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The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of
work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical work done by humans ** House work, housework, or homemaking ** Working animal, an animal tr ...
energy needed to move a unit of
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
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 In physical theories, a test particle, or test charge, is an idealized model of an object whose physical properties (usually mass, charge, or size) are assumed to be negligible except for the property being studied, which is considered to be insuf ...
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 Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...
or a point at
infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
, although any point can be used. In classical
electrostatics Electrostatics is a branch of physics that studies electric charges at rest (static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amber ...
, the electrostatic field is a vector quantity expressed as the gradient of the electrostatic potential, which is a
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers * Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
quantity denoted by or occasionally , equal to the electric potential energy of any
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, ...
at any location (measured in
joule The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force applied ...
s) divided by the
charge Charge or charged may refer to: Arts, entertainment, and media Films * ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * ''Charge!!'', an album by The Aqua ...
of that particle (measured 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). By dividing out the charge on the particle a quotient is obtained that is a property of the electric field itself. In short, an electric potential is the electric potential energy per unit charge. This value can be calculated in either a static (time-invariant) or a dynamic (time-varying)
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 fo ...
at a specific time with the unit joules per coulomb (J⋅C−1) or
volt The volt (symbol: V) is the unit of electric potential, electric potential difference (voltage), and electromotive force in the International System of Units (SI). It is named after the Italian physicist Alessandro Volta (1745–1827). Defi ...
(V). The electric potential at infinity is assumed to be zero. In
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 a ...
, when time-varying fields are present, the electric field cannot be expressed only in terms of a
scalar potential In mathematical physics, scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in trav ...
. Instead, the electric field can be expressed in terms of both the scalar electric potential and the
magnetic vector potential In classical electromagnetism, magnetic vector potential (often called A) is the vector quantity defined so that its curl is equal to the magnetic field: \nabla \times \mathbf = \mathbf. Together with the electric potential ''φ'', the magnetic v ...
. The electric potential and the magnetic vector potential together form a
four-vector In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
, so that the two kinds of potential are mixed under
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 i ...
s. Practically, the electric potential is a
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
in all space, because a spatial derivative of a discontinuous electric potential yields an electric field of impossibly infinite magnitude. Notably, the electric potential due to an idealized
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 up ...
(proportional to , with the distance from the point charge) is continuous in all space except at the location of the point charge. Though electric field is not continuous across an idealized
surface charge Surface charge is a two-dimensional surface with non-zero electric charge. These electric charges are constrained on this 2-D surface, and surface charge density, measured in coulombs per square meter (C•m−2), is used to describe the charge di ...
, it is not infinite at any point. Therefore, the electric potential is continuous across an idealized surface charge. Additionally, an idealized line of charge has electric potential (proportional to , with the radial distance from the line of charge) is continuous everywhere except on the line of charge.


Introduction

Classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
explores concepts such as
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 p ...
,
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
, and
potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple re ...
. Force and potential energy are directly related. A net force acting on any object will cause it to
accelerate In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
. As an object moves in the direction of a force acting on it, its potential energy decreases. For example, the
gravitational potential energy Gravitational energy or gravitational potential energy is the potential energy a massive object has in relation to another massive object due to gravity. It is the potential energy associated with the gravitational field, which is released (conv ...
of a cannonball at the top of a hill is greater than at the base of the hill. As it rolls downhill, its potential energy decreases and is being translated to motion –
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accele ...
. It is possible to define the potential of certain force fields so that the potential energy of an object in that field depends only on the position of the object with respect to the field. Two such force fields are a
gravitational field In physics, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
and an electric field (in the absence of time-varying magnetic fields). Such fields affect objects because of the intrinsic properties (e.g.,
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
or charge) and positions of the objects. An object may possess a property known as
electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
. Since an
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 fo ...
exerts force on a charged object, if the object has a positive charge, the force will be in the direction of the
electric field vector 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 f ...
at the location of the charge; if the charge is negative, the force will be in the opposite direction. The magnitude of force is given by the quantity of the charge multiplied by the magnitude of the electric field vector, , \mathbf, = q , \mathbf, .


