Coulomb's Law: F = kQ₁Q₂/r² Formula, Calculations & Examples

Coulomb's law gives the electrostatic force between two point charges: F = kQ₁Q₂/r², where k = 8.99 × 10⁹ N·m²/C² (Coulomb's constant), Q₁ and Q₂ are the charges in coulombs, and r is their separation in metres. The force is repulsive for like charges and attractive for opposite charges. It follows an inverse square law — exactly like gravity — but is vastly stronger: the electrostatic force between a proton and electron in hydrogen is about 10³⁹ times stronger than the gravitational force between them.

Coulomb's law is the starting point for all of electrostatics. Combined with the superposition principle — forces from multiple charges simply add as vectors — it lets you calculate the net force on any charge in any arrangement. It's also the source of the electric field formula E = F/q = kQ/r², connecting force to field at any point in space.

Coulomb's Law: F = kQ₁Q₂/r²

Where:

• F = electrostatic force (N)
• k = Coulomb's constant = 8.99 × 10⁹ N·m²/C² = 1/(4πε₀)
• Q₁, Q₂ = the two charges (C) — positive for positive charges, negative for negative
• r = centre-to-centre distance between the charges (m)

Alternative form using permittivity of free space ε₀ = 8.854 × 10⁻¹² C²/(N·m²):

Sign convention: If the product Q₁Q₂ is positive (both charges same sign), the force is repulsive — the charges push apart. If Q₁Q₂ is negative (opposite signs), the force is attractive. The magnitude formula gives |F|; direction must be determined from the signs of the charges.

Comparison with Gravity

Coulomb's law and Newton's law of gravitation have identical mathematical forms:

Property	Coulomb's Law	Gravity
Formula	F = kQ₁Q₂/r²	F = Gm₁m₂/r²
Constant	k = 8.99 × 10⁹	G = 6.67 × 10⁻¹¹
Dependence	1/r² (inverse square)	1/r² (inverse square)
Can be repulsive?	Yes (like charges)	No (always attractive)
Relative strength	∼10³⁹ stronger (for proton-electron)	Baseline

The Superposition Principle

When multiple charges are present, the force on any one charge is the vector sum of the individual forces from each other charge. The charges don't interfere with each other's fields — each exerts its Coulomb force independently:

Each force is calculated separately using Coulomb's law, then added as vectors (accounting for direction). This is the superposition principle.

4 Worked Examples

Example 1 — Force between two charges

Problem: Two charges, Q₁ = +4 μC and Q₂ = −6 μC, are separated by 0.3 m. Find the magnitude and nature of the force.

Solution:
F = kQ₁Q₂/r² = 8.99 × 10⁹ × 4 × 10⁻⁶ × 6 × 10⁻⁶ / (0.3)²
F = 8.99 × 10⁹ × 24 × 10⁻¹² / 0.09
F = 8.99 × 10⁹ × 2.667 × 10⁻¹⁰ = 2.40 N (attractive, opposite signs)

Example 2 — Comparing with gravitational force

Problem: A proton and electron in a hydrogen atom are separated by 5.29 × 10⁻¹¹ m (Bohr radius). Find the electrostatic force and compare with the gravitational force. (m_p = 1.67 × 10⁻²⁷ kg, m_e = 9.11 × 10⁻³¹ kg, q = 1.6 × 10⁻¹⁹ C)

Solution:
F_E = kq²/r² = 8.99 × 10⁹ × (1.6 × 10⁻¹⁹)² / (5.29 × 10⁻¹¹)²
= 8.99 × 10⁹ × 2.56 × 10⁻³⁸ / 2.798 × 10⁻²¹ = 8.22 × 10⁻⁸ N
F_G = Gm_pm_e/r² = 6.67 × 10⁻¹¹ × 1.67 × 10⁻²⁷ × 9.11 × 10⁻³¹ / (5.29 × 10⁻¹¹)²
= 3.63 × 10⁻⁴⁷ N
Ratio: F_E/F_G = 8.22 × 10⁻⁸ / 3.63 × 10⁻⁴⁷ = 2.26 × 10³⁹

Example 3 — Three charges in a line

Problem: Three charges on a line: Q₁ = +2 μC at x = 0, Q₂ = +3 μC at x = 0.4 m, Q₃ = −1 μC at x = 0.6 m. Find the net force on Q₃.

