Coulomb's law

[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 square of the 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 – [that were] 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:

[6] The law has been tested extensively, and observations have upheld the law on the scale from 10−16 m to 108 m.[6] 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.

[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.

[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.

Finally, in 1785, the French physicist Charles-Augustin de Coulomb published his first three reports of electricity and magnetism where he stated his law.

The torsion balance consists of a bar suspended from its middle by a thin fiber.

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 two charged balls repelled one another, twisting the fiber through a certain angle, which could be read from a scale on the instrument.

The "continuous charge" version of Coulomb's law is never supposed to be applied to locations for which

There are three conditions to be fulfilled for the validity of Coulomb's inverse square law:[25] 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.

For slow movement, the magnetic force is minimal and Coulomb's law can still be considered approximately correct.

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.

This simple law also correctly accounts for the forces that bind atoms together to form molecules and for the forces that bind atoms and molecules together to form solids and liquids.

Generally, as the distance between ions increases, the force of attraction, and binding energy, approach zero and ionic bonding is less favorable.

Coulomb's law states that the electric field due to a stationary point charge is:

Taking S in the integral form of Gauss's law to be a spherical surface of radius r, centered at the point charge Q, we have

When no acceleration is involved in a particle's history, Coulomb's law can be assumed on any test particle in its own inertial frame, supported by symmetry arguments in solving Maxwell's equation, shown above.

This form of solutions need not obey Newton's third law as is the case in the framework of special relativity (yet without violating relativistic-energy momentum conservation).

Also note that the spherical symmetry for gauss law on stationary charges is not valid for moving charges owing to the breaking of symmetry by the specification of direction of velocity in the problem.

Using the Feynman rules to compute the S-matrix element, we obtain in the non-relativistic limit with

as they arise due to differing normalizations of momentum eigenstate in QFT compared to QM and obtain:

[32] However, the equivalent results of the classical Born derivations for the Coulomb problem are thought to be strictly accidental.

is sufficient to verify that the equality is true taking into account the experimental error.