Coulomb's law

Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law 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. 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 may even be its starting point, as it allowed meaningful discussions of the amount of electric charge in a particle.
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 square of the distance between them. Coulomb discovered that bodies with like electrical charges repel:
Coulomb also showed that oppositely charged bodies attract according to an inverse-square law:
Here, is the Coulomb constant, and 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 bodies. 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. 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. The law has been tested extensively, and observations have upheld the law on the scale from 10⁻¹⁶ m to 10⁸ m.

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, when he noticed that friction could make a piece of amber attract small objects.
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. He coined the Neo-Latin word electricus to refer to the property of attracting small objects after being rubbed. 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.
Early investigators of the 18th century who suspected that the electrical force diminished with distance as the force of gravity did included Daniel Bernoulli 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.
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. In 1767, he conjectured that the force between charges varied as the inverse square of the distance.
Image:Bcoulomb.png|thumb|Coulomb's torsion balance
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.
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. 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 and that of the, 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. 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.

Mathematical form

Coulomb's law states that the electrostatic force experienced by a charge, at position, in the vicinity of another charge, at position, in a vacuum is equal to
where is the displacement vector between the charges, a unit vector pointing from to and the electric constant. Here, is used for the vector notation. The electrostatic force experienced by, according to Newton's third law, is
If both charges have the same sign then the product is positive and the direction of the force on is given by ; the charges repel each other. If the charges have opposite signs then the product is negative and the direction of the force on is the charges attract each other.

System of discrete charges

The law of superposition allows Coulomb's law to be extended to include any number of point charges. The force acting on a point charge due to a system of point charges is simply the vector addition of the individual forces acting alone on that point charge due to each one of the charges. The resulting force vector is parallel to the electric field vector at that point, with that point charge removed.
Force on a small charge at position, due to a system of discrete charges in vacuum is
where is the magnitude of the th charge, is the vector from its position to and is the unit vector in the direction of.

Continuous charge distribution

In this case, the principle of linear superposition is also used. For a continuous charge distribution, an integral over the region containing the charge is equivalent to an infinite summation, treating each infinitesimal element of space as a point charge. The distribution of charge is usually linear, surface or volumetric.
For a linear charge distribution where gives the charge per unit length at position, and is an infinitesimal element of length,
For a surface charge distribution where gives the charge per unit area at position, and is an infinitesimal element of area,
For a volume charge distribution where gives the charge per unit volume at position, and is an infinitesimal element of volume,
The force on a small test charge at position in vacuum is given by the integral over the distribution of charge
The "continuous charge" version of Coulomb's law is never supposed to be applied to locations for which because that location would directly overlap with the location of a charged particle which is not a valid location to analyze the electric field or potential classically. Charge is always discrete in reality, and the "continuous charge" assumption is just an approximation that is not supposed to allow to be analyzed.

Coulomb constant

The constant of proportionality,, in Coulomb's law:
is a consequence of historical choices for units.

The constant is the vacuum electric permittivity. Using the CODATA 2022 recommended value for, the Coulomb constant is

Limitations

There are three conditions to be fulfilled for the validity of Coulomb's inverse square law:

The charges must have a spherically symmetric distribution.
The charges must not overlap.
The charges must be stationary with respect to a non-accelerating frame of reference.

The last of these is known as the electrostatic approximation. When movement takes place, an extra factor is introduced, which alters the force produced on the two objects. This extra part of the force is called the magnetic force. For slow movement, the magnetic force is minimal and Coulomb's law can still be considered approximately correct. A more accurate approximation in this case is, however, the Weber force. When the charges are moving more quickly in relation to each other or accelerations occur, Maxwell's equations and Einstein's theory of relativity must be taken into consideration.

Electric field

An electric field is a vector field that associates to each point in space the Coulomb force experienced by a unit test charge. The strength and direction of the Coulomb force on a charge depends on the electric field established by other charges that it finds itself in, such that. In the simplest case, the field is considered to be generated solely by a single source point charge. More generally, the field can be generated by a distribution of charges who contribute to the overall by the principle of superposition.
If the field is generated by a positive source point charge, the direction of the electric field points along lines directed radially outwards from it, i.e. in the direction that a positive point test charge would move if placed in the field. For a negative point source charge, the direction is radially inwards.
The magnitude of the electric field can be derived from Coulomb's law. By choosing one of the point charges to be the source, and the other to be the test charge, it follows from Coulomb's law that the magnitude of the electric field created by a single source point charge Q at a certain distance from it r in vacuum is given by
A system of n discrete charges stationed at produces an electric field whose magnitude and direction is, by superposition