Electrostatics

The electric potential at a point in a static
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 fo ...
is given by the
line integral In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, alt ...
where is an arbitrary path from some fixed reference point to . In electrostatics, the
Maxwell-Faraday equation Faraday's law of induction (briefly, Faraday's law) is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (emf)—a phenomenon known as electromagnetic inducti ...
reveals that 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 fi ...
\nabla\times\mathbf is zero, making the electric field
conservative Conservatism is a cultural, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in relation to the culture and civilization i ...
. Thus, the line integral above does not depend on the specific path chosen but only on its endpoints, making V_\mathbf well-defined everywhere. The
gradient theorem The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is ...
then allows us to write: This states that the electric field points "downhill" towards lower voltages. By
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 sta ...
, the potential can also be found to satisfy
Poisson's equation 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 th ...
: :\mathbf \cdot \mathbf = \mathbf \cdot \left (- \mathbf V_\mathbf \right ) = -\nabla^2 V_\mathbf = \rho / \varepsilon_0 where is the total
charge density In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in co ...
and \mathbf\cdot denotes the
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
. The concept of electric potential is closely linked with
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...
. A
test charge In physical theories, a test particle, or test charge, is an idealized model of an object whose physical properties (usually mass, charge, or size) are assumed to be negligible except for the property being studied, which is considered to be insuf ...
, , has an electric potential energy, , given by :U_ \mathbf = q\,V. The potential energy and hence, also the electric potential, is only defined up to an additive constant: one must arbitrarily choose a position where the potential energy and the electric potential are zero. These equations cannot be used if \nabla\times\mathbf\neq\mathbf , i.e., in the case of a ''non-conservative electric field'' (caused by a changing
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 ...
; see
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. ...
). The generalization of electric potential to this case is described in the section .


Electric potential due to a point charge

The electric potential arising from a point charge, , at a distance, , from the location of is observed to be V_\mathbf = \frac \frac, where is the
permittivity of vacuum 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 consta ...
, is known as the Coulomb potential, and the ratio, k_e=\frac is known as 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 ...
. The electric potential at any location, \textbf, in a system of point charges is equal to the sum of the individual electric potentials due to every point charge in the system. This fact simplifies calculations significantly, because addition of potential (scalar) fields is much easier than addition of the electric (vector) fields. Specifically, the potential of a set of discrete point charges at points becomes V_\mathbf(\mathbf) = k_e \sum_i \frac, where * \mathbf is a point at which the potential is evaluated. * \mathbf_i is a point at which there is a nonzero charge. * q_i is the charge at the point \mathbf_i . and the potential of a continuous charge distribution becomes V_\mathbf(\mathbf) = k_e \int_R \frac d^3 r'. Where * \mathbf is a point at which the potential is evaluated. * R is a region containing all the points at which the charge density is nonzero. * \mathbf' is a point inside R . * \rho(\mathbf') is the charge density at the point \mathbf' . The equations given above for the electric potential (and all the equations used here) are in the forms required by
SI 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 ...
. In some other (less common) systems of units, such as CGS-Gaussian, many of these equations would be altered.


Generalization to electrodynamics

When time-varying magnetic fields are present (which is true whenever there are time-varying electric fields and vice versa), it is not possible to describe the electric field simply in terms of a scalar potential because the electric field is no longer
conservative Conservatism is a cultural, social, and political philosophy that seeks to promote and to preserve traditional institutions, practices, and values. The central tenets of conservatism may vary in relation to the culture and civilization i ...
: \textstyle\int_C \mathbf\cdot \mathrm\boldsymbol is path-dependent because \mathbf \times \mathbf \neq \mathbf (due to the
Maxwell-Faraday equation Faraday's law of induction (briefly, Faraday's law) is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (emf)—a phenomenon known as electromagnetic inducti ...
). Instead, one can still define a scalar potential by also including the
magnetic vector potential In classical electromagnetism, magnetic vector potential (often called A) is the vector quantity defined so that its curl is equal to the magnetic field: \nabla \times \mathbf = \mathbf. Together with the electric potential ''φ'', the magnetic v ...
. In particular, is defined to satisfy: :\mathbf = \mathbf \times \mathbf where is 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 ...
. By the
fundamental theorem of vector calculus 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 function, smooth, rapidly decaying vector field in three dimensions can b ...
, such an can always be found, since the divergence of the magnetic field is always zero due to the absence of
magnetic monopole In particle physics, a magnetic monopole is a hypothetical elementary particle that is an isolated magnet with only one magnetic pole (a north pole without a south pole or vice versa). A magnetic monopole would have a net north or south "magneti ...
s. Now, the quantity :\mathbf = \mathbf + \frac ''is'' a conservative field, since the curl of \mathbf is canceled by the curl of \frac according to the
Maxwell–Faraday equation Faraday's law of induction (briefly, Faraday's law) is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (emf)—a phenomenon known as electromagnetic inducti ...
. One can therefore write :\mathbf = -\mathbfV - \frac , where is the scalar potential defined by the conservative field . The electrostatic potential is simply the special case of this definition where is time-invariant. On the other hand, for time-varying fields, :-\int_a^b \mathbf \cdot \mathrm\boldsymbol \neq V_ - V_ unlike electrostatics.


