Coulomb's law

Coulomb's law gives the force between two point charges: F = Q₁Q₂ ÷ (4πε₀r²), often written F = kQ₁Q₂ ÷ r² with k = 1 ÷ 4πε₀ ≈ 8.99 × 10⁹ N m² C⁻². The force is proportional to the product of the charges and inversely proportional to the square of their separation r. Like charges repel and unlike charges attract. Two charges of 2.0 µC, 0.30 m apart, push each other apart with a force of about 0.40 N.

The Coulomb's law formula

AQA's A-level data booklet gives Coulomb's law in this form, with ε₀ = 8.85 × 10⁻¹² F m⁻¹. Working out 1 ÷ (4π × 8.85 × 10⁻¹²) gives the Coulomb constant, k ≈ 8.99 × 10⁹ N m² C⁻², so the law is often written:

AQA's specification adds that air can be treated as a vacuum when calculating the force between charges, and that for a charged sphere the charge can be treated as if it were at the centre.

Attraction and repulsion

Put the signs of the charges into the formula:

• Two positive or two negative charges give a positive F: the charges repel.
• One positive and one negative give a negative F: the charges attract.

The force acts along the line joining the charges, and each charge feels a force of the same size, in opposite directions, as Newton's third law requires.

The inverse square law

Because r is squared on the bottom, the force falls off quickly with distance:

Doubling both charges, on the other hand, multiplies the force by 2 × 2 = 4. This inverse square pattern is the same as Newton's law of gravitation, F = Gm₁m₂ ÷ r².

Coulomb's law worked examples

Example 1: two equal charges

Two charges of +2.0 µC are 0.30 m apart. Find the force between them.

Convert: 2.0 µC = 2.0 × 10⁻⁶ C.

Both charges are positive, so the force is repulsive.

Example 2: finding the separation

Charges of 3.0 µC and 5.0 µC repel with a force of 0.090 N. How far apart are they?

Example 3: charged spheres

Two small charged spheres carry +40 nC and −60 nC. Their centres are 10 cm apart. Find the force.

Treat each sphere's charge as if it were at its centre, so r = 0.10 m.

The charges are opposite, so the force is attractive.

Example 4: a proton and an electron

In a simple model of a hydrogen atom, a proton and an electron are 5.3 × 10⁻¹¹ m apart. Each has a charge of magnitude e = 1.60 × 10⁻¹⁹ C. Find the electrostatic force.

Electrostatic force vs gravitational force

AQA's specification asks students to compare the sizes of gravitational and electrostatic forces between subatomic particles. Using the proton and electron from Example 4, with G = 6.67 × 10⁻¹¹ N m² kg⁻², a proton mass of 1.67 × 10⁻²⁷ kg and an electron mass of 9.11 × 10⁻³¹ kg from AQA's data booklet:

The electrostatic force is about 2 × 10³⁹ times bigger. Both follow an inverse square law, so the ratio is the same at any separation. That is why gravity can be ignored inside atoms, and why the electrostatic force holds atoms together.

The Electric Force Calculator works out the force between two charges, or a missing charge or distance, with every step shown.

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Electric Force Calculator

Calculate Coulomb's law electric force

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Now try a question on charges and fields in the course, then press Mark to check it there.

From Coulomb's law to electric field strength

Electric field strength is the force per unit charge, E = F ÷ Q. Dividing Coulomb's law by the charge feeling the force gives the field around a single point charge Q, r metres away, which AQA's data booklet lists for a radial field:

For example, 0.10 m from a 2.0 µC charge, E = 8.99 × 10⁹ × 2.0 × 10⁻⁶ ÷ 0.10² = 1.8 × 10⁶ N C⁻¹. Any charge q placed there would feel a force of qE.

Common mistakes

• Forgetting to square r. The distance is squared; the charges are not.
• Prefixes. µC is × 10⁻⁶, nC is × 10⁻⁹ and pC is × 10⁻¹².
• Using the surface of a sphere. Measure r between the centres of charged spheres.
• Using diameter instead of separation. r is the distance between the two charges.
• Reading the sign wrongly. A negative F means attraction, not a smaller force.

Practise fields in a short timed paper

A-level Physics: Year 2 gives you a short timed paper every day, with 8 questions worth 25 to 35 marks across the course, so electric fields turn up regularly but not in every paper. In AQA's A-level, electric fields are taught in the second year.

Read next

Physics · Electricity and magnetismCoulomb's law calculator: force between two chargesFind the force between two charges with Coulomb's law, F = kQ₁Q₂/r², in C, µC, nC or electron charges, plus field strength, with worked steps.