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Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which re ...
that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventionally called ''electrostatic force'' or Coulomb force. Although the law was known earlier, it was first published in 1785 by French physicist
Charles-Augustin de Coulomb Charles-Augustin de Coulomb (; ; 14 June 1736 – 23 August 1806) was a French officer, engineer, and physicist. He is best known as the eponymous discoverer of what is now called Coulomb's law, the description of the electrostatic force of attra ...
, hence the name. Coulomb's law was essential to the development of the theory of electromagnetism, maybe even its starting point, as it made it possible to discuss the quantity of electric charge in a meaningful way. The law states that the magnitude of the electrostatic
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
of attraction or repulsion between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them. Coulomb studied the repulsive force between bodies having electrical charges of the same sign: Coulomb also showed that oppositely charged bodies obey an inverse-square law of attraction. , F, =k_\text \frac Here, or is the
Coulomb constant The Coulomb constant, the electric force constant, or the electrostatic constant (denoted , or ) is a proportionality constant in electrostatics equations. In SI base units it is equal to .Derived from ''k''e = 1/(4''πε''0) – It was named ...
(), and are the assigned magnitudes of the charges, 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 is repulsive; if they have different signs, the force between them is attractive. Being an
inverse-square law In science, an inverse-square law is any scientific law stating that a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cause for this can be underst ...
, the law is analogous to
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the great ...
's inverse-square
law of universal gravitation Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distance ...
, but gravitational forces are always attractive, while electrostatic forces can be attractive or repulsive. Coulomb's law can be used to derive
Gauss's law In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it s ...
, and vice versa. In the case of a single stationary point charge, 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 The Mediterranean Sea is a sea connected to the Atlantic Ocean, surrounded by the Mediterranean Basin and almost completely enclosed by land: on the north by Western and Southern Europe and Anatolia, on the south by North Africa, and on the ...
knew that certain objects, such as rods of
amber Amber is fossilized tree resin that has been appreciated for its color and natural beauty since Neolithic times. Much valued from antiquity to the present as a gemstone, amber is made into a variety of decorative objects."Amber" (2004). In ...
, could be rubbed with cat's fur to attract light objects like feathers and pieces of paper.
Thales of Miletus Thales of Miletus ( ; grc-gre, Θαλῆς; ) was a Greek mathematician, astronomer, statesman, and pre-Socratic philosopher from Miletus in Ionia, Asia Minor. He was one of the Seven Sages of Greece. Many, most notably Aristotle, regarded h ...
made the first recorded description of
static electricity Static electricity is an imbalance of electric charges within or on the surface of a material or between materials. The charge remains until it is able to move away by means of an electric current or electrical discharge. Static electricity is ...
around 600 BC, when he noticed that
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of t ...
could render a piece of amber magnetic. In 1600, English scientist William Gilbert made a careful study of electricity and magnetism, distinguishing the
lodestone Lodestones are naturally magnetized pieces of the mineral magnetite. They are naturally occurring magnets, which can attract iron. The property of magnetism was first discovered in antiquity through lodestones. Pieces of lodestone, suspe ...
effect from static electricity produced by rubbing amber. He coined the
New Latin New Latin (also called Neo-Latin or Modern Latin) is the revival of Literary Latin used in original, scholarly, and scientific works since about 1500. Modern scholarly and technical nomenclature, such as in zoological and botanical taxonomy ...
word ''electricus'' ("of amber" or "like amber", from '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 Sir Thomas Browne (; 19 October 160519 October 1682) was an English polymath and author of varied works which reveal his wide learning in diverse fields including science and medicine, religion and the esoteric. His writings display a deep curi ...
's ''
Pseudodoxia Epidemica ''Pseudodoxia Epidemica or Enquiries into very many received tenents and commonly presumed truths'', also known simply as ''Pseudodoxia Epidemica'' or ''Vulgar Errors'', is a work by Thomas Browne challenging and refuting the "vulgar" or common ...
'' of 1646. Early investigators of the 18th century who suspected that the electrical
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
diminished with distance as the force of
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the str ...
did (i.e., as the inverse square of the distance) included
Daniel Bernoulli Daniel Bernoulli FRS (; – 27 March 1782) was a Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered for his applications of mathematics to mech ...
and
Alessandro Volta Alessandro Giuseppe Antonio Anastasio Volta (, ; 18 February 1745 – 5 March 1827) was an Italian physicist, chemist and lay Catholic who was a pioneer of electricity and power who is credited as the inventor of the electric battery and the ...
, both of whom measured the force between plates of a
capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of a c ...
, and
Franz Aepinus Franz Ulrich Theodor Aepinus (13 December 172410 August 1802) was a German mathematician, scientist, and natural philosopher residing in the Russian Empire. Aepinus is best known for his researches, theoretical and experimental, in electricity a ...
who supposed the inverse-square law in 1758. Based on experiments with electrically charged spheres,
Joseph Priestley Joseph Priestley (; 24 March 1733 – 6 February 1804) was an English chemist, natural philosopher, separatist theologian, grammarian, multi-subject educator, and liberal political theorist. He published over 150 works, and conducted ...
of England was among the first to propose that electrical force followed an
inverse-square law In science, an inverse-square law is any scientific law stating that a specified physical quantity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cause for this can be underst ...
, similar to
Newton's law of universal gravitation Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distan ...
. 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. 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 Henry Cavendish ( ; 10 October 1731 – 24 February 1810) was an English natural philosopher and scientist who was an important experimental and theoretical chemist and physicist. He is noted for his discovery of hydrogen, which he termed "infl ...
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 + th and that of the 2 − th, 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 Charles-Augustin de Coulomb (; ; 14 June 1736 – 23 August 1806) was a French officer, engineer, and physicist. He is best known as the eponymous discoverer of what is now called Coulomb's law, the description of the electrostatic force of attra ...
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 particle In physics, a charged particle is a particle with an electric charge. It may be an ion, such as a molecule or atom with a surplus or deficit of electrons relative to protons. It can also be an electron or a proton, or another elementary particle, ...
s, and determined that the magnitude of the electric force between two
point charge A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take u ...
s 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 A torsion spring is a spring that works by twisting its end along its axis; that is, a flexible elastic object that stores mechanical energy when it is twisted. When it is twisted, it exerts a torque in the opposite direction, proportional ...
. In Coulomb's experiment, the torsion balance was an insulating rod with a
metal A metal (from Greek μέταλλον ''métallon'', "mine, quarry, metal") is a material that, when freshly prepared, polished, or fractured, shows a lustrous appearance, and conducts electricity and heat relatively well. Metals are typica ...
-coated ball attached to one end, suspended by a
silk Silk is a natural protein fiber, some forms of which can be woven into textiles. The protein fiber of silk is composed mainly of fibroin and is produced by certain insect larvae to form cocoons. The best-known silk is obtained from t ...
thread. The ball was charged with a known charge of
static electricity Static electricity is an imbalance of electric charges within or on the surface of a material or between materials. The charge remains until it is able to move away by means of an electric current or electrical discharge. Static electricity is ...
, 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 of the law

