Why does the force fall as 1/r² instead of 1/r?

In three dimensions the electric field of a point charge spreads over the area of a sphere (∝ r²), so flux conservation (Gauss's law) requires field magnitude ∝ 1/r² and force between two point charges inherits the same scaling.

Does Coulomb's law work inside atoms or metals?

The pairwise formula remains a building block, but atoms and conductors polarize: induced charges reshape the field. Effective forces become distance-dependent in more complex ways, and at very small r quantum mechanics dominates. The simulator's point-charge picture is a clean first model before adding those layers.

How is k related to ε₀?

In SI units k = 1/(4πε₀). Writing F = (1/(4πε₀)) q₁q₂/r² makes the connection to Maxwell's equations and Gauss's law explicit.

Why might displayed forces be rescaled in the interactive view?

Interfaces often rescale vectors for readability while preserving ratios and sign. For quantitative homework, substitute SI values for q and r directly into F = k|q₁q₂|/r².

Coulomb's Law

Two charges with adjustable magnitude and distance. Force vectors shown.

About this model

Coulomb's law quantifies the electrostatic force between two point charges in vacuum: F = k |q₁ q₂| / r², where k ≈ 8.99×10⁹ N·m²/C² is Coulomb's constant, r is the center-to-center separation, and like charges repel while opposite charges attract. The inverse-square dependence mirrors the geometry of electric field lines spreading in three dimensions and is consistent with Gauss's law for a point source. Superposition lets you add vector forces from many charges, but the pairwise law itself assumes point-like carriers and a linear, isotropic medium (here vacuum). The simulator focuses on the scaling of force with distance and charge magnitude, idealizing the charges as points with no polarization of surrounding matter, no quantum or relativistic corrections, and no magnetic effects. Students connect the steep 1/r² growth at small separations to shielding and breakdown in real materials, and they contrast Coulomb's static law with time-varying phenomena treated elsewhere in the catalog.

Who it's for: High school and introductory undergraduate students studying electrostatics, preparing for problems involving superposition, electric fields, and the SI unit system for charge.

Key terms

Coulomb's law
Point charge
Electrostatic force
Coulomb constant
Inverse-square law
Superposition
Vacuum permittivity
Like and opposite charges

How it works

The magnitude of the electrostatic force between two point charges in vacuum is F = k|q₁q₂|/r² with k ≈ 8.99×10⁹ N·m²/C². Like charges repel, unlike attract.

Key equations

F = k |q₁ q₂| / r²

Frequently asked questions

Why does the force fall as 1/r² instead of 1/r?: In three dimensions the electric field of a point charge spreads over the area of a sphere (∝ r²), so flux conservation (Gauss's law) requires field magnitude ∝ 1/r² and force between two point charges inherits the same scaling.
Does Coulomb's law work inside atoms or metals?: The pairwise formula remains a building block, but atoms and conductors polarize: induced charges reshape the field. Effective forces become distance-dependent in more complex ways, and at very small r quantum mechanics dominates. The simulator's point-charge picture is a clean first model before adding those layers.
How is k related to ε₀?: In SI units k = 1/(4πε₀). Writing F = (1/(4πε₀)) q₁q₂/r² makes the connection to Maxwell's equations and Gauss's law explicit.
Why might displayed forces be rescaled in the interactive view?: Interfaces often rescale vectors for readability while preserving ratios and sign. For quantitative homework, substitute SI values for q and r directly into F = k|q₁q₂|/r².

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