TheInfoList

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 * Work (physics), the product of ...

energy needed to move a unit of
electric charge Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respectively). Like ch ...
from a reference point to the specific point in an electric field. More precisely, it is the energy per unit charge for a
test chargeIn physical theories, a test particle, or test charge, is an idealized model of an object whose physical properties (usually mass Mass is the physical quantity, quantity of ''matter'' in a physical body. It is also a measure (mathematics), mea ...
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 harbour and support life. 29.2% of Earth's surface is land consisting of continents and islands. The remaining 70.8% is Water distribution on Earth, covered wit ...
or a point at
infinity Infinity represents something that is boundless or endless, or else something that is larger than any real or natural number In mathematics, the natural numbers are those used for counting (as in "there are ''six'' coins on the table") a ...

, although any point can be used. In classical
electrostatics Electrostatics is a branch of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related enti ...
, the electrostatic field is a vector quantity which is 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 as ...
quantity denoted by ''V'' or occasionally ''φ'', equal to the
electric potential energy Electric potential energy, is a 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 ...
of any
charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Spa ...
at any location (measured in
joule The joule ( ; symbol: J) is a SI derived unit, derived unit of energy in the International System of Units. It is equal to the energy transferred to (or work (physics), work done on) an object when a force of one Newton (unit), newton acts on tha ...

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'' (David Ford album) * Charge (Machel Montano album), ''Charge'' (Mac ...
of that particle (measured in
coulomb The coulomb (symbol: C) is the International System of Units International is an adjective (also used as a noun) meaning "between nations". International may also refer to: Music Albums * International (Kevin Michael album), ''International'' ( ...

s). By dividing out the charge on the particle a quotient is obtained that is a property of the electric field itself. In short, electric potential is the
electric potential energy Electric potential energy, is a 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 ...
per unit charge. This value can be calculated in either a static (time-invariant) or a dynamic (varying with time)
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically-charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' ' ...
at a specific time in units of joules per coulomb (), or
volt The volt (symbol: V) is the SI derived unit, derived unit for electric potential, electric potential difference (voltage), and electromotive force. It is named after the Italian physicist Alessandro Volta (1745–1827). Definition One volt is ...

s (). The electric potential at infinity is assumed to be zero. In
electrodynamics Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is carried by electromagneti ...
, when time-varying fields are present, the electric field cannot be expressed only in terms of a
scalar potential 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 traveling from one position to ...
. Instead, the electric field can be expressed in terms of both the scalar electric potential and the
magnetic vector potential Magnetic vector potential, A, is the vector quantity in classical electromagnetism Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charge Electric charg ...
. The electric potential and the magnetic vector potential together form a
four vector In special relativity, a four-vector (also known as a 4-vector) is an object with four components, which transform in a specific way under Lorentz transformation. Specifically, a four-vector is an element of a four-dimensional vector space consid ...
, so that the two kinds of potential are mixed under
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to pro ...

s. Practically, electric potential is always a
continuous function In mathematics, a continuous function is a function (mathematics), function such that a continuous variation (that is a change without jump) of the argument of a function, argument induces a continuous variation of the Value (mathematics), value o ...
in space; Otherwise, the spatial derivative of it will yield a field with infinite magnitude, which is practically impossible. Even an idealized
point charge A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, l ...
has potential, which is continuous everywhere except the origin. The electric field is not continuous across an idealized
surface charge Surface charge is a two-dimensional surface with non-zero electric charge Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''ne ...
, but it is not infinite at any point. Therefore, the electric potential is continuous across an idealized surface charge. An idealized linear charge has potential, which is continuous everywhere except on the linear charge.

# Introduction

Classical mechanics explores concepts such as
force In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...
,
energy In physics, energy is the physical quantity, quantitative physical property, property that must be #Energy transfer, transferred to a physical body, body or physical system to perform Work (thermodynamics), work on the body, or to heat it. En ...

, and
potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' ...

. Force and potential energy are directly related. A net force acting on any object will cause it to
accelerate In mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among force, matter, and motion. Forces applied to objects result in Displacement ( ...
. As an object moves in the direction in which the force accelerates it, its potential energy decreases. For example, the
gravitational potential energy Gravitational energy or gravitational potential energy is the 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 othe ...
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, being translated to motion, kinetic energy. 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 the
gravitational field In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...

and an electric field (in the absence of time-varying magnetic fields). Such fields must affect objects due to the intrinsic properties of the object (e.g.,
mass Mass is the physical quantity, quantity of ''matter'' in a physical body. It is also a measure (mathematics), measure of the body's ''inertia'', the resistance to acceleration (change of velocity) when a net force is applied. An object's mass ...
or charge) and the position of the object. Objects may possess a property known as
electric charge Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respectively). Like ch ...
and an
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically-charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' ' ...
exerts a force on charged objects. If the charged 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 each electric charge Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. There are two ty ...
at that point while if the charge is negative the force will be in the opposite direction. The magnitude of the force is given by the quantity of the charge multiplied by the magnitude of the electric field vector.

