Point particle

Encyclopedia

A

of particle

s heavily used in physics

. Its defining feature is that it lacks spatial

extension

: being zero-dimensional, it does not take up space. A point particle is an appropriate representation of any object whose size, shape, and structure is irrelevant in a given context. For example, from far enough away, an object of any shape will look and behave as a point-like object.

In the theory of gravity, physicists often discuss a

and no other properties or structure. Likewise, in electromagnetism

, physicists discuss a

.

Sometimes due to specific combinations of properties extended objects behave as point-like even in their immediate vicinity. For example, spherical objects interacting in 3-dimensional space whose interactions are described by the inverse square law behave in such a way as if all their matter were concentrated in their geometric centers. In Newtonian gravitation and classical electromagnetism

, for example, the respective fields

outside of a spherical object are identical to those of a point particle of equal charge/mass located at the center of the sphere.

In quantum mechanics

, the concept of a point particle is complicated by the Heisenberg uncertainty principle: Even an elementary particle

, with no internal structure, occupies a nonzero volume. For example, a 1s

electron

in a hydrogen atom

occupies a volume of ~10

s such as electron

s or quark

s, which have no internal structure, versus composite particles such as proton

s, which do have internal structure: A proton is made of three quarks. Elementary particles are sometimes called "point particles", but this is in a different sense than discussed above. For more details see elementary particle

.

which is infinitely small, or an object which can be thought of as infinitely small. This concept in terms of size is similar to that of point particles, however unlike point particles the object need only be considered infinitely small.

individually. When none of the objects in the system have overlapping center of mass

circumference

s, it is possible to consider the object as a zero-dimensional point mass.

is a discontinuous segment in a probability distribution

. To calculate such point mass, an integration

is carried out over the entire range of the random variable

, on the probability distribution of the continuous part. After equating this integral to 1, the point mass can be found by further calculation.

. A point charge is an electric charge at a mathematical point

with no dimensions.

The fundamental equation

of electrostatics

is Coulomb's law

, which describes the electric force between two point charges. The electric field

associated with a classical point charge increases to infinity as the distance from the point charge decreases towards zero

making energy (thus mass) of point charge infinite. In quantum electrodynamics

, developed in part by Richard Feynman

, the mathematical method of renormalization

eliminates the infinite divergence

of the point charge.

Earnshaw's theorem

states that a collection of point charges cannot be maintained in an equilibrium

configuration solely by the electrostatic interaction of the charges.

In quantum mechanics

, there is a distinction between an elementary particleIn particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...

(also called "point particle") and a composite particle. An elementary particle, such as an electron

, quark

, or photon

, is a particle with no internal structure, whereas a composite particle, such as a proton

or neutron

, has an internal structure (see figure).

However, neither elementary nor composite particles are spatially localized, because of the Heisenberg uncertainty principle. The particle wavepacket always occupies a nonzero volume. For example, see atomic orbital

: The electron is an elementary particle, but its quantum states form three-dimensional patterns.

Nevertheless, there is good reason that an elementary particle is often called a point particle. Even if an elementary particle has a delocalized wavepacket, the wavepacket is in fact a quantum superposition

of quantum states wherein the particle is exactly localized. This is not true for a composite particle, which can never be represented as a superposition of exactly-localized quantum states. It is in this sense that physicists can discuss the intrinsic "size" of a particle: The size of its internal structure, not the size of its wavepacket. The "size" of an elementary particle, in this sense, is exactly zero.

For example, for the electron, experimental evidence shows that the size of an electron is less than 10

, which, despite the name, is unrelated to the actual size of an electron.)

**point particle**is an idealizationIdealization

Idealization is the process by which scientific models assume facts about the phenomenon being modeled that are strictly false. Often these assumptions are used to make models easier to understand or solve. Many times idealizations do not harm the predictive accuracy of the model for one reason or...

of particle

Particle

A particle is, generally, a small localized object to which can be ascribed physical properties. It may also refer to:In chemistry:* Colloidal particle, part of a one-phase system of two or more components where the particles aren't individually visible.In physics:* Subatomic particle, which may be...

s heavily used in physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

. Its defining feature is that it lacks spatial

Dimension

In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

extension

Extension (metaphysics)

In metaphysics, extension is, roughly speaking, the property of "taking up space". René Descartes defines extension as the property of existing in more than one dimension. For Descartes, the primary characteristic of matter is extension, just as the primary characteristic of mind is consciousness...

