Math 2.0

Optics, electromagnetic theory Electrostatics Fact

Coulomb's law

Builds on Electric charge

Statement

Let two point charges q1q_1 and q2q_2 be at rest at distinct points x1x_1 and x2x_2 of ℝ3\R^3 in vacuum. Let ε0>0\varepsilon_0 > 0 be the vacuum permittivity. Then the force exerted by the first charge on the second is F12=14πε0 q1q2∣x2−x1∣2 x2−x1∣x2−x1∣.F_{12} = \frac{1}{4 \pi \varepsilon_0} \, \frac{q_1 q_2}{\abs{x_2 - x_1}^2} \, \frac{x_2 - x_1}{\abs{x_2 - x_1}}.

Remarks

Coulomb's law is a law of physics, established by experiment, and not a theorem. The force is along the line through the two charges, it is repulsive when q1q2>0q_1 q_2 > 0 and attractive when q1q2<0q_1 q_2 < 0, and its size decays like the inverse square of the distance. By Newton's third law F21=−F12F_{21} = -F_{12}, which the formula shows by exchanging the indices. Forces from several charges add as vectors; this superposition principle is part of the law. For example, two charges of 11 coulomb at distance 11 metre repel with a force of about 9×1099 \times 10^9 newtons. The same inverse-square form appears in Newton's law of gravitation, with masses in place of charges and attraction in place of repulsion.

Going deeper

For a student

Coulomb's law is the starting point of electrostatics. Dividing the force on a test charge by its charge gives the electric field of the source charges. The field of a point charge is the gradient of the potential 1/(4πε0r)1/(4 \pi \varepsilon_0 r), which is a multiple of the fundamental solution of the Laplacian. From this one derives Gauss's law and Poisson's equation, which are the differential form of Coulomb's law.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Coulomb%27s_law
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Changed3 November 2026

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