Solution:
F₁₃ = kQ₁Q₃/r₁₃² = 8.99 × 10⁹ × 2 × 10⁻⁶ × 1 × 10⁻⁶ / 0.6² = 0.05 N (attractive, Q₃ pulled toward Q₁, i.e. in −x direction)
F₂₃ = kQ₂Q₃/r₂₃² = 8.99 × 10⁹ × 3 × 10⁻⁶ × 1 × 10⁻⁶ / 0.2² = 0.674 N (attractive, Q₃ pulled toward Q₂, i.e. in −x direction)
F_net = −0.05 − 0.674 = −0.724 N (i.e. 0.724 N in the −x direction)

Example 4 — Equilibrium position

Problem: Charges +4 μC and +9 μC are 1.0 m apart. Where must a −1 μC charge be placed on the line between them to experience zero net force?

Solution:
Let the −1 μC charge be at distance x from the +4 μC charge (so distance (1−x) from +9 μC).
For equilibrium, F from +4 μC = F from +9 μC:
k × 4 × 1 / x² = k × 9 × 1 / (1−x)²
4(1−x)² = 9x²
2(1−x) = 3x (taking positive square roots)
2 − 2x = 3x → x = 2/5 = 0.4 m from the +4 μC charge

Electric Field from Coulomb's Law

The electric field at a distance r from a point charge Q is the force per unit positive test charge:

This connects Coulomb's law directly to electric field and potential. The field from multiple charges is found by vector superposition — add the fields from each charge independently, accounting for direction.

Coulomb's Law in Vector Form

The full vector form of Coulomb's law gives both magnitude and direction:

Where r̂₁₂ is the unit vector pointing from charge 1 to charge 2. If Q₁Q₂ is positive (like charges), F⃗₁₂ points away from Q₁ (repulsion). If negative (opposite charges), it points toward Q₁ (attraction). This single formula handles both cases automatically through the sign of the product Q₁Q₂.

Worked Example 5 — Equilateral triangle arrangement

Problem: Three equal charges Q = +2 μC are placed at the vertices of an equilateral triangle with side 0.3 m. Find the net force on one charge due to the other two.

Solution:
By symmetry, the two forces on charge C (from A and B) have equal magnitude and make equal angles (30°) with the perpendicular bisector of AB.
F_CA = F_CB = kQ²/r² = 8.99 × 10⁹ × (2 × 10⁻⁶)²/0.3² = 8.99 × 10⁹ × 4 × 10⁻¹²/0.09 = 0.399 N
The horizontal components (along AB) cancel by symmetry.
The vertical components both point away from the midpoint of AB (upward for charge C above AB):
F_net = 2 × F_CA × cos30° = 2 × 0.399 × 0.866 = 0.691 N directed away from the midpoint of AB (along the altitude from C).

Electric Field Lines and Force

The electric force on a charge q in an electric field E⃗ is:

For a positive q, F is in the direction of E. For negative q, F is opposite to E. Coulomb's law is really a special case of this: the field created by charge Q at distance r is E = kQ/r², and the force on a test charge q there is F = qE = kQq/r².

Electric field lines show the direction of force on a positive test charge. They start on positive charges and end on negative charges. The density of field lines indicates field strength — closely spaced lines mean strong field. Field lines never cross (because the field direction is unique at every point) and are always perpendicular to equipotential surfaces.

Millikan's Oil Drop Experiment

Coulomb's law underlies one of the most important experiments in physics — Millikan's 1909 measurement of the electron charge. Tiny oil droplets carrying charge were held stationary in an electric field. At equilibrium, the upward electric force balanced gravity:

By measuring the terminal velocity (which gives the droplet radius and hence mass), and knowing the electric field, Millikan could calculate q for each droplet. He found that charge always came in integer multiples of a fundamental unit e = 1.592 × 10⁻¹⁹ C (close to the modern value 1.602 × 10⁻¹⁹ C). This was the first direct evidence for the quantisation of electric charge — that charge doesn't come in arbitrary amounts but in integer multiples of e.