Coulomb's law can be stated as a simple mathematical expression. The scalar form gives the magnitude of the vector of the electrostatic force between two point charges and , but not its direction. If is the distance between the charges, the magnitude of the force is , \mathbf, =k_\text \frac The constant is called the
Coulomb constant The Coulomb constant, the electric force constant, or the electrostatic constant (denoted , or ) is a proportionality constant in electrostatics equations. In SI base units it is equal to .Derived from ''k''e = 1/(4''πε''0) – It was named ...
and is equal to , where is the
electric constant Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...
; . If the product is positive, the force between the two charges is repulsive; if the product is negative, the force between them is attractive.


Vector form of the law

Coulomb's law in vector form states that the electrostatic force \mathbf_1 experienced by a charge, q_1 at position \mathbf_1, in the vicinity of another charge, q_2 at position \mathbf_2, in a vacuum is equal to \mathbf_1 = \frac \frac = \frac \frac where \boldsymbol_ = \boldsymbol_1 - \boldsymbol_2 is the vectorial distance between the charges, \widehat_=\frac a unit vector pointing from q_2 to and \varepsilon_0 the
electric constant Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric const ...
. Here, \mathbf_ is used for the vector notation. The vector form of Coulomb's law is simply the scalar definition of the law with the direction given by the
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vec ...
, parallel with the line ''from'' charge q_2 ''to'' charge q_1. If both charges have the same
sign A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or ...
(like charges) then the product q_1q_2 is positive and the direction of the force on q_1 is given by \widehat_; the charges repel each other. If the charges have opposite signs then the product q_1q_2 is negative and the direction of the force on q_1 is the charges attract each other. The electrostatic force \mathbf_2 experienced by q_2, according to Newton's third law, is


