9 min readConcept sheetSection A MCQ (1 mark); Section B very short answer (2 marks) or short answer (3 marks) numerical on the force between charges; also a NEET favourite

Coulomb's law and the superposition principle

The force between two point charges is proportional to q₁q₂ and inversely proportional to r²; with many charges, add the forces as vectors.

Coulomb's lawF = kq₁q₂/r², with k = 1/4πε₀ = 9 × 10⁹ N m² C⁻² in free space; r = distance between the charges

Permittivity of free spaceε₀ = 8.85 × 10⁻¹² C² N⁻¹ m⁻²

In a mediumF(vacuum)/F(medium) = ε/ε₀ = εᵣ, the relative permittivity (dielectric constant); always more than 1

SuperpositionF = F₁₂ + F₁₃ + F₁₄ + ..., the vector sum of the forces from each other charge

The law in words

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 force acts along the line joining the charges. It is repulsive for like charges and attractive for unlike charges. A point charge is a charged body whose size is very small compared with the distance between the charges. Coulomb found the law in 1785, using a torsion balance. In SI units, F = (1/4πε₀) q₁q₂/r². Here ε₀ is the permittivity of free space. Like gravity, it is an inverse square law, and it holds only for point charges. One coulomb is the charge that repels an equal charge 1 m away in vacuum with a force of 9 × 10⁹ N.

Coulomb's law in vector form

Two point charges q₁ and q₂ are a distance r apart. Let **r̂₁₂** be the unit vector pointing from q₁ towards q₂.

1
Force on q₂ due to q₁: F₂₁ = k(q₁q₂/r²) r̂₁₂, with k = 1/4πε₀.
2
For like charges, q₁q₂ is positive, so F₂₁ points along r̂₁₂, away from q₁: the force is a repulsion.
3
For unlike charges, q₁q₂ is negative, so F₂₁ points back towards q₁: the force is an attraction.
4
Force on q₁ due to q₂: F₁₂ = k(q₁q₂/r²) r̂₂₁, and r̂₂₁ = −r̂₁₂.
5★
So, F₁₂ = −F₂₁: the two forces are equal and opposite, as Newton's third law requires.

★ key step

The superposition principle

When many charges act on one charge, the force between any two of them is not changed by the others. So we find each force separately with Coulomb's law. The net force is the vector sum of these forces. This is the superposition principle: F = F₁₂ + F₁₃ + F₁₄ + ... for the force on q₁. Forces must be added as vectors, by the parallelogram law or by components. For two forces F₁ and F₂ with an angle θ between them, the resultant is F = √(F₁² + F₂² + 2F₁F₂ cos θ).

Worked example: force between two charges

Problem

Two point charges, +2 μC and −3 μC, are 30 cm apart in air. Find the force between them.

Working

Given: q₁ = 2 × 10⁻⁶ C, q₂ = 3 × 10⁻⁶ C (sizes only), r = 0.30 m.
Formula: F = kq₁q₂/r², with k = 9 × 10⁹ N m² C⁻².
F = 9 × 10⁹ × (2 × 10⁻⁶ × 3 × 10⁻⁶) ÷ (0.30)².
F = 9 × 10⁹ × 6 × 10⁻¹² ÷ 0.09 = 0.054 ÷ 0.09 = 0.6 N.
The charges are unlike, so the force is attractive.

Answer

0.6 N, attractive

Worked example: where is the net force zero?

Problem

A charge of +4 μC is at A and a charge of +1 μC is at B, 30 cm away. Where should a third charge be placed so that the net force on it is zero?

Working

The two charges are alike, so the point lies between A and B, nearer the smaller charge.
Let the point be x from A. The two forces must be equal: k(4 μC)q/x² = k(1 μC)q/(0.30 − x)².
So 4/x² = 1/(0.30 − x)². Take square roots: 2/x = 1/(0.30 − x).
2(0.30 − x) = x, so 0.60 = 3x and x = 0.20 m.

Answer

20 cm from A (10 cm from B), on the line between them; the answer does not depend on the size or sign of the third charge

Coulomb force vs gravitational force

Electrostatic (Coulomb) forceActs between charges. Can attract or repel. Depends on the medium between the charges. Very strong: between two protons it is about 10³⁶ times their gravitational pull.

Gravitational forceActs between masses. Always attracts. Does not depend on the medium. Very weak, so it matters only for huge masses such as planets.

Both are inverse-square forces, both act along the line joining the bodies (central forces), and both are conservative.

In real life

Laser printers and photocopiers use Coulomb attraction to pull charged toner powder onto the paper. Dust clings to a TV screen and to plastic fan blades for the same reason.

Exam angle

1 marks: “How does the force between two point charges change if the distance between them is halved?” F ∝ 1/r². Halving r makes the force 2² = 4 times larger.

2 marks: “Two charges are moved from air into oil of relative permittivity 2.5, keeping the same distance. What happens to the force between them?” F(medium) = F(vacuum)/εᵣ, so the force becomes 1/2.5 = 0.4 times its value in air.

3 marks: “State Coulomb's law. Two equal point charges 0.50 m apart in air repel each other with a force of 0.9 N. Find the charge on each.” Law: F = kq₁q₂/r², along the line joining the charges. q² = Fr²/k = 0.9 × 0.25 ÷ (9 × 10⁹) = 2.5 × 10⁻¹¹ C². So q = 5 × 10⁻⁶ C = 5 μC each.

Remember it

Product on top, distance squared below: F = kq₁q₂/r², with k = 9 × 10⁹. Double r, quarter F.