Coulomb interaction

Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law^[1] of physics that calculates the amount of force between two electrically charged particles at rest. This electric force is conventionally called the electrostatic force or Coulomb force.^[2] Although the law was known earlier, it was first published in 1785 by French physicist Charles-Augustin de Coulomb. Coulomb's law was essential to the development of the theory of electromagnetism and maybe even its starting point,^[1] as it allowed meaningful discussions of the amount of electric charge in a particle.^[3]

The law states that the magnitude, or absolute value, of the attractive or repulsive electrostatic force between two point charges is directly proportional to the product of the magnitudes of their charges and inversely proportional to the squared distance between them.^[4] Coulomb discovered that bodies with like electrical charges repel:

It follows therefore from these three tests, that the repulsive force that the two balls – electrified with the same kind of electricity – exert on each other, follows the inverse proportion of the square of the distance.^[5]

Coulomb also showed that oppositely charged bodies attract according to an inverse-square law:

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

Here, $k e$ is a constant, $q 1$ and $q 2$ are the quantities of each charge, and the scalar r is the distance between the charges.

The force is along the straight line joining the two charges. If the charges have the same sign, the electrostatic force between them makes them repel; if they have different signs, the force between them makes them attract.

Being an inverse-square law, the law is similar to Isaac Newton's inverse-square law of universal gravitation, but gravitational forces always make things attract, while electrostatic forces make charges attract or repel. Also, gravitational forces are much weaker than electrostatic forces.^[2] Coulomb's law can be used to derive Gauss's law, and vice versa. In the case of a single point charge at rest, the two laws are equivalent, expressing the same physical law in different ways.^[6] The law has been tested extensively, and observations have upheld the law on the scale from 10⁻¹⁶ m to 10⁸ m.^[6]

History

Ancient cultures around the Mediterranean knew that certain objects, such as rods of amber, could be rubbed with cat's fur to attract light objects like feathers and pieces of paper. Thales of Miletus made the first recorded description of static electricity around 600 BC,^[7] when he noticed that friction could make a piece of amber attract small objects.^[8]^[9]

In 1600, English scientist William Gilbert made a careful study of electricity and magnetism, distinguishing the lodestone effect from static electricity produced by rubbing amber.^[8] He coined the Neo-Latin word electricus ("of amber" or "like amber", from ἤλεκτρον , the Greek word for "amber") to refer to the property of attracting small objects after being rubbed.^[10] This association gave rise to the English words "electric" and "electricity", which made their first appearance in print in Thomas Browne's Pseudodoxia Epidemica of 1646.^[11]

Early investigators of the 18th century who suspected that the electrical force diminished with distance as the force of gravity did (i.e., as the inverse square of the distance) included Daniel Bernoulli^[12] and Alessandro Volta, both of whom measured the force between plates of a capacitor, and Franz Aepinus who supposed the inverse-square law in 1758.^[13]

Based on experiments with electrically charged spheres, Joseph Priestley of England was among the first to propose that electrical force followed an inverse-square law, similar to Newton's law of universal gravitation. However, he did not generalize or elaborate on this.^[14] In 1767, he conjectured that the force between charges varied as the inverse square of the distance.^[15]^[16]

In 1769, Scottish physicist John Robison announced that, according to his measurements, the force of repulsion between two spheres with charges of the same sign varied as $x -2.06$ .^[17]

In the early 1770s, the dependence of the force between charged bodies upon both distance and charge had already been discovered, but not published, by Henry Cavendish of England.^[18] In his notes, Cavendish wrote, "We may therefore conclude that the electric attraction and repulsion must be inversely as some power of the distance between that of the 2 + 1/50th and that of the 2 − 1/50th, and there is no reason to think that it differs at all from the inverse duplicate ratio".

Finally, in 1785, the French physicist Charles-Augustin de Coulomb published his first three reports of electricity and magnetism where he stated his law. This publication was essential to the development of the theory of electromagnetism.^[4] He used a torsion balance to study the repulsion and attraction forces of charged particles, and determined that the magnitude of the electric force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

The torsion balance consists of a bar suspended from its middle by a thin fiber. The fiber acts as a very weak torsion spring. In Coulomb's experiment, the torsion balance was an insulating rod with a metal-coated ball attached to one end, suspended by a silk thread. The ball was charged with a known charge of static electricity, and a second charged ball of the same polarity was brought near it. The two charged balls repelled one another, twisting the fiber through a certain angle, which could be read from a scale on the instrument. By knowing how much force it took to twist the fiber through a given angle, Coulomb was able to calculate the force between the balls and derive his inverse-square proportionality law.

Scalar form

Coulomb's law can be stated as a simple mathematical expression. The scalar form gives the magnitude of the vector of the electrostatic force $F$ between two point charges $q 1$ and $q 2$ , but not its direction. If $r$ is the distance between the charges, the magnitude of the force is

|\mathbf {F} |={\frac {|q_{1}q_{2}|}{4\pi \varepsilon _{0}r^{2}}},

where

ε 0

is the electric constant. If the product

q 1 q 2

is positive, the force between the two charges is repulsive; if the product is negative, the force between them is attractive.^[19]

Vector form

In the image, the vector

F 1

is the force experienced by

q 1

, and the vector

F 2

is the force experienced by

q 2

. When

q 1 q 2 > 0

the forces are repulsive (as in the image) and when

q 1 q 2 < 0

the forces are attractive (opposite to the image). The magnitude of the forces will always be equal.

Coulomb's law in vector form states that the electrostatic force ${\textstyle \mathbf {F} _{1}}$ experienced by a charge, $q_{1}$ at position $\mathbf {r} _{1}$ , in the vicinity of another charge, $q_{2}$ at position $\mathbf {r} _{2}$ , in a vacuum is equal to^[20]

\mathbf {F} _{1}={\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}}}{{\hat {\mathbf {r} }}_{12} \over {|\mathbf {r} _{12}|}^{2}}={\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}}}{\mathbf {r} _{12} \over {|\mathbf {r} _{12}|}^{3}}

where ${\textstyle \mathbf {r_{12}=r_{1}-r_{2}} }$ is the displacement vector between the charges, ${\textstyle {\hat {\mathbf {r} }}_{12}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\mathbf {r} _{12}}{|\mathbf {r} _{12}|}}}$ a unit vector pointing from ${\textstyle q_{2}}$ to ${\textstyle q_{1}}$ , and $\varepsilon _{0}$ the electric constant. Here, ${\textstyle \mathbf {\hat {r}} _{12}}$