📘 Chapter 3 : Coulomb’s Law – Force Between Two Point Charges (Class XII)

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🔷 1. Introduction

The study of electrostatics begins with the fundamental interaction between electric charges. The force between two stationary point charges is governed by a law known as Coulomb’s Law, named after the French physicist Charles-Augustin de Coulomb.

This law is analogous to Newton’s law of gravitation, but unlike gravity, the electric force can be attractive or repulsive, depending on the nature of charges.

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🔷 2. Statement of Coulomb’s Law

The force of attraction or repulsion between two stationary point charges is:

directly proportional to the product of magnitudes of charges and inversely proportional to the square of the distance between them.

Mathematically:

F ∝ (q₁ q₂) / r²

F = k (q₁ q₂ / r²)

Where:

• F = electrostatic force
• q₁, q₂ = point charges
• r = distance between charges
• k = 1 / (4πε₀) = 9 × 10⁹ Nm²/C² (in vacuum)

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🔷 3. Vector Form of Coulomb’s Law

Force is a vector quantity. Hence direction is important.

F⃗₁₂ = (1/4πε₀) (q₁ q₂ / r²) r̂₁₂

Where r̂₁₂ is the unit vector from q₁ to q₂.

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🔷 4. Nature of Electrostatic Force

• Repulsive: if q₁ and q₂ are like charges
• Attractive: if q₁ and q₂ are unlike charges
• Acts along the line joining the charges
• Follows inverse square law

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🔷 5. Medium Dependence

In a medium other than vacuum, Coulomb’s force reduces due to permittivity.

F = (1 / 4πε) (q₁ q₂ / r²)

Where:

• ε = ε₀ εᵣ
• εᵣ = dielectric constant

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📦 6. Important Results (Must Remember)

• Coulomb’s Law: F = k(q₁q₂/r²)
• Force is central: acts along line joining charges
• Inverse square nature: doubling distance reduces force to 1/4
• Superposition applies: net force is vector sum
• Medium reduces force: F ∝ 1/εᵣ

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🔷 7. Physical Interpretation

Coulomb’s law describes how electric field interactions arise at a microscopic level. It shows that electric force is not contact-based but acts through space via electric fields.

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🔷 8. Limitations of Coulomb’s Law

• Valid only for point charges or spherical symmetry
• Charges must be at rest (electrostatics)
• Breaks down at quantum scale distances

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🔷 9. Applications

• Electrostatic force calculation in atoms
• Design of capacitors
• Understanding ionic bonding
• Charge distribution analysis in conductors

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🧠 10. Solved Numerical Questions (Advanced Level)

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🔹 Q1

Two charges q₁ = 2 μC and q₂ = 3 μC are placed 0.5 m apart in vacuum. Find the force.

Solution:

F = k (q₁q₂ / r²)
= 9×10⁹ × (2×10⁻⁶ × 3×10⁻⁶) / (0.5)²
= 9×10⁹ × 6×10⁻¹² / 0.25
= 54×10⁻³ / 0.25
= 0.216 N

Answer: 0.216 N (Repulsive if same sign)

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🔹 Q2

What happens to force if distance is doubled?

Solution:

F ∝ 1/r² → r becomes 2r

F' = F / 4

Answer: Force becomes one-fourth

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🔹 Q3

If medium has dielectric constant 4, how force changes?

Solution:

F' = F / εᵣ = F / 4

Answer: Force reduces to 1/4

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🔹 Q4

Two equal charges repel with force 0.9 N at 0.3 m. Find force at 0.6 m.

Solution:

Distance doubled → force reduces by 4

F' = 0.9 / 4 = 0.225 N

Answer: 0.225 N

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🔷 11. Advanced Conceptual Insight

Coulomb’s law is not just an empirical rule; it is embedded in Maxwell’s equations and emerges from the inverse-square nature of electrostatic fields in 3D space. It is also consistent with quantum electrodynamics where force is mediated by virtual photons.

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🔷 12. Summary

Coulomb’s law forms the foundation of electrostatics. It explains how charged particles interact, how fields are generated, and how matter behaves electrically at both classical and quantum levels.

✨ End of Chapter: Coulomb’s Law ✨