## How Coulomb Made his Equation: Biography of Charles Coulomb & His Equation.

When I looked into the history of Charles Coulomb and his famous equation, I found that even beyond “his” equation, Coulomb was one of the most influential engineers of the 1700s.  And his accomplishments are more impressive when you realize that his background was descent but not considered good enough, his father lost the family fortune, he had a backstabbing boss that almost killed him, and he lived through one of the more turbulent times in history.  Ready for the story? Let’s go!

Back in France Coulomb wrote papers on using calculus to optimize building stability. The encyclopedia says that his paper of 1773 contains “almost an embarrassment of riches [including a]… theory of soil mechanics that remains in use today in basic engineering practice.[7]”

In 1777, the Paris Academy had a competition to create a very sensitive compass with incredibly low friction to measure variations in Earth’s magnetic field during the day[8] as the traditional compass of a small magnet on a pivot had too much friction to record the variation that they were looking for.  Coulomb worked on it for a year and co-won the prize for a thin magnet held up by a thin silk string.  That also inspired Coulomb to work on the physics of friction, which he published in 1781.  An author of a book on friction wrote in 1956, “Coulomb’s contribution to the science of friction were exceptionally great.  Without exaggeration, one can say that he created this science.[9]

Anyway, the Paris Academy was impressed with Coulomb’s compass and installed a version of it at their observatory.  Unfortunately, it turned out to be too sensitive, as a historian put it, the compass “twitched when an assistant sneezed, or when the door opened and trembled when carriages passed the street”.[10]  Coulomb attempted to isolate the needle in a chamber from any drafts but it would still often move when an assistant touched the outside.  He eventually determined that while walking to the compass, the person who was going to measure the angle would often gain a little bit of electric charge which would make the needle sway electrically not magnetically.  Eventually, they grounded the outside of the compass and the needle, which solved the problem.

However, the “twitchy” needle gave Coulomb an idea, maybe he could use a similar setup to measure the electrostatic force.  One of the difficulties with any study of static electricity is that the forces are usually so slight that you can only see the movement of little pieces of fluff or a feather (or if wealthy, gold foil).  Most scales and balances cannot measure such minute forces.  Coulomb decided to study the force of twisting (called torsional force).  He wondered if twisting items worked in a similar manner to springs and to what is called Hooke’s law.

Robert Hooke was an English scientist who lived 100 years earlier than the time of Newton (in fact Hooke claimed that he had proved that the force of gravity depends on one over the distance squared before Newton did but didn’t want to share it, claiming that he would, “conceal it for some time, that others might know how to value it”[11].  Hint: Hooke never revealed it.)  Some laws he did understand and revealed were about springs.  First, Hooke found that for moderate disturbances, the amount springs compress or stretches is in direct proportion to the force on the spring. So, if you double the force, you double the amount the spring stretches, and how much a spring stretches with a given force depends on the strength of the spring, called the spring constant.  Therefore, if you know the spring constant, and you know the amount of displacement, then you can know the force on a spring.  This is how most simple spring scales in people’s bathrooms work today.  Hooke also determined that if you pulled a spring and let go it would oscillate up and down at a steady rate until it stopped.  Therefore, the time for the spring to vibrate back and forth is solely due to the mass on the spring and the spring constant, not the amount you pull on the mass, which means you can use the vibration time to determine the spring constant.

Coulomb then took piano wires and twisted them to measure how they vibrated.  He was pleased to find that the wires twisted back and forth isochronously – or at a constant rate.  This implied that wires twisting follow Hooke’s law – the twist is proportional to the force.  In addition, Coulomb used the time it took for the wire to vibrate back and forth to measure the “twisting” spring constant.  Then, all Coulomb needed was a bar on a piano wire and the amount it moved would tell him the force on the bar.  He had just invented the world’s most precise measuring device, where a degree on the scale was equivalent to 1/100,000th of the weight of a single grain of sand[12]!  Of course, I call it simple, but the actual experiment was incredibly difficult to complete or recreate, and the machine was described by a contemporary as an “all too unsteady twisting machine[13]”.  Still, the torsion machine was then used for all kinds of scientific experiments (including by the quirky Englishman Henry Cavendish in 1798 to determine the Gravitational constant and thus the mass of the Earth).  In fact, torsional balances are *still* used for scientific experiments!

