Coulomb's law: force between two point charges
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 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 measured these forces with a torsion balance. In SI units, F = (1/4πε₀) q₁q₂/r². Here ε₀ is the permittivity of free space. 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
Let q₁ and q₂ be at position vectors **r₁** and **r₂** from an origin O.
- 1The vector from q₁ to q₂ is r₂₁ = r₂ − r₁. Its unit vector is r̂₂₁ = r₂₁/r₂₁.
- 2Force on q₂ due to q₁: F₂₁ = (1/4πε₀)(q₁q₂/r₂₁²) r̂₂₁.
- 3Force on q₁ due to q₂: F₁₂ = (1/4πε₀)(q₁q₂/r₁₂²) r̂₁₂.
- 4Since r̂₁₂ = −r̂₂₁ and r₁₂ = r₂₁, we get F₁₂ = −F₂₁.
- 5★So, the two forces are equal and opposite: Coulomb's law obeys Newton's third law. The same formula works for like and unlike charges; the sign of q₁q₂ sets the direction.
★ key step
Coulomb force vs gravitational force
Both are inverse-square forces, both act along the line joining the bodies (central forces), and both are conservative.
Worked example: force between two charges
Two point charges, +2 μC and −3 μC, are 30 cm apart in air. Find the force between them.
- 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.
0.6 N, attractive
Worked example: spheres touched and separated
Two identical small metal spheres carry +6 μC and −2 μC and are 10 cm apart in air. (a) Find the force between them. (b) They are touched together and put back 10 cm apart. Find the new force.
- (a) F = 9 × 10⁹ × (6 × 10⁻⁶ × 2 × 10⁻⁶) ÷ (0.10)².
- F = 9 × 10⁹ × 12 × 10⁻¹² ÷ 0.01 = 0.108 ÷ 0.01 = 10.8 N, attractive.
- (b) On touching, the total charge (+6 − 2 = +4 μC) is shared equally by the identical spheres: +2 μC each.
- F′ = 9 × 10⁹ × (2 × 10⁻⁶)² ÷ 0.01 = 0.036 ÷ 0.01 = 3.6 N, repulsive.
(a) 10.8 N, attractive; (b) 3.6 N, repulsive (one third of the first force)
Effect of the medium
- In a medium of dielectric constant K: F = (1/4πε₀K) q₁q₂/r².
- The force becomes K times smaller. For water K is about 80, so the force falls about 80 times.
- For air K is almost exactly 1, so air and vacuum give nearly the same force.
Dust clings to a TV screen and to plastic fan blades because they get charged and attract dust by the Coulomb force. Laser printers and photocopiers use the same attraction to pull toner powder onto the paper.
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: “Write Coulomb's law in vector form and show that it obeys Newton's third law.” F₂₁ = (1/4πε₀)(q₁q₂/r₂₁²) r̂₂₁. Since r̂₁₂ = −r̂₂₁, F₁₂ = −F₂₁: the forces are equal and opposite.
3 marks: “Compare the electrostatic force and the gravitational force under three heads.” (1) Electrostatic force can attract or repel; gravity only attracts. (2) Electrostatic force depends on the medium; gravity does not. (3) Electrostatic force is far stronger (about 2.4 × 10³⁹ times for an electron and a proton). Both are inverse-square, central forces.
Product on top, distance squared below: F = kq₁q₂/r², with k = 9 × 10⁹. Double r, quarter F.