: being zero-dimensional, it does not take up space. A point particle is an appropriate representation of any object whose size, shape, and structure is irrelevant in a given context. For example, from far enough away, an object of any shape will look and behave as a point-like object.

In the theory of gravity, physicists often discuss a

**point mass**, meaning a point particle with a nonzero massMass

Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

and no other properties or structure. Likewise, in electromagnetism

Electromagnetism

Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...

, physicists discuss a

**point charge**, a point particle with a nonzero chargeCharge (physics)

In physics, a charge may refer to one of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics. Charges are associated with conserved quantum numbers.-Formal definition:...

.

Sometimes due to specific combinations of properties extended objects behave as point-like even in their immediate vicinity. For example, spherical objects interacting in 3-dimensional space whose interactions are described by the inverse square law behave in such a way as if all their matter were concentrated in their geometric centers. In Newtonian gravitation and classical electromagnetism

Electromagnetism

Electromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...

, for example, the respective fields

Field (physics)

In physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...

outside of a spherical object are identical to those of a point particle of equal charge/mass located at the center of the sphere.

In quantum mechanics

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, the concept of a point particle is complicated by the Heisenberg uncertainty principle: Even an elementary particle

Elementary particle

In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...

, with no internal structure, occupies a nonzero volume. For example, a 1s

Atomic orbital

An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...

electron

Electron

The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

in a hydrogen atom

Hydrogen atom

A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force...

occupies a volume of ~10

^{-30}m^{3}. There is nevertheless a distinction between elementary particleElementary particle

In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...

s such as electron

Electron

The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

s or quark

Quark

A quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...

s, which have no internal structure, versus composite particles such as proton

Proton

The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....

s, which do have internal structure: A proton is made of three quarks. Elementary particles are sometimes called "point particles", but this is in a different sense than discussed above. For more details see elementary particle

Elementary particle

In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...

.

## Point mass

**Point mass**(**pointlike mass**) is an idealistic term used to describe either matterMatter

Matter is a general term for the substance of which all physical objects consist. Typically, matter includes atoms and other particles which have mass. A common way of defining matter is as anything that has mass and occupies volume...

which is infinitely small, or an object which can be thought of as infinitely small. This concept in terms of size is similar to that of point particles, however unlike point particles the object need only be considered infinitely small.

#### Physics

A common use for point mass lies in the analysis of the gravitational force fields. When analyzing the gravitational forces in a system, it becomes impossible to account for every unit of massMass

Mass can be defined as a quantitive measure of the resistance an object has to change in its velocity.In physics, mass commonly refers to any of the following three properties of matter, which have been shown experimentally to be equivalent:...

individually. When none of the objects in the system have overlapping center of mass

Center of mass

In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...

circumference

Circumference

The circumference is the distance around a closed curve. Circumference is a special perimeter.-Circumference of a circle:The circumference of a circle is the length around it....

s, it is possible to consider the object as a zero-dimensional point mass.

#### Mathematics

A point mass in statisticsStatistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

is a discontinuous segment in a probability distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

. To calculate such point mass, an integration

Integral

Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

is carried out over the entire range of the random variable

Random variable

In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

, on the probability distribution of the continuous part. After equating this integral to 1, the point mass can be found by further calculation.

## Point charge

A**point charge**is an idealized model of a particle which has an electric chargeElectric charge

Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

. A point charge is an electric charge at a mathematical point

Point (geometry)

In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...

with no dimensions.

The fundamental equation

Equation

An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...

of electrostatics

Electrostatics

Electrostatics is the branch of physics that deals with the phenomena and properties of stationary or slow-moving electric charges....

is Coulomb's law

Coulomb's law

Coulomb's law or Coulomb's inverse-square law, is a law of physics describing the electrostatic interaction between electrically charged particles. It was first published in 1785 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism...

, which describes the electric force between two point charges. The electric field

Electric field

In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...

associated with a classical point charge increases to infinity as the distance from the point charge decreases towards zero

0 (number)

0 is both a numberand the numerical digit used to represent that number in numerals.It fulfills a central role in mathematics as the additive identity of the integers, real numbers, and many other algebraic structures. As a digit, 0 is used as a placeholder in place value systems...

making energy (thus mass) of point charge infinite. In quantum electrodynamics

Quantum electrodynamics

Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

, developed in part by Richard Feynman

Richard Feynman

Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...

