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Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law ^{ [1] } of physics 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.^{ [2] } Although the law was known earlier, it was first published in 1785 by French physicist Charles-Augustin de Coulomb, hence the name. Coulomb's law was essential to the development of the theory of electromagnetism, maybe even its starting point,^{ [1] } as it made it possible to discuss the quantity of electric charge in a meaningful way.^{ [3] }
The law states that the magnitude of the electrostatic force 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,^{ [4] }
Here, k_{e} or K is Coulomb's constant (k_{e} ≈ 8.988×10^{9} N⋅m^{2}⋅C^{−2}),^{ [1] }q_{1} and q_{2} are the signed 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, the law is analogous to Isaac Newton's inverse-square law of universal gravitation, but gravitational forces are always attractive, while electrostatic forces can be attractive or repulsive.^{ [2] } Coulomb's law can be used to derive Gauss's law, 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.^{ [5] } The law has been tested extensively, and observations have upheld the law on the scale from 10^{−16} m to 10^{8} m.^{ [5] }
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 papers. Thales of Miletus made the first recorded description of static electricity around 600 BC,^{ [6] } when he noticed that friction could render a piece of amber magnetic.^{ [7] }^{ [8] }
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.^{ [7] } He coined the New 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.^{ [9] } 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.^{ [10] }
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 ^{ [11] } 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.^{ [12] }
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.^{ [13] } In 1767, he conjectured that the force between charges varied as the inverse square of the distance.^{ [14] }^{ [15] }
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}.^{ [16] }
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.^{ [17] }
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.^{ [4] } 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.
Coulomb's law can be stated as a simple mathematical expression. The scalar form gives the magnitude of the vector of the electrostatic force F between two point charges q_{1} and q_{2}, but not its direction. If r is the distance between the charges, the magnitude of the force is
The constant k_{e} is called Coulomb's constant and is equal to 1/4πε_{0}, where ε_{0} is the electric constant; k_{e} = 8.988×10^{9} N⋅m^{2}⋅C^{−2}. If the product q_{1}q_{2} is positive, the force between the two charges is repulsive; if the product is negative, the force between them is attractive.^{ [18] }
Coulomb's law in vector form states that the electrostatic force experienced by a charge, at position , in the vicinity of another charge, at position , in a vacuum is equal to^{ [19] }
where is the vectorial distance between the charges, a unit vector pointing from to , and the electric constant.
The vector form of Coulomb's law is simply the scalar definition of the law with the direction given by the unit vector, , parallel with the line from charge to charge .^{ [20] } If both charges have the same sign (like charges) then the product is positive and the direction of the force on is given by ; the charges repel each other. If the charges have opposite signs then the product is negative and the direction of the force on is ; the charges attract each other.
The electrostatic force experienced by , according to Newton's third law, is .
The law of superposition allows Coulomb's law to be extended to include any number of point charges. The force acting on a point charge due to a system of point charges is simply the vector addition of the individual forces acting alone on that point charge due to each one of the charges. The resulting force vector is parallel to the electric field vector at that point, with that point charge removed.
Force on a small charge at position , due to a system of discrete charges in vacuum is^{ [19] }
where and are the magnitude and position respectively of the ith charge, is a unit vector in the direction of , a vector pointing from charges to .^{ [20] }
In this case, the principle of linear superposition is also used. For a continuous charge distribution, an integral over the region containing the charge is equivalent to an infinite summation, treating each infinitesimal element of space as a point charge . The distribution of charge is usually linear, surface or volumetric.
For a linear charge distribution (a good approximation for charge in a wire) where gives the charge per unit length at position , and is an infinitesimal element of length,^{ [21] }
For a surface charge distribution (a good approximation for charge on a plate in a parallel plate capacitor) where gives the charge per unit area at position , and is an infinitesimal element of area,
For a volume charge distribution (such as charge within a bulk metal) where gives the charge per unit volume at position , and is an infinitesimal element of volume,^{ [20] }
The force on a small test charge at position in vacuum is given by the integral over the distribution of charge
where it must be noted that the "continuous charge" version of Coulomb's law is never supposed to be applied to locations for which because that location would directly overlap with the location of a charged particle (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 to be analyzed.
