Coulomb's Law

In one sentence: Coulomb’s law tells you how hard two electric charges push or pull each other. The force gets weaker fast as they move apart, because it drops off with the square of the distance.

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

Read it out loud: “F equals k sub e times the size of q 1 times q 2, all divided by r squared. F is the force, measured in newtons. k sub e is the Coulomb constant. q 1 and q 2 are the two charges, measured in coulombs. r is the distance between them, measured in meters.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Electric charge and the coulomb (C): Electric charge is a property that some particles have, and it comes in two kinds, positive and negative. Electrons carry a negative charge of $-1.602176634 \times 10^{-19}$ C, and protons carry an equal positive charge. The coulomb (C) is the unit of charge. One coulomb is a very large amount of charge, about $6.24 \times 10^{18}$ electrons, so the charges in a lab experiment are usually a tiny fraction of a coulomb.
Powers of ten, micro and centi: A power of ten is 10 multiplied by itself a set number of times. $10^{-6}$ is one millionth, so the prefix micro ( $\mu$ ) turns 1 $\mu$ C into $1 \times 10^{-6}$ C. The prefix centi (c) means $10^{-2}$ , one hundredth, so 10 cm is $10 \times 10^{-2}$ m, which is 0.10 m. Multiplying two powers of ten adds their exponents, so $10^{-6} \times 10^{-6} = 10^{-12}$ .
The newton (N), the unit of force: A force is a push or a pull, and its unit is the newton (N). One newton is the force that gives a 1 kg mass an acceleration of 1 meter per second every second. A 100 gram apple weighs about 1 N, because gravity pulls on it with roughly 1 N. Newtons are the unit of the answer this formula gives.
Squares and inverse squares: A square is a number multiplied by itself, so $r^{2}$ means $r \times r$ . If $r = 0.10$ m, then $r^{2} = 0.10 \times 0.10 = 0.01$ m $^2$ . An inverse square means one divided by the square, so a quantity that falls with the inverse square becomes one quarter when the distance doubles. Squares are needed because the force spreads out in all directions.
Absolute value bars, $|x|$: The bars $|x|$ mean the absolute value of $x$ , which is its size with the sign removed. So $|-3| = 3$ and $|3| = 3$ . In Coulomb’s law the bars make sure the number is positive, and the signs of the charges are used separately to decide whether the force pushes or pulls.
Greek letters pi and epsilon, and the constant $k_e$: The Greek letter $\pi$ (pi) is about 3.14159, the circle’s circumference divided by its diameter. The letter $\varepsilon$ (epsilon) is used for permittivity, a measure of how easily a material lets electric fields through. $\varepsilon_0$ is the value for empty space. The Coulomb constant is $k_e = \frac{1}{4\pi\varepsilon_0}$ , and it is a fixed number of nature.

What each symbol means

Symbol	Meaning
$F$	Force between the two charges, measured in newtons (N). Its size comes from the formula, and the signs of the two charges decide the direction, a push for like charges and a pull for unlike charges. A force of 1 N is about the weight of a 100 gram apple.
$k_e$	The Coulomb constant, a fixed number equal to $8.9875517923 \times 10^{9}\ \text{N}\,\text{m}^2/\text{C}^2$ . It sets the strength of the electric force in SI units, which are newtons, meters and coulombs, and it does not change.
$q_1$	The charge on the first object, measured in coulombs (C). Its sign tells you whether it is positive or negative. A typical lab charge might be one microcoulomb, written $1\ \mu\text{C}$ .
$q_2$	The charge on the second object, measured in coulombs (C). It is read in the same way as $q_1$ , and the two charges are multiplied together in the formula. A typical lab charge might be one microcoulomb, written $1\ \mu\text{C}$ .
$r$	The distance between the two charges, measured in meters (m). The distance is squared in the formula, so it matters a great deal. A distance of 10 cm is written as 0.10 m before it is used.
$\|q_1 q_2\|$	The absolute value of the product of the two charges. The bars mean ‘the size of’, so the sign is removed and only the strength of the product is kept. The result is always positive, in coulombs squared.

