Coulomb Law
Coulomb Law. Coulomb’s law quantifies the electrostatic force between two point charges in a vacuum, expressed as
Theoretical Context
\[
Mathematical Description
\mathbf{F}=k_{\!\mathrm{e}}\frac{q_{1}q_{2}}{r^{2}}\hat{\mathbf r},
\]
where \(q_{1}\) and \(q_{2}\) are the charges, \(r\) is the separation distance, \(\hat{\mathbf r}\) is the unit vector from one charge to the other, and \(k_{\!\mathrm{e}}\!=\!1/(4\pi\epsilon_{\!0})\) is the Coulomb constant. In SI units this constant equals approximately \(8.9875517873681764\times10^{9}\,\text{N·m}^2\text{/C}^2\). The law predicts a repulsive force for like charges and an attractive force for unlike charges, and it implies that the force is radially directed along the line connecting the charges.
The inverse‑square dependence of Coulomb’s law reflects the isotropic expansion of electric field lines from a point source, and it is mathematically equivalent to Gauss’s law for electrostatics. In media other than vacuum, the constant \(k_{\!\mathrm{e}}\) is modified by the relative permittivity \(\epsilon_{\!\mathrm{r}}\) of the material, giving \(k_{\!\mathrm{e}}=\frac{1}{4\pi\epsilon_{0}\epsilon_{\!\mathrm{r}}}\). Experimental verification—including measurements of forces between charges, electron beams, and optical trapping—confirms the law’s validity across a wide range of scales. In dynamic situations involving relativistic velocities, the simple form of Coulomb’s law gives way to the full retarded potential description provided by Maxwell’s equations. Nonetheless, Coulomb’s law remains an essential foundational principle in fields ranging from atomic physics and chemistry to electrical engineering and plasma physics.
Superposition of Electric Forces
When more than two charges are present, the net force on any one charge is the vector sum of the individual Coulomb forces from every other charge. This is the superposition principle. For n charges, the force on charge q₁ due to charges q₂, q₃, ..., qₙ is:
F₁ = Σᵢ kq₁qᵢ/rᵢ² r̂ᵢ
where r̂ᵢ is the unit vector pointing from charge qᵢ to charge q₁. Superposition makes it possible to calculate the force from continuous charge distributions by integrating Coulomb's law over the distribution.
From Coulomb's Law to Gauss's Law
Coulomb's law is the electrostatic equivalent of Newton's law of gravitation. The inverse-square relationship means that the total electric flux through any closed surface equals the enclosed charge divided by ε₀ — this is Gauss's law in integral form: ∮E·dA = Q_enc/ε₀. Gauss's law is one of Maxwell's four equations and is more general than Coulomb's law because it applies even when charges are in motion.
For high-symmetry configurations (spherical, cylindrical, planar), Gauss's law is much easier to use than direct integration of Coulomb's law. For a spherically symmetric charge distribution, the field outside is identical to that of a point charge at the centre.
Experimental Limits on the Inverse-Square Law
The inverse-square law has been tested to extraordinary precision. Modern experiments constrain any deviation from r⁻² to less than 10⁻¹⁶ times the separation — confirming that the photon's rest mass is consistent with zero to high accuracy. Any non-zero photon mass would cause Coulomb's law to deviate, giving a Yukawa-type potential e^(−r/λ)/r instead of 1/r.
Worked Example: Force Between Two Charges
Two point charges q₁ = +3 μC and q₂ = −5 μC are separated by 0.2 m. Find the magnitude and direction of the force.
F = k|q₁||q₂|/r² = 8.99×10⁹ × 3×10⁻⁶ × 5×10⁻⁶ / 0.04 = 8.99×10⁹ × 15×10⁻¹² / 0.04 ≈ 3.37 N. The force is attractive (opposite charges) along the line joining the charges.
Dielectric Materials and Coulomb's Law
In a medium with permittivity ε (rather than vacuum), Coulomb's law becomes F = q₁q₂/(4πεr²) = kq₁q₂/(εᵣr²), where εᵣ = ε/ε₀ is the relative permittivity (dielectric constant). Water has εᵣ ≈ 80 — the electric force between ions in water is 80 times weaker than in vacuum, which is why salts dissolve readily: the reduced electrostatic force no longer holds the crystal together. This principle underlies electrolytic solutions, membrane potentials in biology, and capacitor design.
The concept extends to polarisation: in a dielectric, external electric fields induce dipole moments in the material, partially shielding the original field. The bound surface charge that appears is described by the polarisation P, leading to the displacement field D = ε₀E + P = εE.
Key Equations Summary
Coulomb's force: F = kq₁q₂/r² — magnitude of electrostatic force between point charges. Coulomb's constant: k = 1/(4πε₀) = 8.99×10⁹ N·m²/C². In a medium: F = kq₁q₂/(εᵣr²) where εᵣ is the relative permittivity. Electric field from a point charge: E = kQ/r², directed radially. Superposition: The net force or field from multiple charges is the vector sum of individual contributions. All of these are macroscopic consequences of quantum electrodynamics, which provides the deeper description of electrostatic interactions via photon exchange between charged particles.
References and further reading
- Griffiths, D. J. Introduction to Electrodynamics, 4th ed. Cambridge University Press, 2017.
- Jackson, J. D. Classical Electrodynamics, 3rd ed. Wiley, 1998.