Coulomb's Law and Superposition Principle

Two charged spheres a fixed distance apart push or pull each other — Coulomb measured exactly how hard, and his 1785 result is the quantitative heart of electrostatics.

Definition — Coulomb's law: the force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

The law and its constant

F = k·|q₁q₂| / r², where k = 1/(4πε₀) ≈ 9 × 10⁹ N·m²·C⁻².
Here ε₀ = 8.85 × 10⁻¹² C²·N⁻¹·m⁻² is the permittivity of free space. The force acts along the line joining the charges — repulsive for like charges, attractive for unlike. It obeys Newton's third law: the two charges feel equal and opposite forces.

In a medium and in vectors

Inside a medium of relative permittivity (dielectric constant) K, the force is reduced: F_medium = F_vacuum / K. In vector form the force on charge 2 due to charge 1 is F₂₁ = k·q₁q₂·r̂₂₁ / r², where r̂ points from 1 to 2.

Superposition principle

The net force on any charge is the vector sum of the forces due to every other charge taken one at a time, each computed by Coulomb's law as if the others were absent. Forces add independently — the presence of a third charge does not alter the pairwise force between the first two.

Exam Tricks & Tips

• 🎯 Inverse-square scaling: doubling r cuts the force to one quarter; halving r multiplies it by four.
• 🎯 In a medium divide by K (F drops); water with K ≈ 80 weakens the force ~80-fold.
• 🎯 Always add forces as VECTORS, not magnitudes — use components or the parallelogram/triangle rule.
• 🎯 Two identical conducting spheres sharing charge each end with (q₁ + q₂)/2 after touching — then reapply Coulomb's law.
• 🎯 Compare with gravitation: both are inverse-square, but the electrostatic force is far stronger and can attract or repel.
• ❌ Common mistake: forgetting the vector nature and simply adding the two force magnitudes when the forces are not collinear.

Expected exam pattern

A 2–3 mark numerical using F = kq₁q₂/r², a superposition problem with three charges (often at triangle vertices), or a conceptual item on the effect of a medium (K) and the inverse-square dependence.

Quick recap

F = kq₁q₂/r² with k = 9 × 10⁹, along the line of centres, obeying Newton's third law. In a medium divide by K. For many charges, take the vector sum (superposition). Force scales as 1/r² — a change in distance dominates the answer.

Cover the answer, recall, then check. 10 cards on Coulomb's law.

Q1. State Coulomb's law as a formula.
A1. F = k·|q₁q₂|/r², with k = 1/(4πε₀) ≈ 9 × 10⁹ N·m²·C⁻².

Q2. What is the value of ε₀?
A2. The permittivity of free space, 8.85 × 10⁻¹² C²·N⁻¹·m⁻².

Q3. How does the force change if the distance is doubled?
A3. It becomes one quarter — the force is inverse-square in r.

Q4. How is the force affected inside a medium of dielectric constant K?
A4. It is reduced: F_medium = F_vacuum / K.

Q5. Along what direction does the Coulomb force act?
A5. Along the straight line joining the two point charges.

Q6. Does Coulomb's force obey Newton's third law?
A6. Yes — the two charges exert equal and opposite forces on each other.

Q7. State the superposition principle.
A7. The net force on a charge is the vector sum of the Coulomb forces from all other charges taken individually.

Q8. Two identical conducting spheres with charges q₁ and q₂ touch and separate. Charge on each?
A8. (q₁ + q₂)/2 on each sphere.

Q9. Coulomb's law resembles which other law?
A9. Newton's law of gravitation — both are inverse-square, but Coulomb's is far stronger and can be attractive or repulsive.

Q10. Most common error in superposition problems?
A10. Adding force magnitudes instead of adding the forces as vectors (using components).

Two charged balloons on strings swing apart until the electric push balances gravity. Measure that push carefully and you rediscover Coulomb's law — the exact rule for the force between two charges, and the foundation of all of electrostatics.

Coulomb's law — the core idea

The electrostatic force between two point charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them, directed along the line joining them.

