Coulomb’s Law

Coulomb's law — the force between two point charges is proportional to the product of the charges and inversely proportional to the square of their separation:

F = Q₁ Q₂ / (4 π ε₀ r²)

where ε₀ is the permittivity of free space, 8.85 × 10⁻¹² C² N⁻¹ m⁻². The constant 1/(4πε₀) is about 8.99 × 10⁹ N m² C⁻².

Sign of the result — a positive force means repulsion, a negative force attraction. In practice it is usually easier to find the magnitude and decide the direction from the signs of the charges by inspection.

Field strength of a point charge — dividing by the test charge:

E = Q / (4 π ε₀ r²)

so the field also obeys an inverse square law, exactly as the gravitational field does.

The structural parallel with gravity

The mathematics is identical; only the source quantity and the possibility of two signs differ. Anything you can do with one field you can do with the other.

Example: the enormous difference in strength is worth feeling. For two electrons, the electrostatic repulsion exceeds the gravitational attraction by a factor of about 4 × 10⁴². This is why chemistry, materials and solid matter are governed entirely by electromagnetic forces, and gravity plays no role whatever inside an atom.

Charged spheres behave as point charges — a uniformly charged conducting sphere attracts or repels external charges as though all its charge sat at its centre, just as a spherical mass does for gravity. r is again measured centre-to-centre.

Superposition — with several charges present, the forces add as vectors. Work out each pairwise force separately with Coulomb's law, then combine them with components.