Coulomb's law, explained with examples
Coulomb's law gives the electric force between two charged objects. It is one of the fundamental laws of physics, and it explains everything from static cling to the structure of atoms.
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The law
F = k q1q2 / r²
- F is the size of the electric force, in newtons;
- q1 and q2 are the charges, in coulombs (C);
- r is the distance between their centres, in metres;
- k = 1/(4πε0) ≈ 8.99 × 109 N m²/C², where ε0 is the permittivity of free space.
The law applies to point charges, or to charged spheres measured from their centres. The force acts along the line joining the two charges.
Charge and its unit
Charge is measured in coulombs. One coulomb is a very large amount of charge: the charge on a single electron is only −1.60 × 10−19 C, so a coulomb is the charge of about 6 × 1018 electrons. That is why everyday static charges are measured in microcoulombs (10−6 C) or nanocoulombs (10−9 C), and why converting units correctly is the first step in every Coulomb's law calculation.
Direction: attraction and repulsion
Like charges repel; opposite charges attract. If you substitute the signs of the charges, a positive answer means repulsion and a negative answer means attraction. Many students find it clearer to calculate the size of the force using the magnitudes, then decide the direction from the rule.
The inverse square law
The force depends on 1/r². Doubling the distance makes the force four times smaller; tripling it makes it nine times smaller. Halving the distance makes it four times larger. This inverse square pattern is the same as for gravity and for the intensity of light from a point source, and AP questions love to test it with ratios rather than numbers.
Worked examples
Example 1: two point charges
Two charges of +3.0 μC and −5.0 μC are 0.20 m apart. Find the force between them.
- F = 8.99 × 109 × (3.0 × 10−6)(5.0 × 10−6)/0.20².
- F = 8.99 × 109 × 1.5 × 10−11/0.04 ≈ 3.4 N, attractive because the charges are opposite.
Example 2: ratio reasoning
Two charges exert a force F on each other. One charge is doubled and the distance is tripled. What is the new force?
- Doubling one charge doubles F. Tripling r divides F by 9.
- New force = 2F/9.
Example 3: the hydrogen atom
In a hydrogen atom, the electron and proton are about 5.3 × 10−11 m apart. Find the electric force between them. (e = 1.60 × 10−19 C)
- F = 8.99 × 109 × (1.60 × 10−19)²/(5.3 × 10−11)².
- F ≈ 8.2 × 10−8 N, attractive. Tiny in everyday terms, but enormous for particles this light.
Example 4: three charges in a line
Charges of +2 μC at x = 0, −3 μC at x = 0.1 m and +4 μC at x = 0.3 m lie on a line. Find the net force on the −3 μC charge.
- From the +2 μC charge (0.1 m away): F = 8.99 × 109 × 6 × 10−12/0.01 ≈ 5.39 N, attracting it to the left.
- From the +4 μC charge (0.2 m away): F = 8.99 × 109 × 12 × 10−12/0.04 ≈ 2.70 N, attracting it to the right.
- Net force ≈ 5.39 − 2.70 = 2.7 N to the left.
Forces from several charges
Electric forces add as vectors. To find the net force on one charge, calculate the force from each other charge separately, using magnitudes, decide each direction, and then add the vectors. In a straight line, this means adding and subtracting with signs, as in Example 4. In two dimensions, resolve each force into x and y components, add the components, then find the size and direction of the resultant with Pythagoras and trigonometry.
Example 5: forces at right angles
Three +1.0 μC charges sit at (0, 0), (0, 0.3 m) and (0.4 m, 0). Find the net force on the charge at the origin.
- From the charge at (0, 0.3): F = 8.99 × 109 × 10−12/0.09 ≈ 0.100 N, pushing the origin charge in the −y direction (repulsion).
- From the charge at (0.4, 0): F = 8.99 × 109 × 10−12/0.16 ≈ 0.056 N, in the −x direction.
- Resultant = √(0.100² + 0.056²) ≈ 0.115 N, at about 61° below the negative x-axis, since tanθ = 0.100/0.056.
Electric field from a point charge
The electric field strength at a point is the force per unit positive charge placed there: E = F/q. For a point charge Q, this gives
E = kQ / r²
measured in N/C (equivalently V/m). Field lines point away from positive charges and towards negative charges. Coulomb's law is then simply F = qE for a second charge q placed in that field.
Comparison with gravity
Coulomb's law looks almost identical to Newton's law of gravitation, F = Gm1m2/r². Both are inverse square laws, and both act along the line between centres. The differences are important:
- Gravity is always attractive; electric forces can attract or repel.
- Electric forces are vastly stronger. Between a proton and an electron, the electric force is about 1039 times the gravitational force.
- Gravity dominates at large scales because matter is usually electrically neutral overall, so electric forces cancel while gravitational forces always add.
Common mistakes
- Forgetting to square the distance.
- Leaving charges in microcoulombs or nanocoulombs instead of converting to coulombs.
- Adding forces from several charges as plain numbers without considering direction.
- Using the distance between surfaces of spheres instead of between centres.
Practice questions
Try each one before opening the answer.
Q1. Two 1.0 μC charges are 0.10 m apart. Find the force.
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Q2. The distance between two charges is halved. What happens to the force?
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Q3. Find the electric field 0.5 m from a 2.0 nC point charge.
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Q4. Two charges attract with force F. Both charges are doubled. Find the new force.
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Frequently asked questions
Is Coulomb's law on the AP Physics exams?
Yes. Electric force and Coulomb's law are in AP Physics 2 and in AP Physics C: Electricity and Magnetism, where they lead on to electric fields and Gauss's law.
Is Coulomb's law in A level Physics?
Yes. It is in the electric fields topic in the second year, written F = Q1Q2/(4πε0r²).
Key takeaways
- F = kq1q2/r², with k ≈ 8.99 × 109 N m²/C².
- Like charges repel; opposite charges attract.
- Inverse square: double r, quarter F.
- Add forces from several charges as vectors.
- E = kQ/r² for a point charge, and F = qE.