F = k · q₁q₂ / r²

where k = 1/(4πε₀) = 9 × 10⁹ N·m²/C² and ε₀ = 8.85 × 10⁻¹² C²/N·m² is the permittivity of free space.

Beginner: reading the equation

Double the distance and the force drops to one-quarter (inverse-square). Double one charge and the force doubles. The force is repulsive for like charges and attractive for unlike — captured by the sign of the product q₁q₂.

Intermediate: it is a vector, and Newton's third law holds

Coulomb's law gives a vector force acting along the line joining the charges. The force on charge 1 due to charge 2 is equal and opposite to the force on 2 due to 1: F₁₂ = −F₂₁. Always draw the direction from the physics (repel/attract), then use magnitudes in the formula.

Advanced: role of the medium

Inside a material of dielectric constant (relative permittivity) εr, the force weakens:

F = q₁q₂ / (4πε₀εr r²)

Water (εr ≈ 80) cuts the force to 1/80 of its vacuum value — which is exactly why salts dissociate into ions in water. Also note Coulomb's law strictly applies to point charges (or spheres, treated as if all charge sits at the centre).

The superposition principle

Real problems have many charges. The net force on any charge is the vector sum of the forces exerted by every other charge, each computed independently as if the others were absent.

F_net = F₁ + F₂ + F₃ + …

Forces add as vectors — never add magnitudes blindly. Resolve into components (x and y), sum each, then recombine.

Worked example

Two charges q₁ = +3 μC and q₂ = +6 μC are 30 cm apart in air. Find the force between them.

F = (9 × 10⁹)(3 × 10⁻⁶)(6 × 10⁻⁶) / (0.30)²
F = (9 × 10⁹)(1.8 × 10⁻¹¹) / 0.09
F = 0.162 / 0.09 = 1.8 N, repulsive (both positive).

Real-world / exam application

Coulomb's law explains why an electron stays bound to a proton in hydrogen, why crystals hold their lattice shape, and how ink-jet printers steer charged droplets. In exams it feeds directly into electric field (F = qE), potential energy, and equilibrium-of-charges problems.

Exam tricks & shortcuts

• Coulomb force and gravity share the same inverse-square form, but the electric force is ~10³⁹ times stronger.
• For three collinear charges, a charge placed between two like charges can be in equilibrium; solve F_left = F_right.
• Mnemonic "PPP" — Product over distance-squared, Point charges, Pair (Newton's third law).

Key formulas

• Coulomb's law: F = k·q₁q₂/r², where k = 1/(4πε₀) = 9 × 10⁹ N·m²/C².
• Permittivity of free space: ε₀ = 8.85 × 10⁻¹² C²/(N·m²).
• In a medium: F = q₁q₂/(4πε₀ε_r r²) (ε_r = relative permittivity).
• Superposition: net force = vector sum F⃗ = ΣF⃗ᵢ.

Like charges repel, unlike attract; the medium reduces the force by a factor ε_r.

Worked Example

Problem: Two point charges of +2 μC and +3 μC are placed 30 cm apart in air. Find the magnitude of the electrostatic force between them. (k = 9 × 10⁹ N·m²/C².)

Solution:
Step 1 — Write Coulomb's law for the force between two point charges:
F = k q₁ q₂ / r².

Step 2 — Convert to SI units:
q₁ = 2 × 10⁻⁶ C, q₂ = 3 × 10⁻⁶ C, r = 30 cm = 0.30 m.

Step 3 — Substitute:
F = (9 × 10⁹)(2 × 10⁻⁶)(3 × 10⁻⁶) / (0.30)².

Step 4 — Evaluate step by step:
Numerator = 9 × 10⁹ × 6 × 10⁻¹² = 54 × 10⁻³ = 0.054.
Denominator = 0.09.
F = 0.054 / 0.09 = 0.6 N.

Step 5 — Both charges are positive, so the force is repulsive (directed away from each other).

Answer: The force is 0.6 N, repulsive.