Louis de Broglie

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Louis_de_Broglie

 

De Broglie in 1929

Louis Victor Pierre Raymond, 7th Duc de Broglie (/də ˈbroʊɡli/, also US: /də broʊˈɡliː, də ˈbrɔɪ/;French: [də bʁɔj]^[ or [də bʁœj] ^ⓘ; 15 August 1892 – 19 March 1987) was a French physicist and aristocrat known for his contributions to quantum theory. In his 1924 PhD thesis, he postulated the wave nature of electrons and suggested that all matter has wave properties. This concept is known as the de Broglie hypothesis, an example of wave-particle duality, and forms a central part of the theory of quantum mechanics.

De Broglie won the Nobel Prize in Physics in 1929, after the wave-like behaviour of matter was first experimentally demonstrated in 1927.

The wave-like behaviour of particles discovered by de Broglie was used by Erwin Schrödinger in his formulation of wave mechanics. De Broglie's pilot-wave concept, was presented at the 1927 Solvay Conferences then abandoned, in favor of the quantum mechanics, until 1952 when it was rediscovered and enhanced by David Bohm.

Louis de Broglie was the sixteenth member elected to occupy seat 1 of the Académie française in 1944, and served as Perpetual Secretary of the French Academy of Sciences. De Broglie became the first high-level scientist to call for establishment of a multi-national laboratory, a proposal that led to the establishment of the European Organization for Nuclear Research (CERN).

Biography

Family and education

François-Marie, 1st duc de Broglie (1671–1745) ancestor of Louis de Broglie and Marshal of France under Louis XV of France

Louis de Broglie belonged to the famous aristocratic family of Broglie, whose representatives for several centuries occupied important military and political posts in France. The father of the future physicist, Louis-Alphonse-Victor, 5th duc de Broglie, was married to Pauline d’Armaille, the granddaughter of the Napoleonic General Philippe Paul, comte de Ségur and his wife, the biographer, Marie Célestine Amélie d'Armaillé. They had five children; in addition to Louis, these were: Albertina (1872–1946), subsequently the Marquise de Luppé; Maurice (1875–1960), subsequently a famous experimental physicist; Philip (1881–1890), who died two years before the birth of Louis, and Pauline, Comtesse de Pange (1888–1972), subsequently a famous writer.

Louis was born in Dieppe, Seine-Maritime. As the youngest child in the family, Louis grew up in relative loneliness, read a lot, and was fond of history, especially political. From early childhood, he had a good memory and could accurately read an excerpt from a theatrical production or give a complete list of ministers of the Third Republic of France. For this, he was predicted to become a great statesman in the future.

De Broglie had intended a career in humanities, and received his first degree (licence ès lettres) in history. Afterwards he turned his attention toward mathematics and physics and received a degree (licence ès sciences) in physics. With the outbreak of the First World War in 1914, he offered his services to the army in the development of radio communications.

Military service

After graduation, Louis de Broglie joined the engineering forces to undergo compulsory service. It began at Fort Mont Valérien, but soon, on the initiative of his brother, he was seconded to the Wireless Communications Service and worked on the Eiffel Tower, where the radio transmitter was located. Louis de Broglie remained in military service throughout the First World War, dealing with purely technical issues. In particular, together with Léon Brillouin and brother Maurice, he participated in establishing wireless communications with submarines. Louis de Broglie was demobilized in August 1919 with the rank of adjudant. Later, the scientist regretted that he had to spend about six years away from the fundamental problems of science that interested him.

Scientific and pedagogical career

His 1924 thesis Recherches sur la théorie des quanta (Research on the Theory of the Quanta) introduced his theory of electron waves. This included the wave–particle duality theory of matter, based on the work of Max Planck and Albert Einstein on light. This research culminated in the de Broglie hypothesis stating that any moving particle or object had an associated wave. De Broglie thus created a new field in physics, the mécanique ondulatoire, or wave mechanics, uniting the physics of energy (wave) and matter (particle). He won the Nobel Prize in Physics in 1929 "for his discovery of the wave nature of electrons".

In his later career, de Broglie worked to develop a causal explanation of wave mechanics, in opposition to the wholly probabilistic models which dominate quantum mechanical theory; it was refined by David Bohm in the 1950s. The theory has since been known as the De Broglie–Bohm theory.

In addition to strictly scientific work, de Broglie thought and wrote about the philosophy of science, including the value of modern scientific discoveries. In 1930 he founded the book series Actualités scientifiques et industrielles published by Éditions Hermann.

De Broglie became a member of the Académie des sciences in 1933, and was the academy's perpetual secretary from 1942. He was asked to join Le Conseil de l'Union Catholique des Scientifiques Francais, but declined because he was non-religious. In 1941, he was made a member of the National Council of Vichy France. On 12 October 1944, he was elected to the Académie Française, replacing mathematician Émile Picard. Because of the deaths and imprisonments of Académie members during the occupation and other effects of the war, the Académie was unable to meet the quorum of twenty members for his election; due to the exceptional circumstances, however, his unanimous election by the seventeen members present was accepted. In an event unique in the history of the Académie, he was received as a member by his own brother Maurice, who had been elected in 1934. UNESCO awarded him the first Kalinga Prize in 1952 for his work in popularizing scientific knowledge, and he was elected a Foreign Member of the Royal Society on 23 April 1953.

