₀ is the permittivity of free space, a fundamental property describing how easily an electric field can establish itself through empty space. When charges instead sit inside an insulating medium — oil, water, or any dielectric material — the medium's molecules polarise in response to the electric field, partially cancelling it and reducing the force between the charges. This is accounted for by replacing ε₀ with ε = εᵣε₀, where εᵣ is the medium's relative permittivity (always greater than 1 for real materials), giving a modified Coulomb force F = q₁q₂/(4πεᵣε₀r²) that is weaker than the vacuum value by a factor of εᵣ. Water, with a very high relative permittivity of around 80, dramatically weakens electrostatic forces between ions dissolved in it compared to the same ions in air — a key reason why ionic compounds like table salt dissolve so readily in water, since the reduced attractive force between oppositely charged ions makes it easier for thermal motion and water molecules to pull them apart.

Worked Example: Force Between Two Charged Spheres

Two small spheres carry charges q₁ = 2 µC and q₂ = 3 µC, separated by r = 0.1 m in air (k = 8.99×10⁹ N·m²/C²). Using Coulomb's law: F = k·q₁q₂/r² = 8.99×10⁹ × (2×10⁻⁶)(3×10⁻⁶) / (0.1)² = 8.99×10⁹ × 6×10⁻¹² / 0.01 = 8.99×10⁹ × 6×10⁻¹⁰ = 5.394 N This is a surprisingly large force for such tiny charges — a reminder of just how strong the electric force is compared to gravity at everyday scales. If the separation were doubled to 0.2 m, the inverse-square relationship means the force falls to a quarter: F = 5.394/4 ≈ 1.35 N. If instead the distance stayed the same but both charges doubled (to 4 µC and 6 µC), the force would quadruple to 4 × 5.394 ≈ 21.6 N, since force is proportional to the product of the two charges.

Coulomb's Law vs. Newton's Gravitation: Side-by-Side

Both laws share the identical inverse-square mathematical structure, which is why they're so often taught side by side — but the electric force is vastly stronger at the same distance and same-sized 'charge', and critically, it comes in two signs (positive and negative) that allow both attraction and cancellation, whereas gravitational 'charge' (mass) only ever attracts. This is also why bulk matter is electrically neutral in practice (positive and negative charges cancel almost perfectly) while gravity, having no canceling opposite, always adds up and dominates at large scales like planets and stars.

Worked Example: Electric Field from a Point Charge

Find the electric field strength at r = 0.2 m from a point charge q = 5 µC. Using E = kq/r²: E = (8.99×10⁹ × 5×10⁻⁶) / (0.2)² = 44,950 / 0.04 = 1,123,750 N/C ≈ 1.12×10⁶ N/C This field would exert a force of F = qE on any test charge placed at that point — for a 1 µC test charge, F = 1×10⁻⁶ × 1.12×10⁶ ≈ 1.12 N. Doubling the distance to 0.4 m quarters the field strength (inverse-square law again), to about 2.81×10⁵ N/C, demonstrating that electric field strength falls off with distance exactly as force does, since field strength is just force-per-unit-charge.