System of discrete charges

The
law of superposition The law of superposition is an axiom that forms one of the bases of the sciences of geology, archaeology, and other fields pertaining to geological stratigraphy. In its plainest form, it states that in undeformed stratigraphic sequences, the ...
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 In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ...
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 An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...
vector at that point, with that point charge removed. Force \mathbf on a small charge q at position \mathbf, due to a system of N discrete charges in vacuum is \mathbf(\mathbf) = \sum_^N q_i \frac = \sum_^N q_i , where q_i and \mathbf_i are the magnitude and position respectively of the th charge, \hat_i is a unit vector in the direction of \mathbf_i = \mathbf-\mathbf_i, a vector pointing from charges q_i to q.


Continuous charge distribution

In this case, the principle of
linear superposition The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So tha ...
is also used. For a continuous charge distribution, an
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
over the region containing the charge is equivalent to an infinite summation, treating each
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally refe ...
element of space as a point charge 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 \lambda(\mathbf') gives the charge per unit length at position \mathbf', and d\ell' is an infinitesimal element of length, dq' = \lambda(\mathbf) \, d\ell'. For a surface charge distribution (a good approximation for charge on a plate in a parallel plate
capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of a c ...
) where \sigma(\mathbf') gives the charge per unit area at position \mathbf', and dA' is an infinitesimal element of area, dq' = \sigma(\mathbf)\,dA'. For a volume charge distribution (such as charge within a bulk metal) where \rho(\mathbf') gives the charge per unit volume at position \mathbf', and dV' is an infinitesimal element of volume, dq' = \rho(\boldsymbol)\,dV'. The force on a small test charge q at position \boldsymbol in vacuum is given by the integral over the distribution of charge \mathbf(\mathbf) = \frac\int dq' \frac. where it must be noted that the "continuous charge" version of Coulomb's law is never supposed to be applied to locations for which , \mathbf - \mathbf, = 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 , \mathbf - \mathbf, = 0 to be analyzed.


Coulomb constant

The Coulomb constant is a proportionality factor that appears in Coulomb's law as well as in other electric-related formulas. Denoted k_\text, it is also called the electric force constant or electrostatic constant hence the subscript e. When the electromagnetic theory is expressed in the
International System of Units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
, force is measured in
newtons The newton (symbol: N) is the unit of force in the International System of Units (SI). It is defined as 1 kg⋅m/s, the force which gives a mass of 1 kilogram an acceleration of 1 metre per second per second. It is named after Isaac Newton in ...
, charge in
coulomb The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI). In the present version of the SI it is equal to the electric charge delivered by a 1 ampere constant current in 1 second and to elementary char ...
s and distance in
meter The metre ( British spelling) or meter ( American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pr ...
s. The Coulomb constant is given by k_\text = \frac. The constant \varepsilon_0 is the vacuum electric permittivity (also known as "electric constant") in \mathrm. It should not be confused with \varepsilon_r, which is the dimensionless
relative permittivity The relative permittivity (in older texts, dielectric constant) is the permittivity of a material expressed as a ratio with the electric permittivity of a vacuum. A dielectric is an insulating material, and the dielectric constant of an insula ...
of the material in which the charges are immersed, or with their product \varepsilon_a = \varepsilon_0\varepsilon_r, which is called " absolute permittivity of the material" and is still used in
electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems which use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
. Prior to the 2019 redefinition of the
SI base unit The SI base units are the standard units of measurement defined by the International System of Units (SI) for the seven base quantities of what is now known as the International System of Quantities: they are notably a basic set from which al ...
s, the Coulomb constant was considered to have an exact value: \begin k_\text &= \frac=\frac=c_0^2\times 10^\ \mathrm\\ &= 8.987\,551\,787\,368\,176\,4\times 10^9\ \mathrm. \end Since the 2019 redefinition, the Coulomb constant is no longer exactly defined and is subject to the measurement error in the fine structure constant. As calculated from
CODATA The Committee on Data of the International Science Council (CODATA) was established in 1966 as the Committee on Data for Science and Technology, originally part of the International Council of Scientific Unions, now part of the Internationa ...
2018 recommended values, the Coulomb constant isDerived from – k_\text = 8.987\,551\,792\,3\,(14)\times 10^9\ \mathrm . In
Gaussian units Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs uni ...
and
Heaviside–Lorentz units Heaviside–Lorentz units (or Lorentz–Heaviside units) constitute a system of units (particularly electromagnetic units) within CGS, named for Hendrik Antoon Lorentz and Oliver Heaviside. They share with CGS-Gaussian units the property that th ...
, which are both CGS unit systems, the constant has different,
dimensionless A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
values. In electrostatic units or Gaussian units the unit charge (''esu'' or
statcoulomb The franklin (Fr) or statcoulomb (statC) electrostatic unit of charge (esu) is the physical unit for electrical charge used in the cgs-esu and Gaussian units. It is a derived unit given by : 1 statC = 1 dyn1/2⋅cm = 1 cm3/2⋅g1/2⋅s−1. Tha ...
) is defined in such a way that the Coulomb constant disappears, as it has the value of one and becomes dimensionless. k_\text = 1 In Heaviside–Lorentz units, also called ''rationalized'' ''units'', the Coulomb constant is dimensionless and is equal to k_\text = \frac Gaussian units are more amenable for microscopic problems such as the electrodynamics of individual electrically charged particles. SI units are more convenient for practical, large-scale phenomena, such as engineering applications.


