# Coulomb's Law

> CIE A-Level Physics · 9702 A2
> Source: https://www.owlsprep.com/study/cie-9702-u21-coulomb-s-law/

This module covers Coulomb's law, which describes the electrostatic force between two point charges. You will learn to apply the law to single and multiple charge problems and compare electrostatic and gravitational forces.

**Prerequisites:** [Electric charge and field fundamentals](https://www.owlsprep.com/study/cie-9702-u21-electric-charge-fields/); Vector addition of forces

## Learning objectives

- State Coulomb's law for the force between two point charges
- Calculate electrostatic force between point charges using Coulomb's law
- Find resultant force on a charge from multiple point charges via vector addition
- Compare electrostatic and gravitational forces

## Statement of Coulomb's Law

**Coulomb's Law** — The magnitude of the electrostatic force between two point charges is directly proportional to the product of the charges, and inversely proportional to the square of the distance between their centres. Force is repulsive for like charges and attractive for opposite charges.

*Notation:* F = \frac{1}{4\pi\varepsilon_0} \frac{Q_1 Q_2}{r^2}

*Example:* Two positive charges 1 m apart repel one another

The inverse square relationship means if the distance between two charges doubles, the force decreases to one-quarter of its original value. The constant $\frac{1}{4\pi\varepsilon_0}$ is approximated as $9.0 \times 10^9$ N m² C⁻² for CIE exam calculations.

> **tip**
>
> Force is a vector quantity, so it always has both magnitude and direction.

**Worked example:** What is the magnitude of the force between two point charges of +$1.0 \times 10^{-6}$ C and +$2.0 \times 10^{-6}$ C separated by 0.50 m?

1. State all known values in SI units:
2. $$Q_1 = 1.0 \times 10^{-6} \text{ C}, Q_2 = 2.0 \times 10^{-6} \text{ C}, r = 0.50 \text{ m}, \frac{1}{4\pi\varepsilon_0} = 9.0 \times 10^9 \text{ N m}^2 \text{ C}^{-2}$$
3. Substitute into Coulomb's law:
4. $$F = 9.0 \times 10^9 \times \frac{(1.0 \times 10^{-6})(2.0 \times 10^{-6})}{(0.50)^2}$$
5. Calculate the final magnitude:
6. $$F = 0.072 \text{ N}$$

## Calculating Force Between Two Point Charges

The sign of charges tells you if the force is attractive or repulsive. For magnitude calculations, you can use absolute values of charge. CIE exams always expect answers in SI units (newtons for force).

> **note**
>
> Always check if the question asks for just magnitude or both magnitude and direction of the force.

**Worked example:** A point charge of -$3.0 \times 10^{-9}$ C is placed 15 cm from a point charge of +$2.0 \times 10^{-9}$ C. State the magnitude and nature of the force between them.

1. Convert distance to SI units (metres):
2. $$r = 15 \text{ cm} = 0.15 \text{ m}$$
3. Calculate magnitude using absolute values of charge:
4. $$F = 9.0 \times 10^9 \times \frac{(3.0 \times 10^{-9})(2.0 \times 10^{-9})}{(0.15)^2} = 2.4 \times 10^{-6} \text{ N}$$
5. Determine the nature of the force:
6. One charge is positive and the other negative, so the force is attractive.

## Resultant Force for Multiple Point Charges

When multiple charges act on a single charge, we use the principle of superposition: the total resultant force is the vector sum of the individual forces from each charge acting separately.

**Principle of Superposition** — The total force on a charge is equal to the vector sum of all individual forces exerted by each other charge, calculated separately.

**Worked example:** Three charges are on the x-axis: +2 μC at $x=0$, +3 μC at $x=2$ m, -4 μC at $x=3$ m. Find the resultant force on the +2 μC charge at $x=0$.

1. Calculate force from +3 μC on +2 μC:
2. $$F_{32} = 9.0 \times 10^9 \times \frac{(3 \times 10^{-6})(2 \times 10^{-6})}{2^2} = 1.35 \times 10^{-2} \text{ N}$$
3. Like charges repel, so F₃₂ acts left (negative x-direction).
4. Calculate force from -4 μC on +2 μC:
5. $$F_{42} = 9.0 \times 10^9 \times \frac{(4 \times 10^{-6})(2 \times 10^{-6})}{3^2} = 8.0 \times 10^{-3} \text{ N}$$
6. Opposite charges attract, so F₄₂ acts right (positive x-direction).
7. Add vectors (positive x = right):
8. $$F_{resultant} = -1.35 \times 10^{-2} + 8.0 \times 10^{-3} = -5.5 \times 10^{-3} \text{ N}$$
9. Resultant force has magnitude $5.5 \times 10^{-3}$ N, acting left (negative x-direction).

