Why is Coulomb's Law an inverse-square law?

The electric field from a point charge spreads outward uniformly in all directions. The surface area of a sphere grows as r², so the field strength (and therefore force) falls as 1/r². The same geometric argument applies to gravity and the intensity of light from a point source.

What is the difference between Coulomb's Law and the electric field?

Coulomb's Law gives the force between two specific charges. The electric field E = F/q describes the force per unit charge at a point in space — it exists regardless of whether a test charge is placed there. E = kq/r² for a point charge. Coulomb's Law is F = qE when you place a charge q in a field E.

Does Coulomb's Law work inside materials?

In a material with relative permittivity (dielectric constant) εᵣ, the force is reduced by a factor εᵣ: F = kq₁q₂/(εᵣr²). Water has εᵣ ≈ 80, which dramatically weakens electrostatic interactions — this is why ionic compounds dissolve easily in water.

What are the limits of Coulomb's Law?

Coulomb's Law applies to static (non-moving) charges and point charges. For moving charges, magnetic forces appear and Maxwell's equations are needed. At distances comparable to the size of the charge distribution (not a true point), the law must be integrated over the charge distribution.

Coulomb's Law Calculator — Electric Force Between Charges

Coulomb's Law describes the force between two point charges. Published by Charles-Augustin de Coulomb in 1785, it is the electrostatic analogue of Newton's Law of Gravitation — both forces follow an inverse-square relationship with distance. Coulomb's Law is the foundation of electrostatics, and from it we can derive electric fields, potential, and eventually Maxwell's equations.

Set the charge magnitudes and signs, adjust the separation, and watch how the electric field pattern changes between attraction and repulsion.

Repulsion

35.96 mN

Charge 1 (negative = −, positive = +)

q₁1 μC

q₂1 μC

Distance r50 cm

The Formula

$F = k \frac{q_1 q_2}{r^2}$

Symbol	Quantity	Value/Unit
F	Electrostatic force	N (positive = repulsion, negative = attraction)
k	Coulomb's constant	8.99 × 10⁹ N·m²/C²
q₁, q₂	Charges	Coulombs (C)
r	Distance between charges	m

In SI units, $k = \dfrac{1}{4\pi\varepsilon_0}$ where $\varepsilon_0 = 8.85 \times 10^{-12}$ C²/N·m² is the permittivity of free space.

Sign rule: Like charges (both positive or both negative) → F > 0 → repulsion. Opposite charges → F < 0 → attraction.

Worked Examples

Worked Example

Example 1 — Two protons

Two protons are separated by 1 nm (10⁻⁹ m). Each proton carries charge q = +1.6 × 10⁻¹⁹ C. What is the electrostatic repulsion between them?

$F = k\frac{q^2}{r^2} = \frac{8.99 \times 10^9 \times (1.6 \times 10^{-19})^2}{(10^{-9})^2}$

$F = \frac{8.99 \times 10^9 \times 2.56 \times 10^{-38}}{10^{-18}} \approx 2.3 \times 10^{-10} \text{ N}$

That's 0.23 nN — tiny in everyday terms, but enormous relative to the proton's mass (1.67 × 10⁻²⁷ kg). This is why protons in a nucleus need the strong nuclear force to hold them together against electrostatic repulsion.

Worked Example

Example 2 — Comparing gravity and electrostatics

Compare the gravitational and electrostatic forces between two electrons separated by 1 mm.

Electron charge: q = −1.6 × 10⁻¹⁹ C, mass: m = 9.11 × 10⁻³¹ kg

Electrostatic: $F_E = k q^2 / r^2 = 8.99 \times 10^9 \times (1.6 \times 10^{-19})^2 / (10^{-3})^2 \approx 2.3 \times 10^{-22}$ N

Gravitational: $F_G = G m^2 / r^2 = 6.67 \times 10^{-11} \times (9.11 \times 10^{-31})^2 / (10^{-3})^2 \approx 5.5 \times 10^{-65}$ N

The electrostatic force is about 10⁴³ times stronger than gravity between electrons. Gravity is utterly negligible at the subatomic scale.

Frequently Asked Questions

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