In 1785, Coulomb used his torsional balance on electric charges and experimentally determined that electrical repulsion force has a force that is proportional to one over the distance squared confidently writing: “Fundamental law of Electricity: the repulsive force that the two small balls electrified with the same kind of electricity, follows the inverse proportion of the square of the distance between the centers of the balls”.[14]  It was a bit more difficult to measure attractive forces but the following year Coulomb determined that the attractive force of different charges follows the same relationship and that the electric force was linearly dependent on the size of the charge.  Of course, Coulomb couldn’t directly measure the amount of charge (which is currently measured in Coulombs).  Instead, he touched a charged ball in the balance with a neutral ball of the same size and then removed the outside ball to reduce the charge on the ball in the balance by 2, and found that the force would then also be reduced by 2.  Now Coulomb wasn’t the first person to predict that the electric forces look like gravitational forces, but he was the first to experimentally validate it and the law is justly named Coulomb’s law, the constant, Coulomb’s constant, and the value of a charge is measured in Coulombs in his honor.

Amazingly, Coulomb accomplished all of this while his life was a little, well, chaotic.  In 1783, he was commissioned to determine the feasibility of canal and harbor improvements in Brittany.  However, when he stated that the plan was ill-conceived, the people hoping to profit off of the scheme actually had Coulomb imprisoned in jail for his actions[15]!  He ended up being freed within a week and having the incident improved his reputation and his standing with the government.  However, that didn’t help much when the French Revolution began in 1789 and his supporters were thrown out of power and sometimes, like his friend Lavoisier, lost their heads.  At first, the new leaders were not very happy with a semi-aristocratic scientist like Coulomb, and by 1791 he quit his position in the military and, for a while, hid in the countryside with his very young girlfriend Louise and their baby Charles (who was born in 1790).  By 1795, the new leaders had a change of heart, and Coulomb was invited back to Paris to be an Experimental Physicist at the new Institute of France.  In 1802, Coulomb was made the inspector general of public instruction and, with his new important position, decided to marry Louise and legitimize their now two children (He and Louise had another son in 1797).  By 1806, Coulomb’s illnesses from his time in Martinique caught up with him and he died of a “slow fever” at the ripe age of 70.

Thanks for watching, I have a ton of videos about the history of electricity and quantum mechanics.  I am also making a companion piece about tricks to actually *solve* Coulomb’s problems, in case you want some math (yum, math).

[1] Quoted and translated in Gilmor, C Coulomb and the Evolution of Physics (2017) p. 5

[2] Nollet being his teacher is mentioned in Gilmor, C Coulomb and the Evolution of Physics (2017) p. 16

[3] “Charles Coulomb” from Encyclopedia.com

[4] Quoted and translated in Gilmor, C Coulomb and the Evolution of Physics (2017) p. 24

[5] Quoted and translated in Gilmor, C Coulomb and the Evolution of Physics (2017) p. 25

[6] Gilmor, C Coulomb and the Evolution of Physics (2017) p. 25

[7] “Charles Coulomb” from Encyclopedia.com

[8] p 42 “The Story of Electrical Measurements” Keithley

[9] Quoted in Pickover, C Achimedes to Hawking: Laws of Science (2008) p. 157

[10] p 469-70 “17th

[11] p 45 “Short History of nearly everything”  referring to p219 “the classics of science” Gjertsen

[12] According to Munro, J Pioneers of Electricity (1890) p. 81

[13] ” (p 476 17th)

[14] p 572 Coulomb (1785a) “Premier mémoire sur l’électricité et le magnétisme,” Histoire de l’Académie Royale des Sciences

[15] Found in Munro, J Pioneers of Electricity (1890) p. 80

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