Coulomb's law describes the electrostatic force between two point charges. The law states: F = k|q₁q₂|/r², where F is force in Newtons, k is Coulomb's constant (8.98755×10⁹ N·m²/C²), q₁ and q₂ are charges in Coulombs, and r is distance in meters. The Coulomb constant k = 1/(4πε₀), where ε₀ ≈ 8.854×10⁻¹² F/m is the permittivity of free space. The law is inverse-square, meaning force decreases with the square of distance. Like charges (both positive or both negative) repel; opposite charges attract. Force direction is along the line connecting the charges. Charles-Augustin de Coulomb published this law in 1785 after experimental measurements. It's foundational to electromagnetism and predates Maxwell's equations by decades. In vacuum, k = 8.98755×10⁹; in materials with permittivity ε_r, effective k = k/ε_r (reduced force in denser media). Atomic scale: electron-nucleus attraction (r ~10⁻¹⁰ m) produces ~1.6×10⁻⁸ N force, vastly stronger than gravity. Macroscopic scale: 1 Coulomb is extremely large charge (rarely encountered). Typical charges: static electricity ~μC (10⁻⁶ C), capacitors store mC (10⁻³ C), lightning strikes deliver kC currents. Earth's electric field: ~100 V/m atmosphere maintained by charge separation and thunderstorms. Applications: electrostatic copiers, inkjet printers (charge ink droplets for deflection), and electrostatic precipitators (remove particles via electrostatic attraction).

Advanced concepts: Coulomb's law works for point charges and spherical conductors. For continuous charge distributions, integrate: F = ∫k dq₁dq₂/r². Superposition principle: total force is vector sum of individual pairwise forces. In conductors, charges distribute on surfaces to achieve zero internal field. Shielding: outer charges create field that cancels internal fields (Faraday cage principle). Energy considerations: potential energy U = kq₁q₂/r (negative for attractive pairs). Work to bring charges together equals ΔU. Force and field relationships: E = F/q represents field strength. From Coulomb's law, point charge field: E = kq/r². Historical accuracy: early formulations lacked k factor; modern form established by standardization of units. Modern verifications: experiments confirm inverse-square law to 99.999% accuracy over ranges from 10⁻¹⁵ m (quantum effects) to 10⁵ m (planetary scales). Quantum corrections: at atomic scales, Coulomb + quantum mechanics gives precise spectroscopy matches. Tesla coils demonstrate high-voltage Coulomb forces. Plasma physics applies Coulomb forces to interpret particle collisions and fusion reactions. Semiconductor physics uses Coulomb interaction for band structure calculations. Astrophysics: electrostatic forces negligible compared to gravity in stars but crucial for plasma dynamics.