Coulomb's Law, Explained

After reading this you will be able to compute the electrostatic force between two point charges, tell attraction from repulsion by inspecting the sign, and estimate the field and potential energy around a charge without guessing.

What Coulomb's law describes

Two objects that carry electric charge push or pull on each other, even across empty space. Coulomb's law gives the size and direction of that force for two point charges, meaning charges small enough that their shape does not matter at the distance you care about.

Here is a hook worth keeping in your head. Take two charges of one microcoulomb each, held ten centimeters apart. That sounds tiny: a microcoulomb is a millionth of a coulomb. Yet the force between them is about 0.9 N, roughly the weight of a 90 gram apple. Electric forces are enormous compared to gravity. The same two charges pulling on each other by gravity would need masses of billions of kilograms to match that pull.

The reason we rarely feel these forces is that matter is almost perfectly neutral. Positive and negative charges cancel to extraordinary precision. A small imbalance is all it takes to produce strong effects.

When to use it, and when not

Coulomb's law is exact for two stationary point charges in a vacuum. It stays accurate for charged spheres too, as long as the charge spreads out evenly, because a uniform sphere acts as if all its charge sits at the center. That is a real theorem, not an approximation.

The model starts to break down in three situations. First, when the charges move fast, magnetic forces and radiation enter, and you need the full electromagnetic treatment. Second, when a charge distribution is not a point or a symmetric sphere, you must integrate over the shape. Third, when a material sits between the charges, it partly screens the force. In that case you divide by the relative permittivity of the medium, roughly 80 for water and about 1 for air.

The value k = 8.99×10⁹ N·m²/C² is the vacuum constant. Air is close enough that you can use the same number for most lab work, since air's relative permittivity is about 1.0006.

The formula and the intuition behind it

The magnitude of the force between two point charges is:

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

Here q_1 and q_2 are the two charges in coulombs, r is the distance between them in meters, and k \approx 8.99 \times 10^9 in units of \text{N} \cdot \text{m}^2 / \text{C}^2. The result F comes out in newtons.

The direction follows a simple rule. Like charges repel, opposite charges attract. If you keep the signs on q_1 and q_2 and drop the absolute value bars, a positive product signals repulsion and a negative product signals attraction. The calculator reports the sign for you.

Two features of the formula do most of the work. The force scales with the product of the charges, so doubling either charge doubles the force. And it falls off as the inverse square of distance. Move the charges twice as far apart and the force drops to one quarter. Move them ten times as far and it drops to one hundredth. The constant k is often written as k = \frac{1}{4\pi\varepsilon_0}, where \varepsilon_0 \approx 8.85 \times 10^{-12} is the permittivity of free space.

A worked example with the demo numbers

One positive and one negative microcoulomb, 10 cm apart

Use the tool's default values: q_1 = +1\,\mu\text{C}, q_2 = -1\,\mu\text{C}, and r = 0.1\,\text{m}.

Convert to SI units. One microcoulomb is 10^{-6} C, so q_1 = 1 \times 10^{-6} C and q_2 = -1 \times 10^{-6} C.
Multiply the charges: q_1 q_2 = (10^{-6})(-10^{-6}) = -10^{-12}\,\text{C}^2. The product is negative, so the force is attractive.
Square the distance: r^2 = (0.1)^2 = 0.01\,\text{m}^2.
Divide and multiply by k: F = 8.99 \times 10^9 \times \frac{10^{-12}}{0.01} = 8.99 \times 10^9 \times 10^{-10} = 0.899\,\text{N}.

The force is about 0.9 N of attraction. Each charge feels this same pull toward the other, by Newton's third law. If both charges were positive, the number would be identical but the force would push them apart.

Field and potential energy from the same setup

The calculator also reports the electric field and the potential energy, both derived from the same charges and distance.

The electric field is the force per unit charge that a test charge would feel at a point. For a single source charge q:

E = k \, \frac{|q|}{r^2}

Using q = 1\,\mu\text{C} at r = 0.1 m, the field is E = 8.99 \times 10^9 \times \frac{10^{-6}}{0.01} = 8.99 \times 10^5 N/C. Multiply that field by the second charge and you recover the force: F = E \, q_2 = 8.99 \times 10^5 \times 10^{-6} = 0.899 N. The field is just the force stripped of the second charge, which is why it is useful when you want to describe the effect of one charge everywhere in space.

