# Chapter 2 The Electrostatic Force

In 1687, Sir Isaac Newton published his three laws of motion and his universal law of gravitation, arguably the most successful theory in the whole history of science. Philosophiæ Naturalis Principia Mathematica was written in Latin, was embraced by scientists across all of Europe, and sparked a mathematical revolution. Unifying physics and astronomy, and providing the link between disparate disciplines, many in Europe regarded the Prinicipia as presenting the ultimate solution to the laws of nature.

Despite its success, Newton was worried about several lingering philosophical questions, especially the issue of action at a distance. The primary theme of Newton’s laws of motion, for instance, concerns cause and effect. When one object applies a force on another, according to Newton’s second law, the other object accelerates by an amount proportional to the force applied and inversely proportional to its own mass. This was easy enough to appreciate in the case of contact forces, such as collisions, where one body directly interacts with another. However, gravity works differently. The moon orbits the earth, and the earth’s oceans move in response to the moon. How does this happen? What is it that links the seas to the moon, despite the vast distance between them? Presumably some mechanism conveys this gravitational force, but Newton did not know what it was and would not speculate about it in his formal works, writing:

I have not as yet been able to discover the reason for these properties of gravity from phenomena, and I do not feign hypotheses. For whatever is not deduced from the phenomena must be called a hypothesis; and hypotheses, whether metaphysical or physical, or based on occult qualities, or mechanical, have no place in experimental philosophy. In this philosophy particular propositions are inferred from the phenomena, and afterwards rendered general by induction.[1]

While Newton’s disdain for action at a distance permeated Cambridge, Continental physicists appreciated the mathematical beauty and were not disturbed by action at a distance. After all, God surely made an elegant universe, but why should he be constrained by issues of communication over great distances? More importantly, eighteenth century French mathematicians had the skills, and government backing, to develop intricate theories of celestial mechanics, which were eventually laid out by Pierre-Simon Laplace in his 1798 Traité de Mechanique Céleste (Treatise on Celestial Mechanics).

Meanwhile, in Colonial America, Benjamin Franklin had joined a radical political faction, which, by 1777, was about to loose a revolution against the British Crown. Something had to be done. In a last ditch effort, Benjamin Franklin snuck over to Paris to ask King Louis XVI for help. Let us not forget that, all through the eighteenth century, France and Britain were enmeshed in a great struggle for world dominance. Much like today’s great powers, France’s government used scientific funding to project power and improve their military technology.

Every day, the local magnetic field changes slightly, but by how much and why? Can it be predicted and accounted for by naval navigators? This was the question of the annual scientific Gran Prix of the Académie des Sciences when Franklin arrived in the French capital. This Gran Prix question also launched the second career of a first lieutenant in the French corps of engineers, who, after the Seven Years War, built a new fort on the island of Martinique.

When Charles-Augustin de Coulomb returned to France he continued his military service domestically, where he had time also to apply his craft to his scientific interests, such as how to measure finely the terrestrial magnetic field. Coulomb suspected that the frictional torque of a suspended compass needle could be made negligible if he used fine silk threads. This made the new compass like a pendulum, and Coulomb realized that it would twist back and forth in a similar manner as a pendulum oscillates. Just as the period of a pendulum remains constant with small amplitude, the suspended needle also swings back and forth isochronously. Furthermore, the torque should be directly proportional to the angle of twist, provided that the angle of twist does not exceed the elastic limit of the thread.

This new torsion magnetometer allowed the little known engineer to share the 1777 Gran Prix with an established compass designer. Meanwhile, Franklin showed himself to be a brilliant diplomat, as France went to war the next year and he remained in Paris as the American ambassador. Franklin may have attended the 1781 scientific Gran Prix, which Coulomb won outright with a detailed study of friction in simple machines.

Franklin’s job in France was to be a diplomat, but he was also an eminent scientist. The king put him to work in this capacity, perhaps in the hope that it would help the war effort. In 1784, Franklin chaired a scientific commission that evaluated the technical designs of lightning rods to protect French military outposts.[2] Also authoring the report was Charles-Augustin de Coulomb.

Also in 1784, Coulomb published a report titled Theoretical Research and Experimentation on Torsion and the Elasticity of Metal Wire, which paved the way for perhaps the most fundamental electrical discovery since Franklin’s work as a young man.