, the mathematical method of renormalization

Renormalization

In quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, renormalization is any of a collection of techniques used to treat infinities arising in calculated quantities....

eliminates the infinite divergence

Divergence

In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...

of the point charge.

Earnshaw's theorem

Earnshaw's theorem

Earnshaw's theorem states that a collection of point charges cannot be maintained in a stable stationary equilibrium configuration solely by the electrostatic interaction of the charges. This was first proven by British mathematician Samuel Earnshaw in 1842. It is usually referenced to magnetic...

states that a collection of point charges cannot be maintained in an equilibrium

Mechanical equilibrium

A standard definition of static equilibrium is:This is a strict definition, and often the term "static equilibrium" is used in a more relaxed manner interchangeably with "mechanical equilibrium", as defined next....

configuration solely by the electrostatic interaction of the charges.

## In quantum mechanics

In quantum mechanics

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, there is a distinction between an elementary particle

Elementary particle

(also called "point particle") and a composite particle. An elementary particle, such as an electron

Electron

The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

, quark

Quark

A quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...

, or photon

Photon

In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...

, is a particle with no internal structure, whereas a composite particle, such as a proton

Proton

The proton is a subatomic particle with the symbol or and a positive electric charge of 1 elementary charge. One or more protons are present in the nucleus of each atom, along with neutrons. The number of protons in each atom is its atomic number....

or neutron

Neutron

The neutron is a subatomic hadron particle which has the symbol or , no net electric charge and a mass slightly larger than that of a proton. With the exception of hydrogen, nuclei of atoms consist of protons and neutrons, which are therefore collectively referred to as nucleons. The number of...

, has an internal structure (see figure).

However, neither elementary nor composite particles are spatially localized, because of the Heisenberg uncertainty principle. The particle wavepacket always occupies a nonzero volume. For example, see atomic orbital

Atomic orbital

An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...

: The electron is an elementary particle, but its quantum states form three-dimensional patterns.

Nevertheless, there is good reason that an elementary particle is often called a point particle. Even if an elementary particle has a delocalized wavepacket, the wavepacket is in fact a quantum superposition

Quantum superposition

Quantum superposition is a fundamental principle of quantum mechanics. It holds that a physical system exists in all its particular, theoretically possible states simultaneously; but, when measured, it gives a result corresponding to only one of the possible configurations.Mathematically, it...

of quantum states wherein the particle is exactly localized. This is not true for a composite particle, which can never be represented as a superposition of exactly-localized quantum states. It is in this sense that physicists can discuss the intrinsic "size" of a particle: The size of its internal structure, not the size of its wavepacket. The "size" of an elementary particle, in this sense, is exactly zero.

For example, for the electron, experimental evidence shows that the size of an electron is less than 10

^{-18}m. This is consistent with the expected value of exactly zero. (This should not be confused with the classical electron radiusClassical electron radius

The classical electron radius, also known as the Lorentz radius or the Thomson scattering length, is based on a classical relativistic model of the electron...

, which, despite the name, is unrelated to the actual size of an electron.)

## See also

- Elementary ParticleElementary particle

- QuarkQuarkA quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...
- LeptonLeptonA lepton is an elementary particle and a fundamental constituent of matter. The best known of all leptons is the electron which governs nearly all of chemistry as it is found in atoms and is directly tied to all chemical properties. Two main classes of leptons exist: charged leptons , and neutral...

- Quark
- Charge (physics)Charge (physics)In physics, a charge may refer to one of many different quantities, such as the electric charge in electromagnetism or the color charge in quantum chromodynamics. Charges are associated with conserved quantum numbers.-Formal definition:...

(general concept, not limited to*electric charge*)Electric chargeElectric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two... - Self-propelled particle

## Further reading

- Eric W. Weisstein, "
*Point Charge*". - F. H. J. Cornish, "
*Classical radiation theory and point charges*". Proc. Phys. Soc. 86 427-442, 1965. doi:10.1088/0370-1328/86/3/301 - O. D. Jefimenko, "
*Direct calculation of the electric and magnetic fields of an electric point charge moving with constant velocity*". Am. J. Phys.62 (1994), 79.