Louis became the 7th duc de Broglie in 1960 upon the death without heir of his elder brother, Maurice, 6th duc de Broglie, also a physicist.

In 1961, he received the title of Knight of the Grand Cross in the Légion d'honneur. De Broglie was awarded a post as counselor to the French High Commission of Atomic Energy in 1945 for his efforts to bring industry and science closer together. He established a center for applied mechanics at the Henri Poincaré Institute, where research into optics, cybernetics, and atomic energy were carried out. He inspired the formation of the International Academy of Quantum Molecular Science and was an early member.

Louis never married. When he died on 19 March 1987 in Louveciennes at the age of 94, he was succeeded as duke by a distant cousin, Victor-François, 8th duc de Broglie. His funeral was held 23 March 1987 at the Church of Saint-Pierre-de-Neuilly.

Scientific activity

Physics of X-ray and photoelectric effect

The first works of Louis de Broglie (early 1920s) were performed in the laboratory of his older brother Maurice and dealt with the features of the photoelectric effect and the properties of x-rays. These publications examined the absorption of X-rays and described this phenomenon using the Bohr theory, applied quantum principles to the interpretation of photoelectron spectra, and gave a systematic classification of X-ray spectra. The studies of X-ray spectra were important for elucidating the structure of the internal electron shells of atoms (optical spectra are determined by the outer shells). Thus, the results of experiments conducted together with Alexandre Dauvillier, revealed the shortcomings of the existing schemes for the distribution of electrons in atoms; these difficulties were eliminated by Edmund Stoner. Another result was the elucidation of the insufficiency of the Sommerfeld formula for determining the position of lines in X-ray spectra; this discrepancy was eliminated after the discovery of the electron spin. In 1925 and 1926, Leningrad physicist Orest Khvolson nominated the de Broglie brothers for the Nobel Prize for their work in the field of X-rays.

Matter and wave–particle duality

Main article: De Broglie hypothesis

Studying the nature of X-ray radiation and discussing its properties with his brother Maurice, who considered these rays to be some kind of combination of waves and particles, contributed to Louis de Broglie's awareness of the need to build a theory linking particle and wave representations. In addition, he was familiar with the works (1919–1922) of Marcel Brillouin, which proposed a hydrodynamic model of an atom and attempted to relate it to the results of Bohr's theory. The starting point in the work of Louis de Broglie was the idea of Einstein about the quanta of light. In his first article on this subject, published in 1922, the French scientist considered blackbody radiation as a gas of light quanta and, using classical statistical mechanics, derived the Wien radiation law in the framework of such a representation. In his next publication, he tried to reconcile the concept of light quanta with the phenomena of interference and diffraction and came to the conclusion that it was necessary to associate a certain periodicity with quanta. In this case, light quanta were interpreted by him as relativistic particles of very small mass.

It remained to extend the wave considerations to any massive particles, and in the summer of 1923 a decisive breakthrough occurred. De Broglie outlined his ideas in a short note "Waves and quanta" (French: Ondes et quanta, presented at a meeting of the Paris Academy of Sciences on September 10, 1923), which marked the beginning of the creation of wave mechanics. In this paper and his subsequent PhD thesis, the scientist suggested that a moving particle with energy E and velocity v is characterized by some internal periodic process with a frequency ${\displaystyle E/h}$ (later known as Compton frequency), where ${\displaystyle h}$ is the Planck constant. To reconcile these considerations, based on the quantum principle, with the ideas of special relativity, de Broglie associated wave he called a "phase wave" with a moving body, which propagates with the phase velocity ${\displaystyle c^2/v}$. Such a wave, which later received the name matter wave, or de Broglie wave, in the process of body movement remains in phase with the internal periodic process. Having then examined the motion of an electron in a closed orbit, the scientist showed that the requirement for phase matching directly leads to the quantum Bohr-Sommerfeld condition, that is, to quantize the angular momentum. In the next two notes (reported at the meetings on September 24 and October 8, respectively), de Broglie came to the conclusion that the particle velocity is equal to the group velocity of phase waves, and the particle moves along the normal to surfaces of equal phase. In the general case, the trajectory of a particle can be determined using Fermat's principle (for waves) or the principle of least action (for particles), which indicates a connection between geometric optics and classical mechanics.

This theory set the basis of wave mechanics. It was supported by Einstein, confirmed by the electron diffraction experiments of G P Thomson and Davisson and Germer, and generalized by the work of Erwin Schrödinger.

From a philosophical viewpoint, this theory of matter-waves has contributed greatly to the ruin of the atomism of the past. Originally, de Broglie thought that real wave (i.e., having a direct physical interpretation) was associated with particles. In fact, the wave aspect of matter was formalized by a wavefunction defined by the Schrödinger equation, which is a pure mathematical entity having a probabilistic interpretation, without the support of real physical elements. This wavefunction gives an appearance of wave behavior to matter, without making real physical waves appear. However, until the end of his life de Broglie returned to a direct and real physical interpretation of matter-waves, following the work of David Bohm.