# Electrostatics

The electric potential at a point r in a static
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically-charged particle In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' ' ...
E is given by the
line integral In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
where ''C'' is an arbitrary path from some fixed reference point to $\mathbf$. In electrostatics, the Maxwell-Faraday equation reveals that the
curl cURL (pronounced 'curl') is a computer software project providing a library A library is a curated collection of sources of information and similar resources, made accessible to a defined community for reference or borrowing. It provides ...
$\nabla\times\mathbf$ is zero, making the electric field
conservative Conservatism is a Political philosophy, political and social philosophy promoting traditional social institutions. The central tenets of conservatism may vary in relation to the traditional values or practices of the culture and civilization ...
. Thus, the line integral above does not depend on the specific path ''C'' 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 Conservative vector field, gradient field can be evaluated by evaluating the original scalar field at the endpoints of ...
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 state ...
, 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 t ...
: :$\mathbf \cdot \mathbf = \mathbf \cdot \left \left(- \mathbf V_\mathbf \right \right) = -\nabla^2 V_\mathbf = \rho / \varepsilon_0$ where ''ρ'' is the total
charge density In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is ca ...
and ∇· 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 ou ...
. 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 potential ...

. A
test chargeIn physical theories, a test particle, or test charge, is an idealized model of an object whose physical properties (usually mass Mass is the physical quantity, quantity of ''matter'' in a physical body. It is also a measure (mathematics), mea ...
''q'' has an
electric potential energy Electric potential energy, is a 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 ...
''U''E 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 the curl $\nabla\times\mathbf\neq\mathbf$, 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 the ...

; see
Maxwell's equations Maxwell's 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 equations provide a mathematic ...
). 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 ''Q'', at a distance ''r'' from the charge is observed to be :$V_\mathbf = \frac \frac,$ where ''ε''0 is the permittivity of vacuum. is known as the Coulomb potential. The electric potential for a system of point charges is equal to the sum of the point charges' individual potentials. 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 ''q''''i'' at points r''i'' becomes :$V_\mathbf\left(\mathbf\right) = \frac \sum_i \frac,$ 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 ''ρ''(r) becomes :$V_\mathbf\left(\mathbf\right) = \frac \int_R \frac d^3 r\text{'}.$ 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. :$\mathbf\text{'}$ is a point inside $R$. :$\rho\left(\mathbf\text{'}\right)$ is the charge density at the point $\mathbf\text{'}$. The equations given above for the electric potential (and all the equations used here) are in the forms required by SI units. 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 ''V'' because the electric field is no longer
conservative Conservatism is a Political philosophy, political and social philosophy promoting traditional social institutions. The central tenets of conservatism may vary in relation to the traditional values or practices of the culture and civilization ...
: $\textstyle\int_C \mathbf\cdot \mathrm\boldsymbol$ is path-dependent because $\mathbf \times \mathbf \neq \mathbf$ (due to the Maxwell-Faraday equation). Instead, one can still define a scalar potential by also including the
magnetic vector potential Magnetic vector potential, A, is the vector quantity in classical electromagnetism Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charge Electric charg ...
A. In particular, A is defined to satisfy: :$\mathbf = \mathbf \times \mathbf$ where B 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 the ...

. By the fundamental theorem of vector calculus, such an A 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 Magnetic field lines of a solenoid electromagnet, which are similar to a bar magnet as illustrated below with the iron filings A ...
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. One can therefore write :$\mathbf = -\mathbfV - \frac$ where ''V'' is the scalar potential defined by the conservative field F. The electrostatic potential is simply the special case of this definition where A 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 Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Sp ...
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 Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior throu ...
, 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 t ...
:$\nabla^2 V=-\frac$ just like in electrostatics. However, in the Lorenz gauge, the electric potential is a
retarded potential In electrodynamics, the retarded potentials are the electromagnetic potentials for the electromagnetic field An electromagnetic field (also EM field) is a classical (i.e. non-quantum) field produced by accelerating electric charge Electric ...
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 ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or ...
of electric potential is the
volt The volt (symbol: V) is the SI derived unit, derived unit for electric potential, electric potential difference (voltage), and electromotive force. It is named after the Italian physicist Alessandro Volta (1745–1827). Definition One volt is ...

(in honor of
Alessandro Volta Alessandro Giuseppe Antonio Anastasio Volta (, ; 18 February 1745 – 5 March 1827) was an Italian physicist A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in ...
), which is why a difference in electric potential between two points is known as
voltage Voltage, electric potential difference, electric pressure or electric tension is the difference in electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is the ...

. 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 mechanics, mechanic ...
included a number of different units for electric potential, including the abvolt and the statvolt.

# 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 The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is the amount of work energy needed to move a unit of elec ...

is connected between two different types of metal, it measures not the electric potential difference, but instead the potential difference corrected for the different atomic environments. The quantity measured by a voltmeter is called
electrochemical potential In electrochemistry Electrochemistry is the branch of physical chemistry concerned with the relationship between electrical potential, as a measurable and quantitative phenomenon, and identifiable chemical change, with either electrical potentia ...
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 Thermodynamics is a branch of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (e ...
, while the pure unadjusted electric potential ''V'' is sometimes called
Galvani potential In electrochemistry Electrochemistry is the branch of physical chemistry concerned with the relationship between electrical potential, as a measurable and quantitative phenomenon, and identifiable chemical change, with either electrical potential ...
$\phi$. The terms "voltage" and "electric potential" are a bit ambiguous in that, in practice, they can refer to ''either'' of these in different contexts.