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 (e.g. be point charges, or a charged metal sphere). # The charges must not overlap (e.g. they must be distinct point charges). # The charges must be stationary with respect to each other. The last of these is known as the electrostatic approximation. When movement takes place,
Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
's
theory of relativity The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena i ...
must be taken into consideration, and a result, an extra factor is introduced, which alters the force produced on the two objects. This extra part of the force is called the
magnetic Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particl ...
force, and is described by
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
s. For slow movement, the magnetic force is minimal and Coulomb's law can still be considered approximately correct, but when the charges are moving more quickly in relation to each other, the full
electrodynamics In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
rules (incorporating the magnetic force) must be considered.


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 \mathbf F on a charge q_t depends on the electric field \mathbf established by other charges that it finds itself in, such that \mathbf = q_t \mathbf. In the simplest case, the field is considered to be generated solely by a single source
point charge A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take u ...
. 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 can be derived from
Coulomb's law Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventio ...
. 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 An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...
created by a single source
point charge A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take u ...
''Q'' at a certain distance from it ''r'' in vacuum is given by , \mathbf, = k_\text \frac A system ''N'' of charges q_i stationed at \mathbf_i produces an electric field whose magnitude and direction is, by superposition \mathbf(\mathbf) = \sum_^N q_i \frac


Atomic forces

Coulomb's law holds even within
atom Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, ...
s, correctly describing the
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
between the positively charged
atomic nucleus The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford based on the 1909 Geiger–Marsden gold foil experiment. After the discovery of the neutro ...
and each of the negatively charged
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...
s. This simple law also correctly accounts for the forces that bind atoms together to form
molecule A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
s 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 Ionic bonding is a type of chemical bonding that involves the electrostatic attraction between oppositely charged ions, or between two atoms with sharply different electronegativities, and is the primary interaction occurring in ionic compound ...
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 In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it s ...
cannot be derived from Coulomb's law alone, since Coulomb's law gives the electric field due to an individual
point charge A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take u ...
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, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So tha ...
. The superposition principle says 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). Note that 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.


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 cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was ...
of (see
Helmholtz decomposition In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved int ...
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 A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take u ...
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).


In relativity

Coulomb's law can be used to gain insight of the form of magnetic field generated by moving charges since by special relativity, in certain cases the
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
can be shown to be a transformation of forces caused by the
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field ...
. When no acceleration is involved in a particle's history, it can be viewed to obey Coulomb's law on any test particle in its own inertial frame. Coulomb's law can be expanded to moving test particle to be of the same form, which shows discrepancy in Newton's third law as followed but it is not strictly obeyed in the framework of special relativity (yet without violating relativistic-energy conservation). This assumption can be justified by obtaining the correct form of field equations, that is with respect to agreement with
maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
. 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 In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation ...
of the four force on test charge given by Coulomb's law in the charge's frame of reference and attributing magnetic and electric fields by their definition given by the form of
Lorentz force In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an el ...
. The fields hence found for uniformly moving point charges are given by:\mathbf = \frac q \frac \mathbf\mathbf = \frac q \frac \frac = \fracwhere q is the charge of the point source, \mathbf is the position vector from the point source to the point in space, \mathbf is the velocity vector of the charged particle, \beta is the ratio of speed of the charged particle divided by the speed of light and \theta is the angle between \mathbf and \mathbf. 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 \beta\ll 1) can be applied to electric currents to get the
Biot–Savart law In physics, specifically electromagnetism, the Biot–Savart law ( or ) is an equation describing the magnetic field generated by a constant electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the ...
. These solutions, when expressed in retarded time also correspond to the general solution of
maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
given by solutions of
Liénard–Wiechert potential The Liénard–Wiechert potentials describe the classical electromagnetic effect of a moving electric point charge in terms of a vector potential and a scalar potential in the Lorenz gauge. Stemming directly from Maxwell's equations, these descr ...
, 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. Maxwell's equations can also be manually verified for the above two equations.


