Q.Two point charges $+2\ \mu\text{C}$ and $+3\ \mu\text{C}$ are placed $0.3\ \text{m}$ apart in air. Calculate the force between them.

Coulomb's Law is the single quantitative statement of how two point charges push or pull on each other. Two charges q₁ and q₂ separated by a distance r exert on each other a force of magnitude

F = k · q₁q₂ / r², where k = 1/(4πε₀) ≈ 9 × 10⁹ N·m²·C⁻².

Three ideas live inside that one line, and almost every JEE Main question is really testing one of them.

1 — It is an inverse-square law. The force falls off as 1/r², not 1/r. Double the separation and the force drops to a quarter; halve it and the force quadruples. This is why a graph of F against r is a hyperbola, while F against 1/r² is a straight line through the origin whose slope is k q₁q₂.

2 — It is a vector, and it obeys superposition. The force between two charges points along the line joining them: repulsive for like signs, attractive for unlike signs. When several charges act on one, the net force is the vector sum of the individual Coulomb forces — each computed as if the others were absent. This is the master key to every triangle, square, and collinear-charge problem: never add magnitudes blindly; resolve into components or use the resultant law R = √(F₁² + F₂² + 2F₁F₂cosθ).

3 — Charge is quantized and conserved. Any charge is an integer multiple of the electronic charge, q = ne with e = 1.6 × 10⁻¹⁹ C. When two identical conductors touch, their charge redistributes so each carries the algebraic mean (q₁ + q₂)/2 — signs included. Forgetting the sign here (averaging magnitudes instead) is the most common single mistake in the whole chapter.

The role of the medium. In a medium of dielectric constant K, the force is reduced: F_medium = F_vacuum / K. Equivalently, a separation r in a medium behaves like a larger separation r√K in vacuum.

Why the constant looks the way it does. Writing k = 1/(4πε₀) rather than a bare constant builds in the 4π of spherical geometry, so that later results (Gauss's law, the field of a point charge) come out clean. ε₀, the permittivity of free space, carries the dimensions [M⁻¹L⁻³T⁴A²].

How Coulomb's Law is examined. Beyond direct substitution, it anchors equilibrium problems (a third charge placed for zero net force, charged balls hanging on threads, a charge levitated against gravity), null-point problems (where a test charge feels nothing), optimisation (splitting a charge as Q/2 to maximise the mutual force), and even dynamics (the initial acceleration a = F/m of a released charge, or small oscillations about a symmetric equilibrium). In every case the physics is this one law; the skill is reading which of the three ideas above the problem is probing, and keeping the signs and the r² honest.