The electrostatic potential energy of the pair is:

U = k \, \frac{q_1 q_2}{r}

Note the single power of r, not r^2, and that the signs are kept. For the demo charges, U = 8.99 \times 10^9 \times \frac{-10^{-12}}{0.1} = -0.0899 J. The energy is negative, which tells you the pair is bound: you must add about 0.09 J of work to pull these opposite charges infinitely far apart.

Seeing the inverse-square drop

The single most instructive move is to slide the charges apart and watch the force collapse. The chart below fixes both charges at one microcoulomb and varies the separation from 5 cm to 50 cm.

The force plunges as separation grows. At 5 cm it is about 3.6 N; at 10 cm (the marker) it is 0.9 N, exactly one quarter, because halving the distance quarters the force.

Common mistakes

Most errors are unit slips, not physics errors.

Forgetting the microcoulomb conversion: Entering 1 as coulombs instead of microcoulombs inflates the force by a factor of 10^{12}. Always convert: 1\,\mu\text{C} = 10^{-6} C.
Using centimeters for distance: The formula needs meters. Ten centimeters is 0.1 m, not 10. Since distance is squared, a factor-of-100 slip becomes a factor-of-10000 error in the force.
Squaring the distance in the energy formula: Force goes as 1/r^2, but potential energy goes as 1/r. Mixing them up is common.
Dropping the sign when it matters: For magnitude, absolute values are fine. For potential energy and for deciding attraction versus repulsion, keep the signs.

Coulomb's law treats the charges as points. If two spheres are close compared to their radii, the charge on each shifts toward the other, and the real force exceeds the point-charge estimate. Trust the formula only when the separation is large compared to the object size.

How this connects to other tools

The inverse-square distance law here is the same shape you meet in gravitation and in radiation intensity. If you are propagating measurement uncertainty in charge or distance through this formula, the Error Propagation Calculator handles the squared-distance term correctly. To compare a computed force against a measured one, the Percent Error Calculator gives the gap as a percentage.

For neighboring topics in modern and wave physics, the Photon Energy Calculator covers the quantum of light, the de Broglie Wavelength Calculator gives the wave nature of moving particles, and the Special Relativity Calculator covers what happens when charges move near light speed. For classical wave and optics work, see the Snell's Law & Refraction Calculator, the Lens & Mirror Equation Calculator, and the Doppler Effect Calculator. Radioactive sources involve charge too, handled by the Radioactive Decay & Half-Life Calculator.

Frequently asked questions

Why is the force negative sometimes?

A negative value from the signed formula means attraction, and a positive value means repulsion. The calculator reports both a magnitude and a word so you do not have to read the sign yourself. For two charges of opposite sign, like the demo's +1 and -1 microcoulombs, the product is negative and the force pulls them together.

What is a realistic charge in microcoulombs?

Static electricity on a comb or balloon runs from a fraction of a microcoulomb up to a few microcoulombs. A single electron carries only 1.6 \times 10^{-19} C, so one microcoulomb is about 6.2 \times 10^{12} electrons. That is why microcoulomb charges already produce forces near a newton.

Does the medium between the charges change the force?

Yes. In a medium you divide by the relative permittivity \varepsilon_r. Water's value near 80 cuts the force to about 1/80 of its vacuum size, which is why salts dissolve: water weakens the pull between ions enough for them to separate.

How does Coulomb's law compare to gravity?

Both fall off as 1/r^2, but the electric force is vastly stronger per unit of source. For two protons the electric repulsion beats the gravitational attraction by a factor of about 10^{36}. Gravity dominates on planetary scales only because large bodies are electrically neutral.

Can I use it for more than two charges?

Not directly, but forces add as vectors. Compute the Coulomb force from each source charge separately, then add the vectors. This calculator handles the pairwise step, which is the building block for any arrangement.

Try it yourself in the tool