Coulomb's constant is a proportionality factor that appears in Coulomb's law as well as in other electric-related formulas. Denoted , it is also called the electric force constant or electrostatic constant^{ [22] } hence the subscript . When the electromagnetic theory is expressed in the International System of Units, force is measured in newtons, charge in coulombs and distance in meters. Coulomb's constant is given by . The constant is the vacuum electric permittivity (also known as "electric constant")^{ [23] } in . It should not be confused with , which is the dimensionless relative permittivity of the material in which the charges are immersed, or with their product , which is called "absolute permittivity of the material" and is still used in electrical engineering.
Prior to the 2019 redefinition of the SI base units, the Coulomb constant was considered to have an exact value:
Since the 2019 redefinition,^{ [24] }^{ [25] } the Coulomb constant is no longer exactly defined and is subject to the measurement error in the fine structure constant. As calculated from CODATA 2018 recommended values, the Coulomb constant is^{ [26] }
In Gaussian units and Lorentz–Heaviside units, which are both CGS unit systems, the constant has different, dimensionless values.
In electrostatic units or Gaussian units the unit charge (esu or statcoulomb) is defined in such a way that the Coulomb constant disappears, as it has the value of one and becomes dimensionless.
In Lorentz–Heaviside units, also called rationalizedunits, the Coulomb constant is dimensionless and is equal to
Gaussian units are more amenable for microscopic problems such as the electrodynamics of individual electrically charged particles.^{ [27] } SI units are more convenient for practical, large-scale phenomena, such as engineering applications.^{ [27] }
There are three conditions to be fulfilled for the validity of Coulomb's inverse square law:^{ [28] }
The last of these is known as the electrostatic approximation. When movement takes place, Einstein's theory of relativity 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 force, and is described by magnetic fields. 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 rules (incorporating the magnetic force) must be considered.
An electric field is a vector field that associates to each point in space the Coulomb force experienced by a unit test charge.^{ [19] } The strength and direction of the Coulomb force on a charge depends on the electric field established by other charges that it finds itself in, such that . In the simplest case, the field is considered to be generated solely by a single source point charge. More generally, the field can be generated by a distribution of charges who contribute to the overall by the principle of superposition.
If the field is generated by a positive source point charge , the direction of the electric field points along lines directed radially outwards from it, i.e. in the direction that a positive point test charge would move if placed in the field. For a negative point source charge, the direction is radially inwards.
The magnitude of the electric field 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
A system N of charges stationed at produces an electric field whose magnitude and direction is, by superposition
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.
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 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 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).
Coulomb's law states that the electric field due to a stationary point charge is:
where
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
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^{ [29] }
where δ(r) is the Dirac delta function, the result is
Using the "sifting property" of the Dirac delta function, we arrive at
which is the differential form of Gauss' law, as desired.
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.
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).
Taking S in the integral form of Gauss' law to be a spherical surface of radius r, centered at the point charge Q, we have
By the assumption of spherical symmetry, the integrand is a constant which can be taken out of the integral. The result is
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
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' law.