Dielectric Breakdown and Lightning

Air can withstand an electric field of about 3 × 10⁶ V/m before dielectric breakdown occurs — ions are accelerated enough to ionise neutral molecules, creating a conducting plasma channel. This is lightning: charge accumulates in storm clouds until the electric field between cloud and ground exceeds air's breakdown strength, whereupon a conducting channel (the lightning bolt) forms and charge equalises in milliseconds. The average lightning bolt carries about 5 coulombs of charge over about 30 milliseconds — a current of ~5/0.03 = 167 A, far more than a household circuit's 13 A rating, though for such a brief time that the energy is only about 1 GJ/m of channel.

Shielding and Faraday Cages

A conducting shell completely shields its interior from external electric fields. Any external charge rearranges on the outer surface of the conductor, and the resulting charge distribution cancels the external field inside. This is a Faraday cage — used to protect sensitive electronic equipment from electromagnetic interference, and why you are safe inside a car struck by lightning (the charge distributes over the car's conducting body, leaving the interior field-free). The principle is an exact consequence of Gauss's law (a more powerful generalisation of Coulomb's law) — the net field inside a closed conductor due to all external charges is exactly zero.

Charge Distribution and the Shell Theorem

Coulomb's law applies to point charges. For a uniformly charged sphere, Newton's shell theorem (which also applies to electrostatics via Gauss's law) shows that: outside the sphere, it behaves exactly as if all charge were concentrated at the centre; inside a uniformly charged spherical shell, the field is zero. This is why we can treat the Earth, planets, and atomic nuclei as point charges in many calculations — their spherical charge distribution means the external field is indistinguishable from a point charge at the centre.

Permittivity and Coulomb's Law in Materials

Coulomb's law in a medium with relative permittivity ε_r becomes:

The higher permittivity of the medium (ε_r > 1) reduces the force between charges by the factor ε_r. Water (ε_r = 80) reduces electrostatic forces to 1/80 of their vacuum value — which is why ionic compounds like salt (NaCl) dissolve readily in water: the reduced electrostatic attraction between Na⁺ and Cl⁻ ions can be overcome by thermal motion and hydration forces from water molecules. This is the microscopic basis of water being an excellent solvent for ionic and polar substances.

Exam Approach for Coulomb's Law Problems

Always: (1) identify all charges and their positions; (2) calculate the magnitude of each force using F = kQ₁Q₂/r²; (3) determine the direction of each force from the signs of the charges (like = repel, opposite = attract); (4) add all forces as vectors using components. For three or more charges, draw a diagram first, label each force with its direction, and use the component method. Symmetry arguments can save enormous calculation effort — if two equal charges are symmetrically placed relative to a third, their force components along the axis of symmetry add while perpendicular components cancel.

One final check that trips students up: always verify that your equilibrium position for a three-charge problem makes physical sense. If two like charges are fixed and you're finding the equilibrium position of a third charge between them, the equilibrium must be between the two fixed charges — it cannot be outside them. If the third charge has the opposite sign to one of the fixed charges, equilibrium may be outside the pair. Checking the geometry of your answer against the physics of attractive vs repulsive forces prevents sign errors from giving nonsensical positions.

The superposition principle for electric forces means Coulomb's law scales elegantly to systems of many charges. In a crystal, each ion interacts with every other ion — the total electrostatic energy is called the Madelung energy, and it determines the crystal's stability and lattice energy. Chemists use these Coulomb lattice energy calculations to predict ionic compound properties without any empirical fitting. The same physics governs DNA base pairing (electrostatic attraction between complementary bases), protein folding (charge-charge and dipole interactions), and the binding of drug molecules to receptor sites.

Comparing Coulomb's law and Newton's law of gravitation reveals why gravity dominates at cosmic scales despite being 10³⁹ times weaker: gravity is always attractive (no negative mass exists), so gravitational effects accumulate without cancellation as matter aggregates into planets and stars. Electrostatic forces are strong but matter is overall electrically neutral — positive and negative charges in bulk matter cancel. Only small charge imbalances remain (static electricity), and these are quickly neutralised. At cosmic scales, electrostatics is irrelevant and gravity rules everything.