Coulomb's Law — F = kq₁q₂/r² Explained

High School⏱ 9 min readElectricity & Magnetism

Coulomb's law: the force between electric charges. Inverse-square law, the constant k, and worked examples with unit checks.

Quick answer

The electric force between two point charges is proportional to the product of the charges and inversely proportional to the square of the distance between them. Opposite charges attract; like charges repel.

$F = k\frac{q_1 q_2}{r^2}$

Unit: newton (N); k = 8.988×10⁹ N·m²/C²

Example: Two 1 µC charges 1 m apart repel with F = (9×10⁹)(10⁻⁶)(10⁻⁶)/1² = 0.009 N — about the weight of a gram.

What you will learn

State Coulomb's law and the equation F = kq₁q₂/r²
Explain the inverse-square dependence on distance
Predict attraction vs repulsion from charge signs
Solve two-charge force problems

Charges push and pull

Rub two balloons and hang them side by side — they push apart. Coulomb's law quantifies that push: the electric force between two point charges depends on how much charge each carries and how far apart they are.

It is the electric twin of Newton's law of gravitation — same inverse-square shape, vastly stronger, and with two signs instead of one.

The law

For point charges $q_1$ and $q_2$ separated by distance $r$ :

F = k\frac{q_1 q_2}{r^2}

k = 8.988 × 10⁹ N·m²/C²; use the sign of q₁q₂ for direction

Inverse square: why distance dominates

The $1/r^2$ dependence is the law's most important feature: halve the distance, quadruple the force. Move charges twice as far apart and the force collapses to one quarter.

This extreme sensitivity to distance is why:

Static cling works at contact range but vanishes centimeters away.
Atomic nuclei, though tiny, grip electrons fiercely.
You can usually ignore electric forces between distant everyday objects.

Comparing with gravity

Compare with gravity: $F = Gm_1m_2/r^2$ .

| | Gravity | Coulomb |

|---|---|---|

| Shape | $1/r^2$ | $1/r^2$ |

| Signs | Always attractive | Attract or repel |

| Strength | Extremely weak | ~10³⁶× stronger (proton-electron) |

The identical mathematics is no coincidence — both are field forces spreading through 3D space. But at atomic scales, electricity wins by a landslide: atoms are electric structures, with gravity utterly negligible.

Key idea

Inverse-square laws come from geometry: influence spreading over a sphere's surface (4πr²). Whenever you see 1/r², think 'spreading through space.'

Worked mini-example

Two point charges, $q_1 = +3.0\ \mu\mathrm{C}$ and $q_2 = -2.0\ \mu\mathrm{C}$ , sit $0.50\ \mathrm{m}$ apart. Find the force.

$F = k\frac{|q_1 q_2|}{r^2} = (8.988 \times 10^9)\frac{(3.0 \times 10^{-6})(2.0 \times 10^{-6})}{0.50^2}$

$F = (8.988 \times 10^9)\frac{6.0 \times 10^{-12}}{0.25} \approx 0.22\ \mathrm{N}$

Attractive (opposite signs), about the weight of 22 grams. Unit check: $\mathrm{N·m^2/C^2 \times C^2 / m^2 = N}$ ✓

Variables and units

Symbol	Name	Unit	Meaning
$F$	Electric force	N	Magnitude of the electric force between the charges
$k$	Coulomb constant	N·m²/C²	Electrostatic constant, 8.988×10⁹ N·m²/C²
$q₁, q₂$	Charges	C	The two point charges (include signs)
$r$	Separation	m	Distance between charge centers

Common mistakes

✗ Only opposite charges exert forces on each other.

Like charges interact just as strongly — repulsively. The law covers all sign combinations; the sign of the product sets attract vs repel.

✗ The bigger charge exerts the bigger force.

Newton's third law applies: the forces are exactly equal and opposite. A tiny electron and a huge charged sphere pull on each other identically.

✗ k is just a number with no units.

k carries units (N·m²/C²) that make the equation dimensionally consistent. Constants with units are normal — G in gravity has them too.

Real-world applications

Inkjet printers steer charged droplets with electric fields — the force on each drop follows Coulomb's law.
Electrostatic precipitators in smokestacks charge ash particles and collect them on oppositely charged plates, cleaning exhaust.
Atomic structure is a balance of Coulomb attraction (electrons to nucleus) against quantum rules — the law sets the scale of atoms.

Frequently asked questions

What is the value of k?

k ≈ 8.988 × 10⁹ N·m²/C². It's large because the coulomb is a huge unit — everyday static charges are microcoulombs, keeping forces reasonable.

What are the units of k?

Newton-meters squared per coulomb squared (N·m²/C²) — whatever makes k·q₁q₂/r² come out in newtons.

What if both charges are positive?

It becomes repulsive instead of attractive, with identical magnitude. The formula handles this with signs: same-sign products give positive (repulsive) force.

How is Coulomb's law like Newton's gravity law?

Both are inverse-square laws, both act at a distance. Gravity is always attractive and far weaker; the electric force can attract or repel and dominates at atomic scales.