This article may be too technical for most readers to understand.(October 2020) |
The Coulomb potential admits continuum states (with E > 0), describing electron-proton scattering, as well as discrete bound states, representing the hydrogen atom.^{ [30] } 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 is:
This is to be compared to the:
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
Comparing with the QM scattering, we have to discard the as they arise due to differing normalizations of momentum eigenstate in QFT compared to QM and obtain:
where Fourier transforming both sides, solving the integral and taking at the end will yield
as the Coulomb potential.^{ [31] }
However, the equivalent results of the classical Born derivations for the Coulomb problem are thought to be strictly accidental.^{ [32] }^{ [33] }
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.^{ [30] }
This section may contain an excessive amount of intricate detail that may interest only a particular audience.(October 2020) |
It is possible to verify Coulomb's law with a simple experiment. Consider two small spheres of mass and same-sign charge , hanging from two ropes of negligible mass of length . The forces acting on each sphere are three: the weight , the rope tension and the electric force . In the equilibrium state:
| (1) |
and
| (2) |
| (3) |
Let be the distance between the charged spheres; the repulsion force between them , assuming Coulomb's law is correct, is equal to
| (Coulomb's law) |
so:
| (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 . In the equilibrium state, the distance between the charges will be and the repulsion force between them will be:
| (5) |
We know that and:
Dividing ( 4 ) by ( 5 ), we get:
| (6) |
Measuring the angles and and the distance between the charges and 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:
| (7) |
Using this approximation, the relationship ( 6 ) becomes the much simpler expression:
| (8) |
In this way, the verification is limited to measuring the distance between the charges and check that the division approximates the theoretical value.
An electric 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 for a system of charged particles. Electric fields originate from electric charges and time-varying electric currents. Electric fields and magnetic fields are both manifestations of the electromagnetic field, one of the four fundamental interactions of nature.
In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ε (epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in response to an applied electric field than a material with low permittivity, thereby storing more energy in the material. In electrostatics, the permittivity plays an important role in determining the capacitance of a capacitor.
The electric potential is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in an electric field. More precisely, it is the energy per unit charge for a test charge that is so small that the disturbance of the field under consideration is negligible. Furthermore, the motion across the field is supposed to proceed with negligible acceleration, so as to avoid the test charge acquiring kinetic energy or producing radiation. By definition, the electric potential at the reference point is zero units. Typically, the reference point is earth or a point at infinity, although any point can be used.
In physics and electromagnetism, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field. In its integral form, it states that the flux of the electric field out of an arbitrary closed surface is proportional to the electric charge enclosed by the surface, irrespective of how that charge is distributed. Even though the law alone is insufficient to determine the electric field across a surface enclosing any charge distribution, this may be possible in cases where symmetry mandates uniformity of the field. Where no such symmetry exists, Gauss's law can be used in its differential form, which states that the divergence of the electric field is proportional to the local density of charge.
In physics, specifically electromagnetism, the Biot–Savart law 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 electric current. The Biot–Savart law is fundamental to magnetostatics, playing a role similar to that of Coulomb's law in electrostatics. When magnetostatics does not apply, the Biot–Savart law should be replaced by Jefimenko's equations. The law is valid in the magnetostatic approximation, and consistent with both Ampère's circuital law and Gauss's law for magnetism. It is named after Jean-Baptiste Biot and Félix Savart, who discovered this relationship in 1820.
In physics, screening is the damping of electric fields caused by the presence of mobile charge carriers. It is an important part of the behavior of charge-carrying fluids, such as ionized gases, electrolytes, and charge carriers in electronic conductors . In a fluid, with a given permittivity ε, composed of electrically charged constituent particles, each pair of particles interact through the Coulomb force as
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with the potential field known, one can then calculate electrostatic or gravitational (force) field. It is a generalization of Laplace's equation, which is also frequently seen in physics. The equation is named after French mathematician and physicist Siméon Denis Poisson.
Electrostatics is a branch of physics that studies electric charges at rest.
Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model. The theory provides a description of electromagnetic phenomena whenever the relevant length scales and field strengths are large enough that quantum mechanical effects are negligible. For small distances and low field strengths, such interactions are better described by quantum electrodynamics.
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 units, or often just cgs units. The term "cgs units" is ambiguous and therefore to be avoided if possible: there are several variants of cgs with conflicting definitions of electromagnetic quantities and units.
