Coulomb's Law

Two charged balls on one line. Change the size and the sign of the charges, move them closer or farther apart — and watch the force change: it grows as they approach and turns around when a sign flips. Press “Release the charges” to see the balls speed up under that force.

The idea in 30 seconds

There are two kinds of charge

A red ball is a positive charge “+”, a blue one is a negative charge “−”. Like charges repel, unlike charges attract. You can see it in the arrows: flip a sign and they turn around.

The force depends on the charges and the distance

F = k·|q₁|·|q₂|/r². Twice the charge — twice the force. Twice the distance — four times weaker, three times the distance — nine times weaker. Close up the force is huge, far away it almost vanishes.

Forces always come in pairs

The first charge acts on the second with the same force as the second acts on the first, only in the opposite direction — Newton’s third law. Even if one charge is 40 times bigger than the other, the arrows have the same length.

Try it yourself

Press “r ×2”: the distance doubles and the force drops four times. Compare with the “F · r²” block under the scene — the product has not changed.
Press “q₁ ×2”: the force became twice as large. Now double q₂ with the slider as well — the force has grown four times.
Pick the scene “Big and small” (20 and 0.5 nC). Which ball has the longer arrow? Both are the same — Newton’s third law.
Press “Release the charges” for attraction and for repulsion. When they approach, the force grows and the balls speed up faster and faster; when they fly apart, the force fades and the speed almost stops growing.

Where you meet it

A comb and bits of paper: a comb rubbed on hair picks up a charge of a few nanocoulombs and attracts scraps of paper. It is the same Coulomb’s law: the paper gets an induced charge of the opposite sign on the side closest to the comb.
The atom: the electron is held near the nucleus by the Coulomb force. In a hydrogen atom (r ≈ 5.3·10⁻¹¹ m) it is only ≈ 8·10⁻⁸ N, but for an electron of mass 9.1·10⁻³¹ kg that means an acceleration of about 10²³ m/s².
Electrostatic precipitators in factories and laser printers: charged dust or toner particles are pulled to a plate or a drum of the opposite charge.

Graphs: how the force depends on the distance and the charge

On the left is F(r), the 1/r² hyperbola. The diamonds show “what if r is twice smaller or twice larger”: the force ×4 and ÷4. On the right is F(q₁): a straight line, because the force is directly proportional to the charge. Below is the energy during the motion after “Release the charges”.

Force versus distance F(r): the inverse square

Force versus charge F(q₁): direct proportionality

Energy during the motion: potential turns into kinetic, the total stays constant

Press “Release the charges” — the energy graph will appear here

Ready-made scenes

q₁ first charge 5,0 nC

+ −

q₂ second charge 5,0 nC

+ −

+ + / − − — like signs — repulsion
+ − — opposite signs — attraction

r distance between the centres 0,40 m

Try it in one click

m mass of each ball 0,20 g

Playback speed

Show

field lines and moving dots

glow — the potential of the field

the distance ruler r

The model: two point charges in vacuum, k = 9·10⁹ N·m²/C² (the school value). In the “Release the charges” mode the balls of equal mass m start from rest; there is no gravity or air drag, and the balls touch at r ≈ 0.08–0.12 m. The motion is computed with an exact formula, not step by step.

KEY FORMULAS

Coulomb's law

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

Magnitude of the force with which two point charges in vacuum act on each other

▶ Detailed Explanation

The force with which one point charge acts on another is proportional to the product of the magnitudes |q₁|·|q₂| and inversely proportional to the square of the distance r between them. The signs do not enter the formula — they only set the direction: like signs repel, unlike signs attract (on the scene the arrows F₁ and F₂ either point away from each other or toward each other). The forces on the two charges are equal in magnitude and opposite in direction — Newton's third law. The inverse square is the key part of the formula: half the distance — four times the force, a third of the distance — nine times. Example: q₁ = q₂ = 5 nC = 5·10⁻⁹ C, r = 0.4 m → F = 9·10⁹ · (5·10⁻⁹)² / 0.4² ≈ 1.4·10⁻⁶ N. The law holds for point charges (or balls much smaller than r) in vacuum.

Coulomb's constant

$$k = \frac{1}{4\pi\varepsilon_0} \approx 9\cdot10^{9}$$

The coefficient in Coulomb's law for vacuum: k ≈ 9·10⁹ N·m²/C², ε₀ ≈ 8.85·10⁻¹² F/m

▶ Detailed Explanation

The constant k ties charges, distance and force together in SI: k = 1/(4πε₀) ≈ 9·10⁹ N·m²/C² (more precisely 8.99·10⁹; school problems use 9·10⁹, and the simulator does the same). It is almost twenty orders of magnitude larger than the gravitational constant G ≈ 6.67·10⁻¹¹ N·m²/kg², so electric forces are almost always stronger than gravity: at the atomic scale gravity is negligible, while at the planetary scale charges neutralise each other and gravitation wins. In a medium with permittivity ε the force is ε times weaker: F = k·|q₁q₂| / (ε·r²). For water ε ≈ 81, so in water charges interact almost 80 times more weakly than in vacuum — which is why salt falls apart into ions in water.

