Coulomb’s Law and Quantization of Charge
Coulomb’s law states that the electric 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, F = k|q1q2| ÷ r2, with k = 8.99 × 109 N·m2/C2; electric charge is quantized in integer multiples of the elementary charge e = 1.602 × 10−19 C. In brief: like charges repel and unlike charges attract, along the line joining them. The force on each charge is equal in size and opposite in direction. The constant k = 1 ÷ 4πε0, where ε0 = 8.85 × 10−12 C2/(N·m2) is the permittivity of free space. The electric force in an atom is about 1039 times stronger than the gravitational force between the particles.
Coulomb’s law is the electrical counterpart of Newton’s law of gravitation, and it explains the structure of atoms, molecules and materials. This guide states the law and the quantization of charge, compares electric and gravitational forces, shows how to add forces from several charges, works through numerical examples and corrects common mistakes.
What does Coulomb’s law say?
For two point charges q1 and q2 separated by a distance r in a vacuum (or in air), the magnitude of the force on each is
F = k |q1 q2| ÷ r2, with k = 1 ÷ (4πε0) = 8.99 × 109 N·m2/C2.
The force acts along the line joining the charges: repulsive if the charges have the same sign, attractive if opposite. By Newton’s third law, the forces on the two charges are equal and opposite. As with gravity, the force follows an inverse-square law: doubling the distance makes the force one quarter. In a material medium, the force is reduced by the relative permittivity εr of the medium (for water, about 80), which is why salts dissolve in water: the attraction between the ions is greatly weakened. A uniformly charged sphere acts, outside it, like a point charge at its center.
What is quantization of charge?
All observed free charges are integer multiples of the elementary charge: q = ne, with e = 1.602 × 10−19 C. The electron has charge −e, and the proton +e. Robert Millikan’s oil-drop experiment (1909) showed this, by measuring the charges on tiny oil drops suspended in an electric field: every charge was a multiple of the same smallest value. The quarks inside protons and neutrons have charges of ±e/3 and ±2e/3, but they cannot exist alone. The coulomb is a very large charge: 1 C corresponds to 6.24 × 1018 elementary charges, and typical static charges are in nanocoulombs (nC) or microcoulombs (µC). Charge is also conserved: the total charge of an isolated system never changes.
How are forces from several charges added?
The forces on a charge from several other charges add as vectors (the principle of superposition). The force of one charge on another is not changed by the presence of other charges. To find the net force on a charge, calculate the force from each of the others (magnitude from Coulomb’s law, direction from the signs), resolve into components, and add the components. For charges in a line, take signs for directions.
Electric and gravitational forces compared
| Feature | Electric force | Gravitational force |
|---|---|---|
| Formula | kq1q2 ÷ r2 | Gm1m2 ÷ r2 |
| Acts between | Charges | Masses |
| Attractive or repulsive | Both | Attractive only |
| Relative strength | Very strong | Very weak (about 10−39 as strong between an electron and a proton) |
| Can be shielded? | Yes (by conductors) | No |
Worked example: Coulomb force calculations
- Two charges. q1 = +2.0 µC and q2 = −3.0 µC, 0.30 m apart. F = 8.99 × 109 × (2.0 × 10−6)(3.0 × 10−6) ÷ 0.302 = 8.99 × 109 × 6.0 × 10−12 ÷ 0.090 = 0.60 N, attractive.
- Doubling the distance. At 0.60 m the force is 0.60 ÷ 4 = 0.15 N.
- Hydrogen atom. The electron is at a distance of 5.29 × 10−11 m from the proton. F = 8.99 × 109 × (1.602 × 10−19)2 ÷ (5.29 × 10−11)2 = 8.2 × 10−8 N. The gravitational force between them is G memp ÷ r2 = 3.6 × 10−47 N. The ratio is 2.3 × 1039.
- Three charges in a line. +1.0 µC at x = 0, +2.0 µC at x = 0.10 m, and −1.0 µC at x = 0.30 m. Force on the middle charge: from the first, repulsion to the right: 8.99 × 109 × 1.0 × 10−6 × 2.0 × 10−6 ÷ 0.102 = 1.80 N (to the right). From the third (0.20 m away), attraction to the right: 8.99 × 109 × 2.0 × 10−6 × 1.0 × 10−6 ÷ 0.202 = 0.45 N (to the right). Net force = 1.80 + 0.45 = 2.25 N to the right.
- Number of elementary charges. A charge of 4.8 × 10−9 C: n = 4.8 × 10−9 ÷ 1.602 × 10−19 = 3.0 × 1010 electrons.
How was the law discovered?
Charles-Augustin de Coulomb measured the force between charged spheres in 1785 with a torsion balance, finding the inverse-square dependence. Henry Cavendish had found the same result privately in the 1770s, but his work was unpublished until 1879. Joseph Priestley suggested the inverse-square law in 1767 from the fact that there is no force inside a charged hollow conductor. The modern form with ε0 is part of the SI system, and the law holds in quantum electrodynamics at distances down to about 10−18 m.
Which mistakes are common?
Mistake 1: Putting negative signs inside the magnitude
Use the absolute values of the charges in the formula for the magnitude, and decide the direction from the signs.
Mistake 2: Forgetting to convert micro- and nano-coulombs
1 µC = 10−6 C and 1 nC = 10−9 C.
Mistake 3: Adding forces as scalars
Forces from several charges are vectors. Use components.
Mistake 4: Thinking that the larger charge exerts the larger force
The forces on the two charges are equal in magnitude, whatever their sizes.