Conjecture of an internal clock of the electron

de Broglie presented at the Solvay conference 1927 (third from right in middle row)

In his 1924 thesis, de Broglie conjectured that the electron has an internal clock that constitutes part of the mechanism by which a pilot wave guides a particle. Subsequently, David Hestenes has proposed a link to the zitterbewegung that was suggested by Schrödinger.

While attempts at verifying the internal clock hypothesis and measuring clock frequency are so far not conclusive, recent experimental data is at least compatible with de Broglie's conjecture.

Non-nullity and variability of mass

According to de Broglie, the neutrino and the photon have rest masses that are non-zero, though very low. That a photon is not quite massless is imposed by the coherence of his theory. Incidentally, this rejection of the hypothesis of a massless photon enabled him to doubt the hypothesis of the expansion of the universe.

In addition, he believed that the true mass of particles is not constant, but variable, and that each particle can be represented as a thermodynamic machine equivalent to a cyclic integral of action.

Generalization of the principle of least action

See also: Hamilton's optico-mechanical analogy

In the second part of his 1924 thesis, de Broglie used the equivalence of the mechanical principle of least action with Fermat's optical principle: "Fermat's principle applied to phase waves is identical to Maupertuis' principle applied to the moving body; the possible dynamic trajectories of the moving body are identical to the possible rays of the wave." This latter equivalence had been pointed out by William Rowan Hamilton a century earlier, and published by him around 1830, for the case of light.

Duality of the laws of nature

Far from claiming to make "the contradiction disappear" which Max Born thought could be achieved with a statistical approach, de Broglie extended wave–particle duality to all particles (and to crystals which revealed the effects of diffraction) and extended the principle of duality to the laws of nature.

His last work made a single system of laws from the two large systems of thermodynamics and of mechanics:

When Boltzmann and his continuators developed their statistical interpretation of Thermodynamics, one could have considered Thermodynamics to be a complicated branch of Dynamics. But, with my actual ideas, it's Dynamics that appear to be a simplified branch of Thermodynamics. I think that, of all the ideas that I've introduced in quantum theory in these past years, it's that idea that is, by far, the most important and the most profound.

That idea seems to match the continuous–discontinuous duality, since its dynamics could be the limit of its thermodynamics when transitions to continuous limits are postulated. It is also close to that of Gottfried Wilhelm Leibniz, who posited the necessity of "architectonic principles" to complete the system of mechanical laws.

However, according to him, there is less duality, in the sense of opposition, than synthesis (one is the limit of the other) and the effort of synthesis is constant according to him, like in his first formula, in which the first member pertains to mechanics and the second to optics:

${\displaystyle m{c}^2=h\nu }$

Neutrino theory of light

Main article: Neutrino theory of light

This theory, which dates from 1934, introduces the idea that the photon is equivalent to the fusion of two Dirac neutrinos. In 1938, the concept was challenged as not rotationally invariant and work on the concept was largedly discontinued.

Hidden thermodynamics

De Broglie's final idea was the hidden thermodynamics of isolated particles. It is an attempt to bring together the three furthest principles of physics: the principles of Fermat, Maupertuis, and Carnot.

In this work, action becomes a sort of opposite to entropy, through an equation that relates the only two universal dimensions of the form:

${\displaystyle \frac{\text{action}}{h}=-\frac{\text{entropy}}{k}}$

As a consequence of its great impact, this theory brings back the uncertainty principle to distances around extrema of action, distances corresponding to reductions in entropy.

Coulomb's law

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Coulomb%27s_law 

The magnitude of the electrostatic force F between two point charges q₁ and q₂ is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them. Like charges repel each other, and opposite charges attract each other.

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 maybe even 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:

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.

Coulomb also showed that oppositely charged bodies attract according to an inverse-square law: $${\displaystyle |F|=k_{\text{e}}\frac{|q_1||q_2|}{r^2}}$$

Here, k_e is a constant, q₁ and q₂ 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. 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

Charles-Augustin de Coulomb

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 ("of amber" or "like amber", from ἤλεκτρον [elektron], the Greek word for "amber") 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 (i.e., as the inverse square of the distance) 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.

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 x^−2.06.

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 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. 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

In the image, the vector F₁ is the force experienced by q₁, and the vector F₂ is the force experienced by q₂. When q₁q₂ > 0 the forces are repulsive (as in the image) and when q₁q₂ < 0 the forces are attractive (opposite to the image). The magnitude of the forces will always be equal.

Coulomb's law states that the electrostatic force ${\mathbf{F}}_1$ experienced by a charge, ${\displaystyle q_1}$ at position ${\displaystyle {\mathbf{r}}_1}$, in the vicinity of another charge, ${\displaystyle q_2}$ at position ${\displaystyle {\mathbf{r}}_2}$, in a vacuum is equal to $${\displaystyle {\mathbf{F}}_1=\frac{q_1{q}_2}{4\pi {\epsilon }_0}\frac{{\hat{\mathbf{r}}}_{12}}{{|{\mathbf{r}}_{12}|}^2}}$$

where ${\mathbf{r}}_{\mathbf{12}}\mathbf{=}{\mathbf{r}}_{\mathbf{1}}\mathbf{-}{\mathbf{r}}_{\mathbf{2}}$ is the displacement vector between the charges, ${\hat{\mathbf{r}}}_{12}$ a unit vector pointing from $q_2$ to $q_1$, and ${\displaystyle {\epsilon }_0}$ the electric constant. Here, ${\hat{\mathbf{r}}}_{12}$ is used for the vector notation. The electrostatic force ${\mathbf{F}}_2$ experienced by ${\displaystyle q_2}$, according to Newton's third law, is ${\mathbf{F}}_2=-{\mathbf{F}}_1$.

