Coulomb’s Law · aiwiki.page

Coulomb’s law is a fundamental law of electrostatics describing the force between stationary electric charges. In vacuum, two point charges exert forces along the line joining them, with magnitude proportional to the product of their charge magnitudes and inversely proportional to the square of their separation. Charges with the same sign repel; charges with opposite signs attract. Together with the superposition principle, the law provides a foundation for classical electromagnetism. (openstax.org)

Historical development

The law is named after Charles-Augustin de Coulomb, who reported measurements of electrical forces in 1785. His apparatus used a torsion balance: a lightweight horizontal assembly suspended by a fine wire. Electrical repulsion displaced a charged sphere on the assembly, twisting the wire until its restoring torque balanced the electrical torque. Measuring the twist made it possible to infer very small forces. (aps.org)

Coulomb compared forces at different separations and investigated how force depended on charge. These experiments helped establish a quantitative inverse-square description of electrical interactions, replacing purely qualitative observations of attraction and repulsion with a mathematical relationship. (aps.org)

Mathematical statement and units

For two point charges $q_1$ and $q_2$ , separated by a distance $r>0$ , the force magnitude is

F=k_e\frac{|q_1q_2|}{r^2}, \qquad k_e=\frac{1}{4\pi\varepsilon_0}.

Here $\varepsilon_0$ is the vacuum permittivity. In the International System of Units, charge is measured in coulombs, distance in metres, and force in newtons. The numerical constants are approximately

\varepsilon_0=8.854\times10^{-12}\ \mathrm{F\,m^{-1}}, \qquad k_e=8.988\times10^9\ \mathrm{N\,m^2\,C^{-2}}.

Doubling the separation reduces the force to one-quarter; doubling either charge doubles it. (openstax.org)

A vector expression specifies both magnitude and direction. If $\mathbf r_1$ and $\mathbf r_2$ are the charge positions, the force on charge 2 due to charge 1 is

\mathbf F_{2\leftarrow1} = \frac{q_1q_2}{4\pi\varepsilon_0} \frac{\mathbf r_2-\mathbf r_1} {|\mathbf r_2-\mathbf r_1|^3}.

The signed product $q_1q_2$ determines attraction or repulsion. The force on charge 1 is equal and opposite, consistent with Newton’s third law. (openstax.org)

Superposition and electric fields

For several stationary charges, the superposition principle states that the total force is the vector sum of the individual pairwise forces. Forces must be added by direction and components, rather than by simply summing their magnitudes. (openstax.org)

The corresponding electric field of a point charge $q$ , placed at the origin, is

\mathbf E(\mathbf r) = \frac{q}{4\pi\varepsilon_0} \frac{\mathbf r}{r^3}, \qquad \mathbf F=Q\mathbf E,

where $Q$ is a test charge. A positive source produces an outward field; a negative source produces an inward field. This formulation separates the source’s field from the charge experiencing the force. (feynmanlectures.caltech.edu)

For a continuous charge distribution, summation becomes an integral:

\mathbf E(\mathbf r) = \frac{1}{4\pi\varepsilon_0} \int \rho(\mathbf r') \frac{\mathbf r-\mathbf r'} {|\mathbf r-\mathbf r'|^3}\,d^3r'.

Here $\rho$ is charge per unit volume and $\mathbf r'$ labels source positions. Extended objects generally require this distribution-based treatment rather than replacing their entire charge by a point at their centre. (feynmanlectures.caltech.edu)

Potential, energy, and Gauss’s law

The electrostatic interaction is a conservative force, so its work between fixed endpoints is independent of the path. Choosing zero electric potential at infinity, an isolated point charge has

V(r)=\frac{q}{4\pi\varepsilon_0r}.

The potential energy of two point charges is therefore

U(r)=\frac{q_1q_2}{4\pi\varepsilon_0r}.

Unlike force magnitude, this expression retains the sign of the charge product: opposite charges have negative interaction energy under this reference convention. The field follows from the negative gradient of potential, $\mathbf E=-\nabla V$ . (feynmanlectures.caltech.edu)

Coulomb’s law and superposition imply Gauss’s law: the electric flux through a closed surface equals its enclosed charge divided by $\varepsilon_0$ . Conversely, spherical symmetry and Gauss’s law give the inverse-square point-charge field. A spherically symmetric charge distribution produces, outside its extent, the same field as its total charge concentrated at its centre. These relationships form part of the electrostatic limit of Maxwell’s equations. (feynmanlectures.caltech.edu)

Materials and limits of applicability

In an ideal infinite, homogeneous, isotropic, linear dielectric, the macroscopic interaction between embedded free point charges can be represented by replacing $\varepsilon_0$ with the material permittivity $\varepsilon=\varepsilon_r\varepsilon_0$ . This incorporates the material’s electrical polarization. Boundaries, nonuniform materials, and microscopic local fields require more detailed treatment; dividing every vacuum force by a single dielectric constant is not universally valid. (feynmanlectures.caltech.edu)

The simple law concerns stationary sources. Moving charges generally require electric and magnetic fields, the Lorentz force, and time-dependent Maxwell equations. Changes in electromagnetic influence propagate at the speed of light, not instantaneously between separated charges. (feynmanlectures.caltech.edu)

At atomic scales, Coulomb interaction remains central, but classical trajectories do not adequately describe bound electrons. In quantum mechanics, the attractive potential between an electron and a proton enters the Schrödinger equation for hydrogen, yielding quantum states and discrete energies rather than classical planetary orbits. (feynmanlectures.caltech.edu)