Coulomb's Law in Vector Form — Derivation and Explanation

Scalar form (magnitude only): F = k|q₁||q₂|/r²

Vector form: F₁₂ = k(q₁q₂/r²) r̂₁₂

Where:

• F₁₂ = force on charge 1 due to charge 2 (vector)
• q₁, q₂ = charges (with sign, in coulombs)
• r = distance between charges (in metres)
• r̂₁₂ = unit vector pointing from charge 2 to charge 1
• k = 1/(4πε₀) = 9 × 10⁹ N·m²/C²
• ε₀ = 8.854 × 10⁻¹² C²/(N·m²) (permittivity of free space)

Alternative notation: F₁₂ = (1/4πε₀) × (q₁q₂/r³) × r⃗₁₂

where r⃗₁₂ = r⃗₁ − r⃗₂ (position vector from 2 to 1)

The vector form automatically gives the correct direction:

1. Like charges (same sign, q₁q₂ > 0): F₁₂ = positive × r̂₁₂ → Force is along r̂₁₂ (away from charge 2) → Repulsion ✓

2. Unlike charges (opposite sign, q₁q₂ < 0): F₁₂ = negative × r̂₁₂ → Force is opposite to r̂₁₂ (towards charge 2) → Attraction ✓

Newton's Third Law in vector form: F₁₂ = −F₂₁

The forces are equal in magnitude, opposite in direction.

Superposition Principle: For multiple charges, the total force on charge q₁ is: F₁ = F₁₂ + F₁₃ + F₁₄ + ... (vector sum)

Coulomb's constant: k = 1/(4πε₀) = 9 × 10⁹ N·m²/C²

Permittivity of free space: ε₀ = 8.854 × 10⁻¹² C²/(N·m²) = 8.854 × 10⁻¹² F/m

In a medium of relative permittivity εᵣ: F = kq₁q₂/(εᵣ r²)

For water (εᵣ ≈ 80): Force reduced to 1/80th of the value in vacuum.

Limitations of Coulomb's Law:

1. Valid only for point charges or spherically symmetric charge distributions.
2. Charges must be stationary (or moving slowly).
3. Valid in vacuum/air (medium effect can be included via εᵣ).