If both charges have the same sign (like charges) then the product ${\displaystyle q_1{q}_2}$ is positive and the direction of the force on ${\displaystyle q_1}$ is given by ${\hat{\mathbf{r}}}_{12}$; the charges repel each other. If the charges have opposite signs then the product ${\displaystyle q_1{q}_2}$ is negative and the direction of the force on ${\displaystyle q_1}$ is $-{\hat{\mathbf{r}}}_{12}$; 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 $\mathbf{F}$ on a small charge ${\displaystyle q}$ at position ${\displaystyle \mathbf{r}}$, due to a system of $n$ discrete charges in vacuum is

$${\displaystyle \mathbf{F}(\mathbf{r})=\frac{q}{4\pi {\epsilon }_0}\sum _{i=1}^n{q}_i\frac{{\hat{\mathbf{r}}}_i}{{|{\mathbf{r}}_i|}^2},}$$

where ${\displaystyle q_i}$ is the magnitude of the ith charge, ${\mathbf{r}}_i$ is the vector from its position to ${\displaystyle \mathbf{r}}$ and ${\hat{\mathbf{r}}}_i$ is the unit vector in the direction of ${\displaystyle {\mathbf{r}}_i}$.

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 ${\displaystyle dq}$. The distribution of charge is usually linear, surface or volumetric.

For a linear charge distribution (a good approximation for charge in a wire) where ${\displaystyle \lambda ({\mathbf{r}}^{\prime })}$ gives the charge per unit length at position ${\displaystyle {\mathbf{r}}^{\prime }}$, and ${\displaystyle d{\ell }^{\prime }}$ is an infinitesimal element of length, $${\displaystyle d{q}^{\prime }=\lambda ({\mathbf{r}}^{\prime })d{\ell }^{\prime }.}$$

For a surface charge distribution (a good approximation for charge on a plate in a parallel plate capacitor) where ${\displaystyle \sigma ({\mathbf{r}}^{\prime })}$ gives the charge per unit area at position ${\displaystyle {\mathbf{r}}^{\prime }}$, and ${\displaystyle d{A}^{\prime }}$ is an infinitesimal element of area, $${\displaystyle d{q}^{\prime }=\sigma ({\mathbf{r}}^{\prime })d{A}^{\prime }.}$$

For a volume charge distribution (such as charge within a bulk metal) where ${\displaystyle \rho ({\mathbf{r}}^{\prime })}$ gives the charge per unit volume at position ${\displaystyle {\mathbf{r}}^{\prime }}$, and ${\displaystyle d{V}^{\prime }}$ is an infinitesimal element of volume, $${\displaystyle d{q}^{\prime }=\rho ({\mathit{r}}^{\mathbf{\prime }})d{V}^{\prime }.}$$

The force on a small test charge ${\displaystyle q}$ at position ${\displaystyle \mathit{r}}$ in vacuum is given by the integral over the distribution of charge $${\displaystyle \mathbf{F}(\mathbf{r})=\frac{q}{4\pi {\epsilon }_0}\int d{q}^{\prime }\frac{\mathbf{r}-{\mathbf{r}}^{\prime }}{|\mathbf{r}-{\mathbf{r}}^{\prime }{|}^3}.}$$

The "continuous charge" version of Coulomb's law is never supposed to be applied to locations for which ${\displaystyle |\mathbf{r}-{\mathbf{r}}^{\prime }|=0}$ because that location would directly overlap with the location of a charged particle (e.g. electron or proton) 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 ${\displaystyle |\mathbf{r}-{\mathbf{r}}^{\prime }|=0}$ to be analyzed.

Coulomb constant

The constant of proportionality, ${\displaystyle \frac{1}{4\pi {\epsilon }_0}}$, in Coulomb's law: $${\displaystyle {\mathbf{F}}_1=\frac{q_1{q}_2}{4\pi {\epsilon }_0}\frac{{\hat{\mathbf{r}}}_{12}}{{|{\mathbf{r}}_{12}|}^2}}$$ is a consequence of historical choices for units.

The constant ${\displaystyle {\epsilon }_0}$ is the vacuum electric permittivity. Using the CODATA 2022 recommended value for ${\displaystyle {\epsilon }_0}$, the Coulomb constant is $${\displaystyle k_{\text{e}}=\frac{1}{4\pi {\epsilon }_0}=8.9875517862(14)\times {10}^9\mathrm{N}\cdot {\mathrm{m}}^2\cdot {\mathrm{C}}^{-2}}$$

Limitations

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

1. The charges must have a spherically symmetric distribution (e.g. be point charges, or a charged metal sphere).
2. The charges must not overlap (e.g. they must be distinct point charges).
3. The charges must be stationary with respect to a nonaccelerating 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

Main article: Electric field

If two charges have the same sign, the electrostatic force between them is repulsive; if they have different sign, the force between them is attractive.