Coulomb potential


Quantum field theory

The Coulomb potential admits continuum states (with ''E'' > 0), describing electron-proton
scattering Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including ...
, 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(, \mathbf \rangle \to , \mathbf'\rangle) is: \mathcal(, \mathbf \rangle \to , \mathbf'\rangle) - 1 = 2\pi \delta(E_p - E_)(-i)\int d^3\mathbf r \, V(\mathbf r) e^ This is to be compared to the: \int \frac e^ \langle p',k , S, p,k \rangle 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 m_0 \gg , \mathbf p, \langle p',k , S , p,k \rangle , _ = -i\frac(2m)^2\delta(E_ - E_)(2\pi)^4\delta(\mathbf p - \mathbf p') Comparing with the QM scattering, we have to discard the (2m)^2 as they arise due to differing normalizations of momentum eigenstate in QFT compared to QM and obtain: \int V(\mathbf r)e^d^3\mathbf r = \frac where Fourier transforming both sides, solving the integral and taking \varepsilon \to 0 at the end will yield V(r) = \frac 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 In particle, atomic and condensed matter physics, a Yukawa potential (also called a screened Coulomb potential) is a potential named after the Japanese physicist Hideki Yukawa. The potential is of the form: :V_\text(r)= -g^2\frac, where is a ...
, which is the case where the exchanged boson – the photon – has no rest mass.


Simple experiment to verify Coulomb's law

It is possible to verify Coulomb's law with a simple experiment. Consider two small spheres of mass m and same-sign charge q, hanging from two ropes of negligible mass of length l. The forces acting on each sphere are three: the weight mg, the rope tension \mathbf T and the electric force \mathbf F. In the equilibrium state: and Dividing () by (): Let \mathbf L_1 be the distance between the charged spheres; the repulsion force between them \mathbf F_1, assuming Coulomb's law is correct, is equal to so: 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. 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: We know that \mathbf F_2 = mg \tan \theta_2 and: \frac=mg \tan \theta_2 Dividing () by (), we get: Measuring the angles \theta_1 and \theta_2 and the distance between the charges \mathbf L_1 and \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: Using this approximation, the relationship () becomes the much simpler expression: In this way, the verification is limited to measuring the distance between the charges and check that the division approximates the theoretical value.


See also

*
Biot–Savart law In physics, specifically electromagnetism, the Biot–Savart law ( or ) is an equation describing the magnetic field generated by a constant electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the ...
* Darwin Lagrangian *
Electromagnetic force In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of ...
*
Gauss's law In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, (or sometimes simply called Gauss's theorem) is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it s ...
*
Method of image charges The method of image charges (also known as the method of images and method of mirror charges) is a basic problem-solving tool in electrostatics. The name originates from the replacement of certain elements in the original layout with imaginary c ...
*
Molecular modelling Molecular modelling encompasses all methods, theoretical and computational, used to model or mimic the behaviour of molecules. The methods are used in the fields of computational chemistry, drug design, computational biology and materials scie ...
*
Newton's law of universal gravitation Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distan ...
, which uses a similar structure, but for mass instead of charge * Static forces and virtual-particle exchange


References


Related reading

* * * * * * *


External links


''Coulomb's Law''
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Project PHYSNET


��a chapter from an online textbook

��a game created by the Molecular Workbench software

Walter Lewin, ''8.02 Electricity and Magnetism, Spring 2002: Lecture 1'' (video). MIT OpenCourseWare. License: Creative Commons Attribution-Noncommercial-Share Alike. {{Authority control Electromagnetism Electrostatics Force Scientific laws