Study Guide

Coulomb's Law

CIE A-Level PhysicsΒ· 21.2 Coulomb's lawΒ· 20 min read

1. Statement of Coulomb's Lawβ˜…β˜…β˜†β˜†β˜†β± 5 min

πŸ“˜ Definition

Coulomb's Law

F=14πΡ0Q1Q2r2F = \frac{1}{4\pi\varepsilon_0} \frac{Q_1 Q_2}{r^2}

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.

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 is approximated as N m² C⁻² for CIE exam calculations.

πŸ“ Worked Example

What is the magnitude of the force between two point charges of + C and + C separated by 0.50 m?

  1. 1

    State all known values in SI units:

  2. 2
    Q1=1.0Γ—10βˆ’6 C,Q2=2.0Γ—10βˆ’6 C,r=0.50 m,14πΡ0=9.0Γ—109 N m2 Cβˆ’2Q_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. 3

    Substitute into Coulomb's law:

  4. 4
    F=9.0Γ—109Γ—(1.0Γ—10βˆ’6)(2.0Γ—10βˆ’6)(0.50)2F = 9.0 \times 10^9 \times \frac{(1.0 \times 10^{-6})(2.0 \times 10^{-6})}{(0.50)^2}
  5. 5

    Calculate the final magnitude:

  6. 6
    F=0.072 NF = 0.072 \text{ N}

2. Calculating Force Between Two Point Chargesβ˜…β˜…β˜…β˜†β˜†β± 7 min

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).

πŸ“ Worked Example

A point charge of - C is placed 15 cm from a point charge of + C. State the magnitude and nature of the force between them.

  1. 1

    Convert distance to SI units (metres):

  2. 2
    r=15 cm=0.15 mr = 15 \text{ cm} = 0.15 \text{ m}
  3. 3

    Calculate magnitude using absolute values of charge:

  4. 4
    F=9.0Γ—109Γ—(3.0Γ—10βˆ’9)(2.0Γ—10βˆ’9)(0.15)2=2.4Γ—10βˆ’6 NF = 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. 5

    Determine the nature of the force:

  6. 6

    One charge is positive and the other negative, so the force is attractive.

3. Resultant Force for Multiple Point Chargesβ˜…β˜…β˜…β˜…β˜†β± 8 min

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.

πŸ“˜ Definition

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.

πŸ“ Worked Example

Three charges are on the x-axis: +2 ΞΌC at , +3 ΞΌC at m, -4 ΞΌC at m. Find the resultant force on the +2 ΞΌC charge at .

  1. 1

    Calculate force from +3 ΞΌC on +2 ΞΌC:

  2. 2
    F32=9.0Γ—109Γ—(3Γ—10βˆ’6)(2Γ—10βˆ’6)22=1.35Γ—10βˆ’2 NF_{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. 3

    Like charges repel, so F₃₂ acts left (negative x-direction).

  4. 4

    Calculate force from -4 ΞΌC on +2 ΞΌC:

  5. 5
    F42=9.0Γ—109Γ—(4Γ—10βˆ’6)(2Γ—10βˆ’6)32=8.0Γ—10βˆ’3 NF_{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. 6

    Opposite charges attract, so Fβ‚„β‚‚ acts right (positive x-direction).

  7. 7

    Add vectors (positive x = right):

  8. 8
    Fresultant=βˆ’1.35Γ—10βˆ’2+8.0Γ—10βˆ’3=βˆ’5.5Γ—10βˆ’3 NF_{resultant} = -1.35 \times 10^{-2} + 8.0 \times 10^{-3} = -5.5 \times 10^{-3} \text{ N}
  9. 9

    Resultant force has magnitude N, acting left (negative x-direction).

Exam tip:

Always draw a labelled diagram of the charge arrangement to check force directions before adding vectors.

4. Comparison of Electrostatic and Gravitational Forceβ˜…β˜…β˜†β˜†β˜†β± 5 min

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

(permittivity)

(gravitational constant)

πŸ“ Worked Example

Compare the magnitude of electrostatic repulsion and gravitational attraction between two protons separated by m. kg, C, N m² kg⁻².

  1. 1

    Calculate electrostatic force:

  2. 2
    Fe=9.0Γ—109(1.6Γ—10βˆ’19)2(1.0Γ—10βˆ’15)2β‰ˆ230 NF_e = 9.0 \times 10^9 \frac{(1.6 \times 10^{-19})^2}{(1.0 \times 10^{-15})^2} \approx 230 \text{ N}
  3. 3

    Calculate gravitational force:

  4. 4
    Fg=6.67Γ—10βˆ’11(1.67Γ—10βˆ’27)2(1.0Γ—10βˆ’15)2β‰ˆ1.9Γ—10βˆ’34 NF_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. 5

    Find the ratio:

  6. 6
    FeFgβ‰ˆ1.2Γ—1036\frac{F_e}{F_g} \approx 1.2 \times 10^{36}
  7. 7

    Electrostatic force is approximately times stronger than gravity for this system.

5. Common Pitfalls

Wrong move:

Forgetting to convert distance from centimetres to metres before substitution.

Why:

The constant uses SI units, so r must be in metres to get the correct force magnitude.

Correct move:

Always check all quantities are in SI units: charge in coulombs, distance in metres, force will be in newtons.

Wrong move:

Adding only magnitudes of forces for multiple charge problems.

Why:

Electrostatic force is a vector, so direction matters when calculating resultant force.

Correct move:

Find the direction of each individual force first, then add them as vectors.

Wrong move:

Using distance between surfaces of charged spheres instead of distance between centres.

Why:

Uniformly charged spheres behave as point charges at their centres, so r is distance between centres.

Correct move:

For any spherical charge, treat the charge as concentrated at the centre for Coulomb's law calculations.

Wrong move:

Claims force halves when distance doubles for an inverse square law.

Why:

Inverse square means , so doubling r quarters F, not halves.

Correct move:

Remember the inverse square relationship: force changes with the square of the reciprocal of distance.

Wrong move:

Mixing up direction because of charge sign errors.

Why:

It is easy to confuse attraction and repulsion when working with negative charges.

Correct move:

Always confirm force direction by checking if charges are like or opposite, regardless of magnitude calculation.

6. Quick Reference Cheatsheet

Concept

Key Formula / Fact

Coulomb's Law (magnitude)

Approximate constant

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 β†’ force = original magnitude

vs Gravity

Inverse square for both; electrostatics can repel, much stronger

When this came up on past exams

AI-estimated based on syllabus patterns β€” cross-check with official past papers for accuracy. Use only as revision-focus signals.

  • 2022 Β· 22

    Force between two protons

  • 2023 Β· 21

    Resultant force on three collinear charges

  • 2021 Β· 12

    Compare gravity and Coulomb force

Going deeper

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.