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 $\mathbf{F}$ on a charge $q_t$ depends on the electric field $\mathbf{E}$ established by other charges that it finds itself in, such that $\mathbf{F}=q_t\mathbf{E}$. 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 $q$, 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 $q_t$ would move if placed in the field. For a negative point source charge, the direction is radially inwards.

The magnitude of the electric field E 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 E created by a single source point charge Q at a certain distance from it r in vacuum is given by $${\displaystyle |\mathbf{E}|=k_{\text{e}}\frac{|q|}{r^2}}$$

A system of n discrete charges ${\displaystyle q_i}$ stationed at ${\mathbf{r}}_i=\mathbf{r}-{\mathbf{r}}_i$ produces an electric field whose magnitude and direction is, by superposition $${\displaystyle \mathbf{E}(\mathbf{r})=\frac{1}{4\pi {\epsilon }_0}\sum _{i=1}^n{q}_i\frac{{\hat{\mathbf{r}}}_i}{{|{\mathbf{r}}_i|}^2}}$$

Atomic forces

See also: Coulomb explosion

Coulomb's law holds even within atoms, correctly describing the force between the positively charged atomic nucleus and each of the negatively charged electrons. 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. As the magnitude of opposing charges increases, energy increases and ionic bonding is more favorable.

Relation to Gauss's law

Deriving Gauss's law from Coulomb's law

Strictly speaking, Gauss's law cannot be derived from Coulomb's law alone, since Coulomb's law gives the electric field due to an individual, electrostatic point charge only. However, Gauss's law can be proven from Coulomb's law if it is assumed, in addition, that the electric field obeys the superposition principle. The superposition principle states that the resulting field is the vector sum of fields generated by each particle (or the integral, if the charges are distributed smoothly in space).

Outline of proof

Coulomb's law states that the electric field due to a stationary point charge is: $${\displaystyle \mathbf{E}(\mathbf{r})=\frac{q}{4\pi {\epsilon }_0}\frac{{\mathbf{e}}_r}{r^2}}$$ where

• e_r is the radial unit vector,
• r is the radius, |r|,
• ε₀ is the electric constant,
• q is the charge of the particle, which is assumed to be located at the origin.

Using the expression from Coulomb's law, we get the total field at r by using an integral to sum the field at r due to the infinitesimal charge at each other point s in space, to give $${\displaystyle \mathbf{E}(\mathbf{r})=\frac{1}{4\pi {\epsilon }_0}\int \frac{\rho (\mathbf{s})(\mathbf{r}-\mathbf{s})}{|\mathbf{r}-\mathbf{s}{|}^3}{\mathrm{d}}^3\mathbf{s}}$$ where ρ is the charge density. If we take the divergence of both sides of this equation with respect to r, and use the known theorem

$${\displaystyle \mathrm{\nabla }\cdot \left(\frac{\mathbf{r}}{|\mathbf{r}{|}^3}\right)=4\pi \delta (\mathbf{r})}$$ where δ(r) is the Dirac delta function, the result is $${\displaystyle \mathrm{\nabla }\cdot \mathbf{E}(\mathbf{r})=\frac{1}{{\epsilon }_0}\int \rho (\mathbf{s})\delta (\mathbf{r}-\mathbf{s}){\mathrm{d}}^3\mathbf{s}}$$

Using the "sifting property" of the Dirac delta function, we arrive at $${\displaystyle \mathrm{\nabla }\cdot \mathbf{E}(\mathbf{r})=\frac{\rho (\mathbf{r})}{{\epsilon }_0},}$$ which is the differential form of Gauss's law, as desired.

Since Coulomb's law only applies to stationary charges, there is no reason to expect Gauss's law to hold for moving charges based on this derivation alone. In fact, Gauss's law does hold for moving charges, and, in this respect, Gauss's law is more general than Coulomb's law.

Proof (without Dirac Delta)

Let ${\displaystyle \mathrm{\Omega }\subseteq R^3}$ be a bounded open set, and $${\displaystyle {\mathbf{E}}_0(\mathbf{r})=\frac{1}{4\pi {\epsilon }_0}{\int }_{\mathrm{\Omega }}\rho ({\mathbf{r}}^{\prime })\frac{\mathbf{r}-{\mathbf{r}}^{\prime }}{{\Vert \mathbf{r}-{\mathbf{r}}^{\prime }\Vert }^3}\mathrm{d}{\mathbf{r}}^{\prime }\equiv \frac{1}{4\pi {\epsilon }_0}{\int }_{\mathrm{\Omega }}e(\mathbf{r}\mathbf{,}{\mathbf{r}}^{\prime })\mathrm{d}{\mathbf{r}}^{\prime }}$$ be the electric field, with ${\displaystyle \rho ({\mathbf{r}}^{\prime })}$ a continuous function (density of charge).

It is true for all ${\displaystyle \mathbf{r}\ne {\mathbf{r}}^{\prime }}$ that ${\displaystyle {\mathrm{\nabla }}_{\mathbf{r}}\cdot \mathbf{e}(\mathbf{r}\mathbf{,}{\mathbf{r}}^{\prime })=0}$.