The formula piece by piece

$q_1$: Start with the charge on one object. $q_1$ is its charge in coulombs, with a sign that says whether it is positive or negative. The other object has its own charge, $q_2$ , which is used in the same way.
$|q_1 q_2|$: Multiply the two charges and keep only the size. Doubling either charge doubles the product, and so it doubles the force. The bars make the number positive, so the signs can be used to find the direction instead.
$r^2$: Square the distance. $r^{2}$ means $r \times r$ , measured in square meters. The force spreads out over the surface of a sphere around the charge, and that surface grows with the square of the radius.
$\frac{|q_1 q_2|}{r^2}$: The charge product divided by the squared distance. Dividing by $r^{2}$ means that the farther apart the charges are, the smaller the force becomes, and it falls off faster than the distance does.
$k_e$: The Coulomb constant, which multiplies everything else. It turns the ratio of charges and distance into a force in newtons, so its units are newton square meters per square coulomb.
$F = k_e \frac{|q_1 q_2|}{r^2}$: The whole formula in one sentence: the electric force between two point charges equals the Coulomb constant times the size of the product of the charges, divided by the square of the distance between them.
$F = \frac{1}{4\pi\varepsilon_0} \frac{|q_1 q_2|}{r^2}$: A standard second form, written with the permittivity of free space $\varepsilon_0$ . Since $k_e = \frac{1}{4\pi\varepsilon_0}$ , the two forms are the same law. It is common in physics books that use $\varepsilon_0$ throughout.

What the formula is doing, and why

Electric charge comes in two kinds, positive and negative. Charges of the same sign push each other apart, and charges of opposite sign pull each other together. Coulomb’s law gives the size of that push or pull for two point charges, which are charged objects small enough that their size does not matter. The force $F$ is measured in newtons (N), and the distance $r$ is measured in meters (m).

Why multiply the charges? The force depends on how much charge is on each object. If one charge doubles, the force on each object doubles, and if both charges double, the force becomes four times bigger. Multiplying the two charges, $|q_1 q_2|$ , captures this. The bars around the product mean only its size is used, so the number is always positive. The signs decide the direction: repulsion for like charges, attraction for unlike charges.

Why is the distance squared? The charge sends its influence out in all directions, and at distance $r$ that influence covers the surface of a sphere. The area of a sphere is $4\pi r^{2}$ , and it grows with the square of the radius. A force spread over a bigger area is weaker. Double the distance and the sphere has four times the area, so the force is one quarter of its old value. Triple the distance and it is one ninth. This pattern is an inverse-square law.

The constant $k_e = 8.9875517923 \times 10^{9}\ \text{N}\,\text{m}^2/\text{C}^2$ sets the units and the strength. Gravity has the same inverse-square shape, with masses in place of charges, but gravity only pulls. Coulomb’s law can push or pull. The formula is exact for point charges in empty space, and inside a material the force is weaker, because the material’s charges shield the field. For an object with charge spread over it, the distance is measured from the center of the object.

An everyday picture

Think of a flashlight shining on a wall. The farther the wall is, the wider the same light is spread over it, so each square centimeter gets less light. A point charge sends its influence out in every direction, over the surface of a growing sphere. A sphere’s surface grows with the square of its radius, so at twice the distance the same influence is spread over four times the area, and the force is one quarter.

Worked example: Two 1 microcoulomb charges 10 cm apart

Two positive point charges, each with $q = 1\ \mu\text{C}$ , which is $1 \times 10^{-6}$ C, sit 10 cm apart. Find the force between them. Then move them to 20 cm apart and find the new force. Use $k_e = 8.9875517923 \times 10^{9}\ \text{N}\,\text{m}^2/\text{C}^2$ .

