Electrostatic Force Formula: F = kq₁q₂/r² with Examples

The electrostatic force formula is $F = \frac{kq_1q_2}{r^2}$ , known as Coulomb's law: the force between two point charges is proportional to the product of the charges and inversely proportional to the square of the distance between them. Like charges repel and unlike charges attract.

The constant is $k = \frac{1}{4\pi\varepsilon_0} = 8.99\times10^{9}$ N m²/C², with $\varepsilon_0 = 8.854\times10^{-12}$ C²/(N m²). US courses often call this simply the electric force formula or electrical force formula. Whatever the name, it is the starting point of electrostatics and leads directly to the electric field.

Quick facts

Formula
$F = \frac{kq_1q_2}{r^2}$
Constant
k = 8.99 × 10⁹ N m²/C² (often rounded to 9 × 10⁹)
SI unit
newton (N)
Dimensional formula
[M¹ L¹ T⁻²]
Type
Vector along the line joining the charges
In a medium
$F = \frac{kq_1q_2}{K r^2}$ , K = dielectric constant

Parts of the electrostatic force formula

Meaning of each term in Coulomb's law
Symbol	Meaning	SI unit
$F$	Size of the force on either charge	N
$k$	Coulomb constant, $\frac{1}{4\pi\varepsilon_0}$	N m²/C²
$q_1, q_2$	The two point charges	C
$r$	Distance between their centres	m

Both charges feel forces of the same size in opposite directions, as Newton's third law requires. If you keep the signs of q₁ and q₂, a positive product means repulsion and a negative product means attraction. Many teachers prefer to put in magnitudes only and decide the direction from like or unlike charges.

How to use the formula for electric force

Convert charges to coulombs: 1 μC = 10⁻⁶ C, 1 nC = 10⁻⁹ C.
Convert the distance to metres and square it.
Multiply k by the two charge magnitudes and divide by r².
State the direction: repulsive for like charges, attractive for unlike.

With three or more charges, find each pair force separately and add them as vectors. The calculator above handles a single pair.

Worked examples

Example 1: two small charged spheres

Two small spheres carrying 4.0 μC and 5.0 μC sit 50 cm apart in air. Work out the force between them.

q₁q₂ = 2.0 × 10⁻¹¹ C², r² = 0.25 m².

$F = \frac{(8.99\times10^{9})(2.0\times10^{-11})}{0.25} \approx 0.72\ \text{N}$

Both are positive, so the force is repulsive.

Example 2: proton and electron in hydrogen

Find the attraction between a proton and an electron 5.29 × 10⁻¹¹ m apart.

$F = \frac{(8.99\times10^{9})(1.602\times10^{-19})^2}{(5.29\times10^{-11})^2} = \frac{(8.99\times10^{9})(2.566\times10^{-38})}{2.798\times10^{-21}}$

F ≈ 8.2 × 10⁻⁸ N, attractive.

Example 3: the same pair in water

The spheres in Example 1 are placed in water, dielectric constant 80, at the same distance. Find the new force.

$F_{\text{water}} = \frac{0.72}{80} = 9.0\times10^{-3}\ \text{N}$

How the force scales

Changing one quantity at a time
Change	Effect on F
Double one charge	F doubles
Double both charges	F becomes 4 times
Double the distance	F becomes one quarter
Halve the distance	F becomes 4 times
Move into a medium of dielectric constant K	F divides by K

These ratio questions are fast marks in objective papers. Write $F \propto \frac{q_1q_2}{r^2}$ and compare the before and after values instead of computing each force in full.

Comparison with gravity

Coulomb's law has the same inverse square shape as the gravitational force formula, but with two key differences. Gravity only attracts, while charges can attract or repel. And the electric force is far stronger: between a proton and an electron it is about 10³⁹ times the gravitational pull. That is why gravity is ignored inside atoms.

Coulomb's law is exact only for point charges or uniformly charged spheres measured from their centres.

Units, dimensions and common mistakes

Force is in newtons, and the dimensions of force work out to [M¹ L¹ T⁻²]. From $k = \frac{Fr^2}{q_1q_2}$ the unit of k is N m²/C².

Forgetting to square r, or squaring the charge but not the distance.
Leaving charges in μC: a factor of 10⁻¹² is easy to drop.
Adding forces from different charges without directions.
Writing ε₀ for the medium; use Kε₀ in a dielectric.

The electricity hub groups more electrostatics formulas.

Frequently asked questions

What is the formula for electric force between two charges?

F = kq₁q₂ / r², with k = 8.99 × 10⁹ N m²/C². This is Coulomb's law.

Which unit should an electrostatic force answer carry?

The newton (N), the same as any force.

Is electrostatic force a vector?

Yes. It acts along the line joining the two charges, away from each other for like charges and toward each other for unlike charges.

What is the value of 1/(4πε₀)?

About 8.99 × 10⁹ N m²/C², often rounded to 9 × 10⁹ in exam problems.

What is the difference between electrostatic force and gravitational force?

Both follow an inverse square law. Electric force can attract or repel and is enormously stronger; gravity only attracts.