Consider now a compact set ${\displaystyle V\subseteq R^3}$ having a piecewise smooth boundary ${\displaystyle \mathrm{\partial }V}$ such that ${\displaystyle \mathrm{\Omega }\cap V=\mathrm{\varnothing }}$. It follows that ${\displaystyle e(\mathbf{r}\mathbf{,}{\mathbf{r}}^{\prime })\in C^1(V\times \mathrm{\Omega })}$ and so, for the divergence theorem:

$${\displaystyle {\oint }_{\mathrm{\partial }V}{\mathbf{E}}_0\cdot d\mathbf{S}={\int }_V\mathrm{\nabla }\cdot {\mathbf{E}}_{0}dV}$$

But because ${\displaystyle e(\mathbf{r}\mathbf{,}{\mathbf{r}}^{\prime })\in C^1(V\times \mathrm{\Omega })}$,

$${\displaystyle \mathrm{\nabla }\cdot {\mathbf{E}}_0(\mathbf{r})=\frac{1}{4\pi {\epsilon }_0}{\int }_{\mathrm{\Omega }}{\mathrm{\nabla }}_{\mathbf{r}}\cdot e(\mathbf{r}\mathbf{,}{\mathbf{r}}^{\prime })\mathrm{d}{\mathbf{r}}^{\prime }=0}$$ for the argument above (${\displaystyle \mathrm{\Omega }\cap V=\mathrm{\varnothing }⟹\mathrm{\forall }\mathbf{r}\in V\mathrm{\forall }{\mathbf{r}}^{\prime }\in \mathrm{\Omega }\mathbf{r}\ne {\mathbf{r}}^{\prime }}$ and then ${\displaystyle {\mathrm{\nabla }}_{\mathbf{r}}\cdot \mathbf{e}(\mathbf{r}\mathbf{,}{\mathbf{r}}^{\prime })=0}$)

Therefore the flux through a closed surface generated by some charge density outside (the surface) is null.

Now consider ${\displaystyle {\mathbf{r}}_0\in \mathrm{\Omega }}$, and ${\displaystyle B_R({\mathbf{r}}_0)\subseteq \mathrm{\Omega }}$ as the sphere centered in ${\displaystyle {\mathbf{r}}_0}$ having ${\displaystyle R}$ as radius (it exists because ${\displaystyle \mathrm{\Omega }}$ is an open set).

Let ${\displaystyle {\mathbf{E}}_{B_R}}$ and ${\displaystyle {\mathbf{E}}_C}$ be the electric field created inside and outside the sphere respectively. Then,

${\displaystyle {\mathbf{E}}_{B_R}=\frac{1}{4\pi {\epsilon }_0}{\int }_{B_R({\mathbf{r}}_0)}e(\mathbf{r}\mathbf{,}{\mathbf{r}}^{\prime })\mathrm{d}{\mathbf{r}}^{\prime }}$, ${\displaystyle {\mathbf{E}}_C=\frac{1}{4\pi {\epsilon }_0}{\int }_{\mathrm{\Omega }\setminus B_R({\mathbf{r}}_0)}e(\mathbf{r}\mathbf{,}{\mathbf{r}}^{\prime })\mathrm{d}{\mathbf{r}}^{\prime }}$ and ${\displaystyle {\mathbf{E}}_{B_R}+{\mathbf{E}}_C={\mathbf{E}}_0}$

$${\displaystyle \mathrm{\Phi }(R)={\oint }_{\mathrm{\partial }{B}_R({\mathbf{r}}_0)}{\mathbf{E}}_0\cdot d\mathbf{S}={\oint }_{\mathrm{\partial }{B}_R({\mathbf{r}}_0)}{\mathbf{E}}_{B_R}\cdot d\mathbf{S}+{\oint }_{\mathrm{\partial }{B}_R({\mathbf{r}}_0)}{\mathbf{E}}_C\cdot d\mathbf{S}={\oint }_{\mathrm{\partial }{B}_R({\mathbf{r}}_0)}{\mathbf{E}}_{B_R}\cdot d\mathbf{S}}$$

The last equality follows by observing that ${\displaystyle (\mathrm{\Omega }\setminus B_R({\mathbf{r}}_0))\cap B_R({\mathbf{r}}_0)=\mathrm{\varnothing }}$, and the argument above.

The RHS is the electric flux generated by a charged sphere, and so:

$${\displaystyle \mathrm{\Phi }(R)=\frac{Q(R)}{{\epsilon }_0}=\frac{1}{{\epsilon }_0}{\int }_{B_R({\mathbf{r}}_0)}\rho ({\mathbf{r}}^{\prime })\mathrm{d}{\mathbf{r}}^{\prime }=\frac{1}{{\epsilon }_0}\rho ({\mathbf{r}}_c^{\prime })|B_R({\mathbf{r}}_0)|}$$ with ${\displaystyle r_c^{\prime }\in B_R({\mathbf{r}}_0)}$

Where the last equality follows by the mean value theorem for integrals. Using the squeeze theorem and the continuity of ${\displaystyle \rho }$, one arrives at:

$${\displaystyle \mathrm{\nabla }\cdot {\mathbf{E}}_0({\mathbf{r}}_0)=\underset{R\to 0}{lim}\frac{1}{|B_R({\mathbf{r}}_0)|}\mathrm{\Phi }(R)=\frac{1}{{\epsilon }_0}\rho ({\mathbf{r}}_0)}$$

Deriving Coulomb's law from Gauss's law

Strictly speaking, Coulomb's law cannot be derived from Gauss's law alone, since Gauss's law does not give any information regarding the curl of E (see Helmholtz decomposition and Faraday's law). However, Coulomb's law can be proven from Gauss's law if it is assumed, in addition, that the electric field from a point charge is spherically symmetric (this assumption, like Coulomb's law itself, is exactly true if the charge is stationary, and approximately true if the charge is in motion).