Convert the distance to meters. Centi means one hundredth, so $10\ \text{cm} = 10 \times 10^{-2}\ \text{m} = 0.10\ \text{m}$ . Then $r^{2} = 0.10 \times 0.10 = 0.01\ \text{m}^2$ . The formula needs meters, and skipping this step would change the answer by a factor of 10 000.
Multiply the charges. $|q_1 q_2| = (1 \times 10^{-6}\ \text{C}) \times (1 \times 10^{-6}\ \text{C}) = 1 \times 10^{-12}\ \text{C}^2$ . The front numbers multiply to 1. The exponents add, $-6 + (-6) = -12$ , which gives $10^{-12}$ . The unit is coulomb squared, because each charge is in coulombs.
Divide by the squared distance. $\frac{1 \times 10^{-12}}{0.01} = \frac{1 \times 10^{-12}}{1 \times 10^{-2}} = 1 \times 10^{-10}$ . Dividing by a power of ten subtracts the exponents: $-12 - (-2) = -10$ . Dividing by 0.01 is the same as multiplying by 100, which is why the answer is a much smaller number.
Multiply by the Coulomb constant. $F = 8.9875517923 \times 10^{9} \times 1 \times 10^{-10}$ . Multiply the front numbers, which gives 8.9875517923, and add the exponents, $9 + (-10) = -1$ . So $F = 8.9875517923 \times 10^{-1} \approx 0.899\ \text{N}$ , because $10^{-1}$ means one tenth.
Double the distance. At 20 cm, $r = 0.20$ m and $r^{2} = 0.04\ \text{m}^2$ . That is four times the value at 10 cm, so the force is one quarter: $F = \frac{0.899\ \text{N}}{4} \approx 0.225\ \text{N}$ . Doubling the distance quartered the force, which is the inverse-square law at work.
Find the direction. Both charges are positive, so they are like charges. Like charges repel, so the force pushes the two objects apart. The formula gives the size, and the signs give the direction.
Check the units. The constant has units of newtons times square meters per square coulomb. Multiplying by coulombs squared and dividing by square meters leaves $\frac{\text{N m}^2}{\text{C}^2} \times \frac{\text{C}^2}{\text{m}^2} = \text{N}$ . So the answer is in newtons, as a force should be.

Answer: At 10 cm the two charges push each other apart with a force of about 0.899 N. Like charges repel, so this force is repulsive. At 20 cm the force is about 0.225 N, one quarter of the first.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Forgetting to convert centimeters to meters. Using 10 instead of 0.10 m makes $r^{2}$ 10 000 times too big, so the force comes out 10 000 times too small.
Forgetting to square the distance. The force depends on $r^{2}$ , not on $r$ , so doubling the distance does not halve the force. It cuts the force to one quarter.
Reading the sign of the charges as part of the size. Use the size of the product for the number, then decide push or pull from the signs.
Applying the formula to large or irregular objects without care. It is exact for point charges, and for spheres with evenly spread charge when the distance is measured from their centers.

Where you will meet it

Chemistry, where the attraction between electrons and the nucleus, and between ions, helps explain how atoms bond.
Static electricity, such as a balloon sticking to a wall or hair standing up after it is rubbed.
Inkjet printing, where electric forces steer tiny charged droplets onto the paper.
Electrostatic spray painting, where charged paint droplets are drawn toward a grounded object.

Questions people ask

Why do like charges repel and unlike charges attract?

This is an observed fact of nature, and Coulomb’s law describes it rather than explaining its cause. The formula gives the size of the force, and the signs of the charges give its direction.

What is a coulomb?

The SI unit of charge. A single electron carries about $1.6 \times 10^{-19}$ C, so one coulomb is a very large amount of charge, about $6.24 \times 10^{18}$ electrons.

Why is the distance squared?

The force spreads out over the surface of a sphere around the charge, and a sphere’s surface grows with the square of its radius. A bigger sphere has more area, so the same influence is spread more thinly.

Is Coulomb's law the same as the law of gravity?

The shape is the same, with the square of the distance on the bottom. Gravity uses masses and only attracts, while Coulomb’s law uses charges and can push or pull.

Ohm's Law: Ohm’s law says the push on a wire, called voltage, equals the flow through it, called current, times how hard the wire resists that flow, called resistance.
RC Time Constant: The RC time constant tells you how long a capacitor takes to fill up through a resistor. You get it by multiplying the resistance in ohms by the capacitance in farads, and the answer comes out in seconds.
Hydrogen Energy Levels: An electron in a hydrogen atom can sit only on certain energy steps, and when it jumps between steps it gives off or takes in light of one exact color.