In plasmas and electrolytes, the Debye length, is a measure of a charge carrier's net electrostatic effect in a solution and how far its electrostatic effect persists. With each Debye length the charges are increasingly electrically screened and the electric potential decreases in magnitude by 1/e. A Debye sphere is a volume whose radius is the Debye length. Debye length is an important parameter in plasma physics, electrolytes, and colloids. The corresponding Debye screening wave vector for particles of density , charge at a temperature is given by in Gaussian units. Expressions in MKS units will be given below. The analogous quantities at very low temperatures are known as the Thomas–Fermi length and the Thomas–Fermi wave vector. They are of interest in describing the behaviour of electrons in metals at room temperature.
Electric field work is the work performed by an electric field on a charged particle in its vicinity.
Electric potential energy is a potential energy that results from conservative Coulomb forces and is associated with the configuration of a particular set of point charges within a defined system. An object may have electric potential energy by virtue of two key elements: its own electric charge and its relative position to other electrically charged objects.
The Coulomb constant, the electric force constant, or the electrostatic constant (denoted k_{e}, k or K) is a proportionality constant in electrostatics equations. In SI base units it is equal to 8.9875517923(14)×10^{9} kg⋅m^{3}⋅s^{−4}⋅A^{−2}. It was named after the French physicist Charles-Augustin de Coulomb (1736–1806) who introduced Coulomb's law.
Vacuum permittivity, commonly denoted ε_{0}, 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 constant, or the distributed capacitance of the vacuum. It is an ideal (baseline) physical constant. Its CODATA value is:
In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density is the quantity of charge per unit volume, measured in the SI system in coulombs per cubic meter (C⋅m^{−3}), at any point in a volume. Surface charge density (σ) is the quantity of charge per unit area, measured in coulombs per square meter (C⋅m^{−2}), at any point on a surface charge distribution on a two dimensional surface. Linear charge density (λ) is the quantity of charge per unit length, measured in coulombs per meter (C⋅m^{−1}), at any point on a line charge distribution. Charge density can be either positive or negative, since electric charge can be either positive or negative.
A Gaussian surface is a closed surface in three-dimensional space through which the flux of a vector field is calculated; usually the gravitational field, electric field, or magnetic field. It is an arbitrary closed surface S = ∂V used in conjunction with Gauss's law for the corresponding field by performing a surface integral, in order to calculate the total amount of the source quantity enclosed; e.g., amount of gravitational mass as the source of the gravitational field or amount of electric charge as the source of the electrostatic field, or vice versa: calculate the fields for the source distribution.
Lorentz–Heaviside units constitute a system of units within CGS, named for Hendrik Antoon Lorentz and Oliver Heaviside. They share with CGS-Gaussian units the property that the electric constant ε_{0} and magnetic constant µ_{0} do not appear, having been incorporated implicitly into the electromagnetic quantities by the way they are defined. Heaviside-Lorentz units may be regarded as normalizing ε_{0} = 1 and µ_{0} = 1, while at the same time revising Maxwell's equations to use the speed of light c instead.
The method of image charges is a basic problem-solving tool in electrostatics. The name originates from the replacement of certain elements in the original layout with imaginary charges, which replicates the boundary conditions of the problem.
Axial multipole moments are a series expansion of the electric potential of a charge distribution localized close to the origin along one Cartesian axis, denoted here as the z-axis. However, the axial multipole expansion can also be applied to any potential or field that varies inversely with the distance to the source, i.e., as . For clarity, we first illustrate the expansion for a single point charge, then generalize to an arbitrary charge density localized to the z-axis.
Il résulte donc de ces trois essais, que l'action répulsive que les deux balles électrifées de la même nature d'électricité exercent l'une sur l'autre, suit la raison inverse du carré des distances. Translation: 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 obey an inverse-square law of attraction.— Coulomb (1785b) "Second mémoire sur l’électricité et le magnétisme," Histoire de l’Académie Royale des Sciences, pages 578–611