Outline of proof

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 $${\displaystyle {\oint }_S\mathbf{E}\cdot d\mathbf{A}=\frac{Q}{{\epsilon }_0}}$$

By the assumption of spherical symmetry, the integrand is a constant which can be taken out of the integral. The result is $${\displaystyle 4\pi r^2\hat{\mathbf{r}}\cdot \mathbf{E}(\mathbf{r})=\frac{Q}{{\epsilon }_0}}$$ where r̂ is a unit vector pointing radially away from the charge. Again by spherical symmetry, E points in the radial direction, and so we get $${\displaystyle \mathbf{E}(\mathbf{r})=\frac{Q}{4\pi {\epsilon }_0}\frac{\hat{\mathbf{r}}}{r^2}}$$ which is essentially equivalent to Coulomb's law. Thus the inverse-square law dependence of the electric field in Coulomb's law follows from Gauss's law.

In relativity

Coulomb's law can be used to gain insight into the form of the magnetic field generated by moving charges since by special relativity, in certain cases the magnetic field can be shown to be a transformation of forces caused by the electric field. 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. Coulomb's law can be expanded to moving test particles to be of the same form. This assumption is supported by Lorentz force law which, unlike Coulomb's law is not limited to stationary test charges. Considering the charge to be invariant of observer, the electric and magnetic fields of a uniformly moving point charge can hence be derived by the Lorentz transformation of the four force on the test charge in the charge's frame of reference given by Coulomb's law and attributing magnetic and electric fields by their definitions given by the form of Lorentz force. The fields hence found for uniformly moving point charges are given by:$${\displaystyle \mathbf{E}=\frac{q}{4\pi {ϵ}_0{r}^3}\frac{1-{\beta }^2}{(1-{\beta }^2{\sin }^2\theta {)}^{3/2}}\mathbf{r}}$$$${\displaystyle \mathbf{B}=\frac{q}{4\pi {ϵ}_0{r}^3}\frac{1-{\beta }^2}{(1-{\beta }^2{\sin }^2\theta {)}^{3/2}}\frac{\mathbf{v}\times \mathbf{r}}{c^2}=\frac{\mathbf{v}\times \mathbf{E}}{c^2}}$$where ${\displaystyle q}$ is the charge of the point source, ${\displaystyle \mathbf{r}}$ is the position vector from the point source to the point in space, ${\displaystyle \mathbf{v}}$ is the velocity vector of the charged particle, ${\displaystyle \beta }$ is the ratio of speed of the charged particle divided by the speed of light and ${\displaystyle \theta }$ is the angle between ${\displaystyle \mathbf{r}}$ and ${\displaystyle \mathbf{v}}$.

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). Note that the expression for electric field reduces to Coulomb's law for non-relativistic speeds of the point charge and that the magnetic field in non-relativistic limit (approximating ${\displaystyle \beta \ll 1}$) can be applied to electric currents to get the Biot–Savart law. These solutions, when expressed in retarded time also correspond to the general solution of Maxwell's equations given by solutions of Liénard–Wiechert potential, due to the validity of Coulomb's law within its specific range of application. 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. Agreement with Maxwell's equations can also be manually verified for the above two equations.

Coulomb potential

See also: Electric potential

Quantum field theory

The most basic Feynman diagram for QED interaction between two fermions

The Coulomb potential admits continuum states (with E > 0), describing electron-proton scattering, as well as discrete bound states, representing the hydrogen atom. It can also be derived within the non-relativistic limit between two charged particles, as follows:

Under Born approximation, in non-relativistic quantum mechanics, the scattering amplitude $\mathcal{A}(|\mathbf{p}⟩\to |{\mathbf{p}}^{\prime }⟩)$ is: $${\displaystyle \mathcal{A}(|\mathbf{p}⟩\to |{\mathbf{p}}^{\prime }⟩)-1=2\pi \delta (E_p-E_{p^{\prime }})(-i)\int d^3\mathbf{r}V(\mathbf{r})e^{-i(\mathbf{p}-{\mathbf{p}}^{\prime })\mathbf{r}}}$$ This is to be compared to the: $${\displaystyle \int \frac{d^{3}k}{(2\pi {)}^3}{e}^{ik{r}_0}⟨p^{\prime },k|S|p,k⟩}$$ where we look at the (connected) S-matrix entry for two electrons scattering off each other, treating one with "fixed" momentum as the source of the potential, and the other scattering off that potential.

