Coulomb’s Law

Coulomb's Law is the fundamental rule governing the electric force between two stationary, electrically charged objects. Just as gravity pulls every mass toward every other mass, the electric force either attracts or repels charged particles — the key difference being that charge comes in two flavors, positive and negative, while mass has only one. Two unlike charges attract each other; two like charges repel. Discovered experimentally by Charles-Augustin de Coulomb in 1785 using a delicate torsion balance, this law is the cornerstone of electrostatics. The law states that the magnitude of the electric force between two point charges is directly proportional to the product of the magnitudes of those charges and inversely proportional to the square of the distance between them. That 'inverse-square' relationship is crucial: double the distance and the force drops to one-quarter of its original value; triple the distance and it falls to one-ninth. This same inverse-square pattern appears in Newton's law of gravity, and for a deep geometric reason — both forces spread out over the surface area of an expanding sphere, which grows as r². Embedded in Coulomb's Law is a constant of proportionality that encodes how strongly the medium between the charges responds to them. In a vacuum this is expressed either as Coulomb's constant k ≈ 8.99 × 10⁹ N·m²/C² or equivalently through the permittivity of free space ε₀ ≈ 8.85 × 10⁻¹² C²/(N·m²). Permittivity tells us how easily an electric field can establish itself through a given medium; a higher permittivity means the medium 'shields' charges from each other more effectively, reducing the force. Understanding Coulomb's Law unlocks the entire study of electric fields, electric potential, and ultimately circuits and electromagnetic waves.

Key takeaways
  • What Is a Point Charge?
  • The Proportionality to Charge
  • The Inverse-Square Dependence on Distance
  • The Role of Permittivity
  • Direction of the Force

What Is a Point Charge?

A 'point charge' is an idealization: a charged object whose physical size is negligible compared to the distances involved. In practice, a proton, electron, or even a small charged sphere can be treated as a point charge when you are far enough away. Coulomb's Law applies exactly to point charges and approximately to spherical charge distributions, where all the charge can be treated as concentrated at the center.

The Proportionality to Charge

If you double the charge on one object (say, add more electrons), the force doubles. If you double both charges, the force quadruples. Mathematically, F ∝ |q₁||q₂|. Think of charge like 'electric strength' — more charge means a bigger 'grip' on the other particle. The absolute value signs remind us that we compute the magnitude of the force separately from its direction.

The Inverse-Square Dependence on Distance

The force obeys F ∝ 1/r². Imagine the electric influence of a charge radiating outward uniformly in all directions, like light from a lamp. At distance r, that influence is spread over a sphere of surface area 4πr². As r doubles, the area quadruples, so the intensity (force per unit area) drops to one-quarter. This geometric dilution over spherical surfaces is why inverse-square laws appear throughout physics whenever something spreads equally in three dimensions.

The Role of Permittivity

The constant k = 1/(4πε₀) connects the proportionalities to a quantitative equality. The permittivity of free space ε₀ characterizes the vacuum itself — specifically, how much electric flux a given charge produces. In a material medium with permittivity ε = εᵣε₀ (where εᵣ is the relative permittivity or dielectric constant), the force becomes F = kq₁q₂/(εᵣr²), always weaker than in vacuum. Water has εᵣ ≈ 80, meaning electric forces between ions dissolved in water are 80 times weaker than they would be in a vacuum — a fact crucial to understanding chemistry and biology.

Direction of the Force

Coulomb's Law in vector form is F⃗₁₂ = k(q₁q₂/r²)r̂₁₂, where r̂₁₂ is a unit vector pointing from charge 1 toward charge 2. If the product q₁q₂ is positive (like charges), the force on charge 2 points away from charge 1 — repulsion. If q₁q₂ is negative (unlike charges), the force points toward charge 1 — attraction. Newton's third law holds: the force on charge 1 due to charge 2 is equal in magnitude and opposite in direction.

The Key Equation

F=k∣q1∣∣q2∣r2=14πε0∣q1∣∣q2∣r2F = k\frac{|q_1||q_2|}{r^2} = \frac{1}{4\pi\varepsilon_0}\frac{|q_1||q_2|}{r^2}
  • F — Electric Force (Newton (N)): The magnitude of the electrostatic force between the two charges.
  • k — Coulomb's Constant (N·m²/C²): k ≈ 8.99 × 10⁹ N·m²/C²; equal to 1/(4πε₀). Sets the scale of electric force in SI units.
  • q_1,\, q_2 — Point Charges (Coulomb (C)): The electric charges of the two objects. Can be positive or negative; only magnitudes enter the force magnitude formula.
  • r — Separation Distance (Meter (m)): The straight-line distance between the centers of the two point charges.
  • \varepsilon_0 — Permittivity of Free Space (C²/(N·m²)): ε₀ ≈ 8.85 × 10⁻¹² C²/(N·m²). Describes how well a vacuum supports an electric field; appears in the denominator so larger permittivity means weaker force.

Worked Example

Problem. Two small charged spheres are held 0.30 m apart in air (treat as vacuum). Sphere A carries a charge of +4.0 μC and Sphere B carries a charge of −2.0 μC. (a) Calculate the magnitude of the electric force between them. (b) Is the force attractive or repulsive?

F≈0.80 N (attractive)F \approx 0.80\text{ N (attractive)}

Answer. The magnitude of the electric force is approximately 0.80 N, and the force is attractive.

Figure
Diagram
Coulomb's Law: Inverse-Square Electric Force+q₁ = +4 μC−q₂ = −2 μCr = 0.30 mF on q₁F on q₂AttractiveF = k|q₁||q₂| / r² ≈ 0.80 N
Origins · 1 minCharles-Augustin de Coulomb and how this was discovered
CD
Charles-Augustin de Coulomb
1736–1806 · French

Coulomb invented the torsion balance — a remarkably sensitive device that detects tiny forces by measuring how much a thin wire twists. Between 1785 and 1791 he used it to perform meticulous experiments on electric and magnetic forces, establishing that the electric force between charges follows an exact inverse-square law. His quantitative precision set a new standard for experimental physics and earned the SI unit of charge, the coulomb (C), his name. Before his electric force experiments, Coulomb spent years as a military engineer building fortifications in Martinique. The grueling tropical work damaged his health but sharpened his talent for precision measurement — skills he later turned to physics with revolutionary results.

  • Electric Field — Coulomb's Law describes the force between two specific charges; the electric field concept extends this to describe the force a single source charge would exert on any test charge placed anywhere in space.
  • Electric Potential and Potential Energy — The electric potential energy between two charges is derived directly from integrating Coulomb's force over distance, giving U = kq₁q₂/r — a concept essential for understanding circuits and capacitors.
  • Newton's Law of Universal Gravitation — Gravity follows the same inverse-square form as Coulomb's Law. Comparing the two laws reveals both the deep geometric reason for inverse-square behavior and just how much stronger the electric force is than gravity.
  • Superposition of Electric Forces — When more than two charges are present, Coulomb's Law is applied to each pair separately and the resulting force vectors are added — this superposition principle is the key to calculating forces in real charge distributions.