Pursuing an analogy with Newton’s Law of Gravity, Coulomb demonstrated experimentally that the force between two stationary point charges acts along the line joining them and is inversely proportional to the square of their mutual distance, or in symbols:  $\vec{F}\propto \frac{{{q}_{1}}\,{{q}_{2}}}{{{r}^{2}}}\hat{r}\,$.

As in his diagram (above), Coulomb replaced the silk thread in his compass with a piano wire, as it had a more linear rotational spring constant. He then suspended a lead cylinder in a pot of water and observed the rate at which the amplitude of the torsional motions decreased.

In this manner, Coulomb determined the force of resistance of the water to the cylinder, equivalent to a torque of , which was beyond the sensitivities of previous instruments.[3]

He then applied his technique to measure the force between electric charges, by using a waxed silk thread to suspend a bar holding small charged spheres at each end. In spite of significant experimental error, Coulomb announced his conclusion that the force between the charges obeyed an inverse square law similar to Newton’s universal law of gravity.[4]

Before the recent ascent of computational integration techniques, applying Coulomb’s law was quite difficult in practice. However, since Coulomb’s law follows the same inverse square law as Newton’s Law of Gravity, Laplace and others could apply the physics of celestial mechanics to the field of electrostatics. This led to what was known as the astronomical model of electricity, and is now the basis for the field of electrostatics.

In the end, Franklin gained the help of Louis XVI and the French government. Thanks to France’s involvement, the war against Britain was a great success for the American revolutionaries. France got little out of it, however, except a worsening financial situation and perhaps a strong whiff of Franklin’s subversive political ideas. In 1785 Franklin returned to America and by 1789 political unrest took over France. In 1793, Louis XVI lost his head and the young Napoleon Bonaparte sieged Toulon.

Coulomb retired from the military, and laid low during the French Revolution. After the French Revolution, Coulomb came out of retirement to help with the development of the metric system and public education.   He remained active in research until his death at age 70.

Meanwhile, across the English channel, Michael Faraday picked up where Franklin’s Leyden jar experiments had left off. In particular, Faraday conducted experiments using spherical Leyden jars with interchangeable insulators made a various materials. To explain his experiments, Faraday suggested that the positive and negative charge in the insulator remain bound together, but become somewhat separated, so he called these materials dielectrics. Written in clear English without mathematical complexity, his well illustrated three volume treatise[5] documents each experiment, along with how they fit together. Eventually he forms a coherent theory involving long chains of microscopic dipoles linking cause to effect.

Regardless of the microscopic details, Faraday could map out these lines of electric force from positive to negative charge. The electric force imparted on another charge would simply be related to the number of lines of force per area, which he eventually called the electric field. His idea also nicely explained the inverse-square nature of Coulomb’s law, as the number of radial field lines per area would decrease as the square of the distance from a spherical charge.

Using his spherical Leyden jar, Faraday also showed that the constant in Coulomb’s law varied as a function of each dielectric material. Thus, every substance would have a corresponding property, called the permittivity, that measured how much the internal charges are displaced when exposed to an electric field, which in turn would convey the electric force.

While it did not have anything to do with gravity directly, the idea of an analogous gravitational field, which was somehow conveyed through matter, gave hope to the British physics community that it was only a matter of time until mechanisms would be found for every force that appears to act at a distance.

[1] Isaac Newton, The Principia: Mathematical Principles of Natural Philosophy (Philosophiae Naturalis Principia Mathematica) Third ed. (1726), trans. I. Bernard Cohen and Anne Whitman (California: University of California Press, 1999), p. 943.

[2] MM. Franklin, Le Roy, Coulomb, Delaplace and l’Abbé Rochon, Académie Royale des Sciences, minutes, tome CIII, fol. 90 v°-95 r° (session for Saturday April 24, 1784). From The Papers of Benjamin Franklin, Yale University Press, posted on franklinpapers.org.

[3] J.L. Heilbron, Electricity in the17th and 18th centuries: a study of early Modern physics. (University of California Press, 1979): 469-70.

[4] Coulomb, Mémoires des Académie des Sciences, (1785): 229-638. These are a series of papers. It is interesting to note that the English scientist Henry Cavendish (1731-1810) obtained the same result, but using a different apparatus. Unlike Coulomb, Cavendish did not publish his result. J.C. Maxwell published this, along with other pioneering work done by Cavendish, nearly a hundred years later in 1879.

[5] Michael Faraday, Experimental Researches in Electricity, 3 vols. (London: Taylor, 1839, 1844; Taylor and Francis, 1855)