Gauge freedom

The electrostatic potential could have any constant added to it without affecting the electric field. In electrodynamics, the electric potential has infinitely many degrees of freedom. For any (possibly time-varying or space-varying) scalar field, \psi , we can perform the following
gauge transformation In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...
to find a new set of potentials that produce exactly the same electric and magnetic fields: :V^\prime = V - \frac :\mathbf^\prime = \mathbf + \nabla\psi Given different choices of gauge, the electric potential could have quite different properties. In the
Coulomb gauge In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct co ...
, the electric potential is given by
Poisson's equation 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 th ...
:\nabla^2 V=-\frac just like in electrostatics. However, in the
Lorenz gauge In electromagnetism, the Lorenz gauge condition or Lorenz gauge, for Ludvig Lorenz, is a partial gauge fixing of the electromagnetic vector potential by requiring \partial_\mu A^\mu = 0. The name is frequently confused with Hendrik Lorentz, who ha ...
, the electric potential is a
retarded potential In electrodynamics, the retarded potentials are the electromagnetic potentials for the electromagnetic field generated by time-varying electric current or charge distributions in the past. The fields propagate at the speed of light ''c'', so the ...
that propagates at the speed of light and is the solution to an inhomogeneous wave equation: :\nabla^2 V - \frac\frac = -\frac


Units

The
SI derived unit SI derived units are units of measurement derived from the seven base units specified by the International System of Units (SI). They can be expressed as a product (or ratio) of one or more of the base units, possibly scaled by an appropriate po ...
of electric potential is the
volt The volt (symbol: V) is the unit of electric potential, electric potential difference (voltage), and electromotive force in the International System of Units (SI). It is named after the Italian physicist Alessandro Volta (1745–1827). Defi ...
(in honor of
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 ...
), denoted as V, which is why the electric potential difference between two points in space is known as a
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
. Older units are rarely used today. Variants of the
centimetre–gram–second system of units The centimetre–gram–second system of units (abbreviated CGS or cgs) is a variant of the metric system based on the centimetre as the unit of length, the gram as the unit of mass, and the second as the unit of time. All CGS mechanical units a ...
included a number of different units for electric potential, including the
abvolt The abvolt (abV) is the unit of potential difference in the CGS-EMU system of units. It corresponds to in the SI system and 1/ statvolt ≈ in the CGS-ESU system. A potential difference of 1 abV will drive a current of one abampere through ...
and the
statvolt The statvolt is a unit of voltage and electrical potential used in the CGS-ESU and gaussian systems of units. In terms of its relation to the SI units, one statvolt corresponds to exactly  , i.e. to 299.792458 volts. The statvolt is also ...
.


Galvani potential versus electrochemical potential

Inside metals (and other solids and liquids), the energy of an electron is affected not only by the electric potential, but also by the specific atomic environment that it is in. When a
voltmeter A voltmeter is an instrument used for measuring electric potential difference between two points in an electric circuit. It is connected in parallel. It usually has a high resistance so that it takes negligible current from the circuit. Ana ...
is connected between two different types of metal, it measures the potential difference corrected for the different atomic environments. The quantity measured by a voltmeter is called
electrochemical potential In electrochemistry, the electrochemical potential (ECP), ', is a thermodynamic measure of chemical potential that does not omit the energy contribution of electrostatics. Electrochemical potential is expressed in the unit of J/ mol. Introductio ...
or
fermi level The Fermi level of a solid-state body is the thermodynamic work required to add one electron to the body. It is a thermodynamic quantity usually denoted by ''µ'' or ''E''F for brevity. The Fermi level does not include the work required to remove ...
, while the pure unadjusted electric potential, , is sometimes called the
Galvani potential In electrochemistry, the Galvani potential (also called Galvani potential difference, or inner potential difference, Δφ, delta phi) is the electric potential difference between two points in the bulk of two phases. These phases can be two differ ...
, \phi. The terms "voltage" and "electric potential" are a bit ambiguous but one may refer to ''either'' of these in different contexts.


See also

*
Absolute electrode potential Absolute electrode potential, in electrochemistry, according to an IUPAC definition, is the electrode potential of a metal measured with respect to a universal reference system (without any additional metal–solution interface). Definition Acco ...
*
Electrochemical potential In electrochemistry, the electrochemical potential (ECP), ', is a thermodynamic measure of chemical potential that does not omit the energy contribution of electrostatics. Electrochemical potential is expressed in the unit of J/ mol. Introductio ...
*
Electrode potential In electrochemistry, electrode potential is the electromotive force of a galvanic cell built from a standard reference electrode and another electrode to be characterized. By convention, the reference electrode is the standard hydrogen electrode (S ...


References


Further reading

* * * * * {{Authority control Potentials Electrostatics Physical quantities Voltage Electromagnetism