# 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].

opposite charges have an attractive force

negative charges have a repulsive force

positive charges have a repulsive force

neutral charges have no force

All charge comes from electrons, protons, or rare exotic particles.

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). You can see the randomly placed charges spontaneously form atoms.

They love making atoms. Try poking the simulations with your mouse.

## 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.

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

$$F$$ = electrostatic force [N, Newton, kg m/s²] vector
$$k_e$$ = 8.987 × 10 9 = Coulomb's constant [N m²/C²]
$$q$$ = charge [C, Coulomb]
$$r$$ = distance between the center of each charge [m, meters]

Valid for stationary point source charges at macroscopic sizes

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 $$F = \frac{k_{e}q_{1}q_{2}}{r^{2}}$$ $$F = \frac{(8.987 \times 10^{9})(-1.602 \times 10^{-19})(1.602 \times 10^{-19})}{1^{2}}$$ $$F = 2.30 \times 10^{-28}N$$

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

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

The electrostatic force is 10 39 times stronger!
or
1,000,000,000,000,000,000,000,000,000,000,000,000,000!

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.

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

## Limits of Coulomb's Law

Coulomb's law is a good approximation of nature, but like most classical equations it has it's limits.

Coulomb's law assumes that force is applied instantly at a distance. This works fine for stationary charges, but when a charge is moving it doesn't take into account the delay we see from the speed of light. This issue was fixed by Maxwell's equations in 1861.

Coulomb's law breaks down at the atomic scale. A better model of charged particles comes from quantum electrodynamics. To learn more I recommend QED, a book by Richard Feynman.

## Conductivity

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 simulation electrons are removed from the right and added to the left to produce a messy looking flow of charge.

Why is the current higher when the atoms are closer together?

A gap in the wire will reduce the flow of electrons.

gap =                  separation =                 rows =

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 ceramics

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 water

insulators - very high resistance
vacuum, nonmetals: gases, plastics, silk, fur

Question: Why are metals more conductive than nonmetals?
solution
In metals some electrons are free to move between atoms. In nonmetals the electrons are locked up in covalent bonds so they resist the electrostatic force.

If you heat a nonmetal up so much that electrons can leave the nucleus you get a plasma. A plasma is very conductive.

## Static Electricity

Static electricity occurs when there is an imbalance of electrons and protons. A lasting charge separation can only occur in insulating materials, because in conductors the charges pair up very quickly.

Static electricity effects are much stronger and longer lasting in low humidity. This is because water molecules increase the conductivity of air, allowing more separated charges to return.

It is quite easy to separate a couple trillion electrons from their protons by walking with socks on a carpet. Lightning is produced in a similar way when a cloud with rising air ends up with an unbalanced distribution of charge.

A TriboElectric Series lists which materials will become electrically charged after they are rubbed together.

Question: If you rubbed polystyrene foam (styrofoam) on a cat, static electricity would cause them stick together. Which would gain a positive charge? Use the TriboElectric series to decide.
solution
The cat would end up with a positive charge. This means the cat would lose electrons to the styrofoam.
Question: Rubbing glass on plastic wrap can get super staticky. Use the TriboElectric series to decide which gets the positive charge.
solution
The glass would get the positive charge.