Coulomb's Law

Coulomb's law lets us calculate the electrostatic force. It is similar to the force of gravity, but while gravity shapes things on a large scale, the electrostatic force shapes the small. It explains concepts like: electricity, chemistry, and light!

Electric Charge

Electric charge determines the magnitude and direction of the electrostatic force. Charge has a role in the electrostatic force similar to mass's role in gravity. To measure charge we use units called Coulombs [C].

The direction of the electrostatic force is attractive if the charges are opposite.

- +

The direction of the force is repulsive if the charges are the same.

- -


+ +

Neutrally charged particles don't produce or experience net electrostatic forces. Examples: neutrons, neutrinos, photons, atoms.


Most electric charge can be thought of as an imbalance of electrons and protons. The electrostatic force draws electrons and protons together in atoms, but you can separate a couple trillion electrons from their protons by walking with socks on a carpet. A TriboElectric Series lists which materials will become electrically charged after they are rubbed together.

- electron:
charge = -1.602 × 10-19 C
mass = 9.109 × 10−31 kg
+ proton:
charge = +1.602 × 10-19 C
mass = 1.672 × 10−27 kg

This simulation runs Coulomb's law (like charges repel, opposites attract). Hopefully, you see "atoms" spontaneously form from randomly placed charges. Try poking it with your mouse.

What role does distance play in the electrostatic force?


Conductive materials allow electric charges to easily move through them. If a charge is applied to one part of a conductive material the charge will quickly spread out because like charges repel.

In this sim fresh electrons are added to the left side to induce a flow.

In chemistry, elements are roughly divided into metals, metalloids and nonmetals. Metals are held together by loosely sharing their outer valence electrons. The cloud of free flowing electrons give metals most of their shared characteristics, like conductivity.

The ease of electron flow is loosely ranked below. Check out wikipedia for more detail on conductivity.

superconductors - zero resistance!
(certain low temperature materials)

conductors - very low resistance
(metals, plasma)

semi-conductors - medium resistance
(metalloids: carbon and silicon)

electrolytes - medium resistance
(a solvent with dissolved ions: sea water, drinking water, soda)

insulators - very high resistance
(vacuum, gases, nonmetals)

Coulomb's Law

After the success of Newton's universal gravitation in 1687 several scientists hypothesized that static electricity worked in a similar fashion, but the French physicist Charles-Augustin de Coulomb is given credit for first publishing the law in 1785.

Historical Background
Since the time of ancient Greece in 600 BCE, people knew that rubbing amber against wool would make the amber attracted to the wool. However, they did not understand why this happened. This question baffled scientists for a long time. By the mid 1700’s things started to change. At the time, static electricity was the only electricity acknowledged in the scientific community. However, Benjamin Franklin started testing reactivity. He used metals to observe electrical charge. Later, Charles Coulomb, a French scientist, verified the Law of Attraction by using hairs and wires to create a torsion balance. He developed and published his findings and law, resulting in what we now know today as Coulomb’s Law.
-Aja Two Crows
F q q r

$$ F = \frac{k_{e}q_{1}q_{2}}{r^{2}} $$

\(F\) = electrostatic force [N, Newton, kg m/s²] vector
\(k_e\) = 8.987 × 109 = Coulomb's constant [N m²/C²] scalar
\(q\) = charge [C, Coulomb] scalar
\(r\) = distance between the center of each charge [m, meters] scalar

Example: What is the electrostatic force between a 4.30μC charge and a 10.08μC charge at a distance of 0.03m?
solution $$ F = \frac{k_{e}q_{1}q_{2}}{r^{2}} $$ $$ F = \frac{(8.987 \times 10^{9})(4.30 \times 10^{-6})(10.08 \times 10^{-6})}{0.03^{2}} $$ $$F = 432.8N$$
Example: What is the electrostatic force between an electron and a proton at a distance of 0.2m? How much acceleration will the electron experience?
solution $$ F = \frac{k_{e}q_{1}q_{2}}{r^{2}} $$ $$ F = \frac{(8.987 \times 10^{9})(-1.602 \times 10^{-19})(1.602 \times 10^{-19})}{0.2^{2}} $$ $$F = 5.76 \times 10^{-27}N$$
$$F = ma$$ $$a = \frac{F}{m} $$ $$a = \frac{5.76 \times 10^{-27}}{9.109 \times 10^{-31}} $$ $$a = 6323.4 \small \frac{m}{s^{2}}$$
Example: Compare the gravitational force to the electrostatic force for a proton and an electron 1m apart.
solution $$ F = \frac{k_{e}q_{1}q_{2}}{r^{2}} $$ $$ F = \frac{(8.987 \times 10^{9})(-1.602 \times 10^{-19})(1.602 \times 10^{-19})}{1^{2}} $$ $$F = 2.30 \times 10^{-28}N$$

$$ F = \frac{GM_{1}M_{2}}{r^{2}} $$ $$ F = \frac{(6.674 \times 10^{-11})(1.672 \times 10^{-27}) (9.109 \times 10^{-31})}{1^{2}}$$ $$F = 1.02 \times 10^{-67}N$$

$$10^{-28} \quad vs. \quad 10^{-67}$$

The electrostatic force is 1039 stronger!

Example: You rub a balloon on a dry erase board and pull 20nC off the board onto the balloon. The balloon sticks to the board. Estimate the maximum possible mass of the balloon if the centers of the charges are 5mm apart?
solution $$\scriptsize n=nano=10^{-9} \quad \quad m=milli = 10^{-3}$$ $$ F = \frac{k_{e}q_{1}q_{2}}{r^2} $$ $$ F = \frac{(8.987 \times 10^{9}) (20 \times 10^{-9})(-20 \times 10^{-9})}{(5 \times 10^{-3})^{2}} $$ $$ F = 0.144N $$

The maximum mass would be when the force of gravity is balanced with the electrostatic force.

$$\sum F=ma$$ $$F_{e}-F_{g}=ma$$ $$0.144-m(9.8)=0$$ $$m(9.8)=0.144$$ $$m=0.0147kg$$