> **Exam tip:** Always draw a labelled diagram of the charge arrangement to check force directions before adding vectors.

## Comparison of Electrostatic and Gravitational Force

Both Coulomb's law and Newton's law of gravitation follow the inverse square law, but they have key differences that are frequently tested in CIE exams:

| Property | Electrostatic Force | Gravitational Force |
| --- | --- | --- |
| Depends on | Charge | Mass |
| Direction | Attractive OR repulsive | Only attractive |
| Relative strength | Much stronger for small particles | Much weaker |
| Constant | $\varepsilon_0$ (permittivity) | $G$ (gravitational constant) |

**Worked example:** Compare the magnitude of electrostatic repulsion and gravitational attraction between two protons separated by $1.0 \times 10^{-15}$ m. $m_p = 1.67 \times 10^{-27}$ kg, $e = 1.6 \times 10^{-19}$ C, $G = 6.67 \times 10^{-11}$ N m² kg⁻².

1. Calculate electrostatic force:
2. $$F_e = 9.0 \times 10^9 \frac{(1.6 \times 10^{-19})^2}{(1.0 \times 10^{-15})^2} \approx 230 \text{ N}$$
3. Calculate gravitational force:
4. $$F_g = 6.67 \times 10^{-11} \frac{(1.67 \times 10^{-27})^2}{(1.0 \times 10^{-15})^2} \approx 1.9 \times 10^{-34} \text{ N}$$
5. Find the ratio:
6. $$\frac{F_e}{F_g} \approx 1.2 \times 10^{36}$$
7. Electrostatic force is approximately $10^{36}$ times stronger than gravity for this system.

## Common pitfalls

- **Wrong:** Forgetting to convert distance from centimetres to metres before substitution.
  - Why it fails: The constant $9 \times 10^9$ uses SI units, so r must be in metres to get the correct force magnitude.
  - Correct: Always check all quantities are in SI units: charge in coulombs, distance in metres, force will be in newtons.
- **Wrong:** Adding only magnitudes of forces for multiple charge problems.
  - Why it fails: Electrostatic force is a vector, so direction matters when calculating resultant force.
  - Correct: Find the direction of each individual force first, then add them as vectors.
- **Wrong:** Using distance between surfaces of charged spheres instead of distance between centres.
  - Why it fails: Uniformly charged spheres behave as point charges at their centres, so r is distance between centres.
  - Correct: For any spherical charge, treat the charge as concentrated at the centre for Coulomb's law calculations.
- **Wrong:** Claims force halves when distance doubles for an inverse square law.
  - Why it fails: Inverse square means $F \propto 1/r^2$, so doubling r quarters F, not halves.
  - Correct: Remember the inverse square relationship: force changes with the square of the reciprocal of distance.
- **Wrong:** Mixing up direction because of charge sign errors.
  - Why it fails: It is easy to confuse attraction and repulsion when working with negative charges.
  - Correct: Always confirm force direction by checking if charges are like or opposite, regardless of magnitude calculation.

## Cheatsheet

| Concept | Key Formula / Fact |
| --- | --- |
| Coulomb's Law (magnitude) | $F = \frac{1}{4\pi\varepsilon_0} \frac{Q_1 Q_2}{r^2}$ |
| Approximate constant | $\frac{1}{4\pi\varepsilon_0} = 9.0 \times 10^9$ N m² C⁻² |
| Force direction | Like charges repel, opposite charges attract |
| Multiple charges | Resultant force = vector sum of individual forces |
| Inverse square rule | Double distance → force = $1/4$ original magnitude |
| vs Gravity | Inverse square for both; electrostatics can repel, much stronger |

## What's next

Coulomb's law is the fundamental foundation for all further topics in electric fields for CIE A-Level Physics, including electric field strength, electric potential, and capacitance. Mastering the inverse square relationship and vector addition of electrostatic forces here makes solving more complex problems significantly easier, as almost all other electric field concepts derive directly from this core law. Next, you will learn to calculate electric field strength for point charges, which relies on the exact same inverse square relationship you have covered in this module, so mastering Coulomb's law is critical for exam success.

- [Electric Field Strength](https://www.owlsprep.com/study/cie-9702-u21-electric-field-strength/)
- [Electric Potential](https://www.owlsprep.com/study/cie-9702-u21-electric-potential/)
- [Charged particle motion in E-fields](https://www.owlsprep.com/study/cie-9702-u21-charged-particle-motion-in-e/)

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