Interaction through the electric field

$$E = k\,\frac{|Q|}{r^2},\qquad F = |q|\,E$$

One charge creates an electric field, the other feels a force in it: “action at a distance” is replaced by the field

▶ Detailed Explanation

Coulomb's law is conveniently written through the field. A charge Q at a distance r creates an electric field of strength E = k·|Q|/r² (unit: N/C), and a charge q in that field feels a force F = |q|·E. Substitute E and you get Coulomb's law back. The field explains how charges “know” about each other: the interaction is carried by the field at the speed of light, not instantly. On the scene the field lines are exactly the picture of the field: the denser the lines, the stronger the field. Halve r and the field at the second charge grows four times, and so does the force F = qE. Example: Q = 5 nC, r = 0.4 m → E = 9·10⁹ · 5·10⁻⁹ / 0.4² ≈ 280 N/C; on an equal charge q = 5 nC the force is F = 280 · 5·10⁻⁹ ≈ 1.4·10⁻⁶ N — the same number as in the Measurements panel.

Comparison with Newton's law of gravitation

$$F_g = G\frac{m_1m_2}{r^2},\qquad G \approx 6.67\cdot10^{-11}$$

Both laws are inverse-square, but the electric force is many orders of magnitude larger, and gravity only attracts

▶ Detailed Explanation

Coulomb's law and Newton's gravitation have the same structure: a product of the “sources” on top, the square of the distance at the bottom. The constants and the signs differ: k ≈ 9·10⁹ versus G ≈ 6.67·10⁻¹¹ (for G the mass is in kg, N·m²/kg²), and electric charges come in two signs and can repel, while mass only attracts. Example: two balls of 1 g each carrying 1 µC at a distance of 1 cm interact with a force F = 9·10⁹ · (10⁻⁶)² / (10⁻²)² = 90 N. The weight of each ball is only 0.0098 N — the electric force is about 9000 times larger. And their gravitational attraction G·m²/r² ≈ 6.7·10⁻¹³ N is about 10¹⁴ times smaller than the Coulomb force. For two protons the ratio is ≈ 10³⁶.

Potential energy of two charges

$$E_{\text{p}} = k\,\frac{q_1 q_2}{r}$$

Interaction energy of a pair of point charges: positive for repulsion, negative for attraction

▶ Detailed Explanation

The interaction energy of two point charges is Ep = k·q₁q₂/r, with the charges taken with their signs. For repulsion (q₁q₂ > 0) the energy is positive and grows as the charges approach: moving like charges together takes work. For attraction (q₁q₂ < 0) the energy is negative: the closer the charges, the lower it is. When the charges are released, potential energy turns into kinetic energy: Ep + Ek = const. That is what happens in the “Release the charges” mode: when the balls fly apart Ep falls to zero and all the energy becomes kinetic, so each ball's speed tends to v = √(k|q₁q₂| / (m·r₀)). Example: q₁ = q₂ = 5 nC, r = 0.4 m → Ep = 9·10⁹ · 25·10⁻¹⁸ / 0.4 ≈ 5.6·10⁻⁷ J.

What is going on. Two small charged balls sit on one line. The red one carries a positive charge “+”, the blue one a negative charge “−”. Each ball acts on the other with a force — the Coulomb force. If the signs are the same (+ + or − −), the balls repel; if they differ, they attract. On the scene the force is shown by the arrows F₁ and F₂: green for attraction, orange for repulsion. The coloured lines are the electric field of the charges: they leave “+” and enter “−”, and the dots running along them show the direction of the field.

Coulomb's law: F = k·|q₁|·|q₂| / r². The force is proportional to the product of the charges and inversely proportional to the square of the distance between them. Twice the charge — twice the force. Twice the distance — four times weaker, three times the distance — nine times weaker. Checking it is easy: the “F · r²” block under the scene shows that at any distance the product stays equal to k·|q₁|·|q₂| as long as the charges are unchanged. The constant k ≈ 9·10⁹ N·m²/C² is huge, but the charges here are tiny — nanocoulombs (1 nC = 10⁻⁹ C, about 6 billion electrons), so the force comes out in micronewtons. For comparison: two charges of 1 C each at a distance of 1 m would repel with 9·10⁹ N — the weight of about 900 thousand tonnes.

Newton's third law. The first charge acts on the second with the same force as the second acts on the first, only in the opposite direction. That is why the arrows F₁ and F₂ always have the same length — even if one charge is 40 times bigger than the other (the “Big and small” scene). The big charge is not “stronger”: the force depends on the product q₁·q₂, and the pair shares it.

The “Release the charges” mode. Balls of equal mass m start from rest. With attraction the force grows as they approach, so the balls speed up faster and faster and collide. With repulsion the force falls with distance, and the speed tends to a limit. The motion is computed with exact formulas from energy conservation, not step by step. The potential energy of the pair Ep = k·q₁q₂/r turns into kinetic energy while their sum stays constant — you can see it on the energy graph.

How to use it. Drag either ball with the mouse or your finger, move the q₁, q₂, r sliders. The buttons “r ×2”, “r ÷2”, “q₁ ×2” and “sign of q₂” run an experiment in one click and explain the result at once. The “Measurements” tab shows the forces, the field, a “weight equivalent” and a prediction of the motion. Check yourself: with q₁ = q₂ = 5 nC and r = 0.40 m the force is about 1.4·10⁻⁶ N, and at r = 0.20 m it is already 5.6·10⁻⁶ N.