In electrostatics (vacuum), the magnitude of the force F between two point charges q1 and q2 separated by distance r is proportional to the product of the charges and inversely proportional to the square of the separation: |F| = (1/(4π ε0))·|q1 q2|/r^2, directed along the line joining them (attractive for opposite signs, repulsive for like signs). The law defines the fundamental inverse-square interaction between static point charges in the classical continuum model.

Principle

Electrostatic interaction between point charges is a central inverse-square force proportional to the product of charges and mediated by the Coulomb constant 1/(4π ε0); it establishes the electric field of an isolated point charge as E(r) = (1/(4π ε0))·q r̂/r^2.

Demonstration

Illustrative scenario — Two small charged spheres: Situation: Two small, well-separated conducting spheres carry charges q1 and q2 in vacuum, with separation r much larger than sphere radii. Recognition: Treat each sphere as an approximate point charge. Action: Compute magnitude and direction of force using |F| = (1/(4π ε0))|q1 q2|/r^2 along the line connecting centers. Consequence: The spheres accelerate toward or away from each other according to force direction; the result approximates actual force when charge distribution is sufficiently localized relative to separation.

Misapplication

Applying Coulomb's point-charge formula to extended or closely spaced charge distributions without integrating over charge density, or failing to include medium permittivity (using ε0 instead of ε of the material), produces quantitatively incorrect forces—this is a category error in domain and scaling assumptions.

Consequence

Forms the basis for the electric field and potential concepts in electrostatics, underlies solution methods (superposition, Gauss's law in symmetric cases), and determines forces in particle-scale and macroscopic problems when its assumptions hold.

Reversal

When charges move at relativistic speeds or currents are present, magnetic forces described by the Lorentz force law become relevant and Coulomb's static law alone is insufficient; within matter, screening, polarization, or nontrivial permittivity modify the effective interaction, requiring replacement of ε0 by the medium permittivity and possibly additional modeling.

Boundary

Clearly within: Two isolated point charges (or well-separated small conductors) in vacuum or homogeneous linear dielectric where r ≫ object sizes. Boundary case: Uniformly charged sphere — outside the sphere force equals that of a point charge at the center, inside it does not. Clearly outside: Time-varying charge/current problems, moving charges with significant magnetic interactions, and condensed-matter situations with collective screening effects.

Semantic Tension

Tension between the particle-centric inverse-square description (discrete charges interacting pairwise) and field-centric descriptions (electric field as a continuum quantity); both are equivalent in electrostatics but lead to different practical calculational approaches.

Synthesis

Coulomb's law is the static, inverse-square limit of electromagnetic interaction in the classical continuum: it defines the point-charge field and potential and, together with superposition, underpins electrostatic analysis, but it must be extended or replaced when motion, material response, or quantum effects are significant.