Q.Two point charges of $+3\times10^{-6}\ \text{C}$ and $-5\times10^{-6}\ \text{C}$ are placed $0.4\ \text{m}$ apart in vacuum. Calculate the magnitude of the force between them and state whether it is attractive or repulsive.

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.