Using the Feynman rules to compute the S-matrix element, we obtain in the non-relativistic limit with ${\displaystyle m_0\gg |\mathbf{p}|}$ $${\displaystyle ⟨p^{\prime },k|S|p,k⟩{|}_{conn}=-i\frac{e^2}{|\mathbf{p}-{\mathbf{p}}^{\prime }{|}^2-i\epsilon }(2m{)}^2\delta (E_{p,k}-E_{p^{\prime },k})(2\pi {)}^4\delta (\mathbf{p}-{\mathbf{p}}^{\prime })}$$

Comparing with the QM scattering, we have to discard the ${\displaystyle (2m{)}^2}$ as they arise due to differing normalizations of momentum eigenstate in QFT compared to QM and obtain: $${\displaystyle \int V(\mathbf{r})e^{-i(\mathbf{p}-{\mathbf{p}}^{\prime })\mathbf{r}}{d}^3\mathbf{r}=\frac{e^2}{|\mathbf{p}-{\mathbf{p}}^{\prime }{|}^2-i\epsilon }}$$ where Fourier transforming both sides, solving the integral and taking ${\displaystyle \epsilon \to 0}$ at the end will yield $${\displaystyle V(r)=\frac{e^2}{4\pi r}}$$ as the Coulomb potential.

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

The Coulomb potential, and its derivation, can be seen as a special case of the Yukawa potential, which is the case where the exchanged boson – the photon – has no rest mass.

Verification

Experiment to verify Coulomb's law.

It is possible to verify Coulomb's law with a simple experiment. Consider two small spheres of mass ${\displaystyle m}$ and same-sign charge ${\displaystyle q}$, hanging from two ropes of negligible mass of length ${\displaystyle l}$. The forces acting on each sphere are three: the weight ${\displaystyle mg}$, the rope tension ${\displaystyle \mathbf{T}}$ and the electric force ${\displaystyle \mathbf{F}}$. In the equilibrium state:

$${\displaystyle \mathbf{T}\sin {\theta }_1={\mathbf{F}}_1}$$

1

and

$${\displaystyle \mathbf{T}\cos {\theta }_1=mg}$$

2

Dividing (1) by (2):

$${\displaystyle \frac{\sin {\theta }_1}{\cos {\theta }_1}=\frac{F_1}{mg}\Rightarrow F_1=mg\tan {\theta }_1}$$

3

Let ${\displaystyle {\mathbf{L}}_1}$ be the distance between the charged spheres; the repulsion force between them ${\displaystyle {\mathbf{F}}_1}$, assuming Coulomb's law is correct, is equal to

$${\displaystyle F_1=\frac{q^2}{4\pi {\epsilon }_0{L}_1^2}}$$

Coulomb's law

so:

$${\displaystyle \frac{q^2}{4\pi {\epsilon }_0{L}_1^2}=mg\tan {\theta }_1}$$

4

If we now discharge one of the spheres, and we put it in contact with the charged sphere, each one of them acquires a charge $\frac{q}{2}$. In the equilibrium state, the distance between the charges will be ${\mathbf{L}}_2<{\mathbf{L}}_1$ and the repulsion force between them will be:

$${\displaystyle F_2=\frac{{(\frac{q}{2})}^2}{4\pi {\epsilon }_0{L}_2^2}=\frac{\frac{q^2}{4}}{4\pi {\epsilon }_0{L}_2^2}}$$

5

We know that ${\displaystyle {\mathbf{F}}_2=mg\tan {\theta }_2}$ and: $${\displaystyle \frac{\frac{q^2}{4}}{4\pi {\epsilon }_0{L}_2^2}=mg\tan {\theta }_2}$$ Dividing (4) by (5), we get:

$${\displaystyle \frac{\left(\frac{q^2}{4\pi {\epsilon }_0{L}_1^2}\right)}{\left(\frac{\frac{q^2}{4}}{4\pi {\epsilon }_0{L}_2^2}\right)}=\frac{mg\tan {\theta }_1}{mg\tan {\theta }_2}\Rightarrow 4{\left(\frac{L_2}{L_1}\right)}^2=\frac{\tan {\theta }_1}{\tan {\theta }_2}}$$

6

Measuring the angles ${\displaystyle {\theta }_1}$ and ${\displaystyle {\theta }_2}$ and the distance between the charges ${\displaystyle {\mathbf{L}}_1}$ and ${\displaystyle {\mathbf{L}}_2}$ is sufficient to verify that the equality is true taking into account the experimental error. In practice, angles can be difficult to measure, so if the length of the ropes is sufficiently great, the angles will be small enough to make the following approximation:

$${\displaystyle \tan \theta \approx \sin \theta =\frac{\frac{L}{2}}{\ell }=\frac{L}{2\ell }\Rightarrow \frac{\tan {\theta }_1}{\tan {\theta }_2}\approx \frac{\frac{L_1}{2\ell }}{\frac{L_2}{2\ell }}}$$

7

Using this approximation, the relationship (6) becomes the much simpler expression:

$${\displaystyle \frac{\frac{L_1}{2\ell }}{\frac{L_2}{2\ell }}\approx 4{\left(\frac{L_2}{L_1}\right)}^2\Rightarrow \frac{L_1}{L_2}\approx 4{\left(\frac{L_2}{L_1}\right)}^2\Rightarrow \frac{L_1}{L_2}\approx \sqrt[3]{4}}$$

8

In this way, the verification is limited to measuring the distance between the charges and checking that the division approximates the theoretical value.