Editing Electric potential

{{Short description|Line integral of the electric field}}
{{distinguish|Voltage}}
{{Infobox physical quantity
| name =Electric potential 
| image = File:VFPt metal balls largesmall potential+contour.svg|thumb|upright=1.2
| caption = Electric potential around two oppositely charged conducting spheres. Purple represents the highest potential, yellow zero, and cyan the lowest potential. The electric [[field line]]s are shown leaving perpendicularly to the surface of each sphere.
| unit = [[volt]]
| otherunits = [[statvolt]]
| symbols = ''V'', ''φ''
| baseunits = V = kg⋅m<sup>2</sup>⋅s<sup>−3</sup>⋅A<sup>−1</sup>
| dimension = '''M''' '''L'''<sup>2</sup> '''T'''<sup>−3</sup> '''I'''<sup>−1</sup>
| extensive = yes
| derivations =
}}
{{Electromagnetism|Electrostatics}}

'''Electric potential''' (also called the ''electric field potential'', potential drop, the '''electrostatic potential''') is defined as the amount of [[work (physics)|work]]/[[energy]] needed per unit of [[electric charge]] to move the charge from a reference point to a specific point in an electric field. More precisely, the electric potential is the energy per unit charge for a [[test charge]] that is so small that the disturbance to the field (due to the test charge's own field) under consideration is negligible. 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 (electricity)|earth]] or a point at [[infinity]], although any point can be used. 

In classical [[electrostatics]], the electrostatic field is a vector quantity expressed as the gradient of the electrostatic potential, which is a [[scalar (physics)|scalar]] quantity denoted by {{math|''V''}} or occasionally {{math|''φ''}},<ref>{{cite book |last=Goldstein |first=Herbert |date=June 1959 |title=Classical Mechanics |location=United States |publisher=Addison-Wesley |page=383 |author-link=Herbert Goldstein |isbn=0201025108}}</ref> equal to the [[electric potential energy]] of any [[charged particle]] at any location (measured in [[joule]]s) divided by the [[electric charge|charge]] of that particle (measured in [[coulomb]]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]] at a specific time with the unit joules per coulomb (J⋅C<sup>−1</sup>) or [[volt]] (V). The electric potential at infinity is assumed to be zero.

In [[electrodynamics]], when time-varying fields are present, the electric field cannot be expressed only as a [[scalar potential]]. Instead, the electric field can be expressed as both the scalar electric potential and the [[magnetic vector potential]].<ref>{{Cite book|title=Introduction to Electrodynamics|last=Griffiths|first=David J.|year=1999|publisher=Pearson Prentice Hall|isbn=978-81-203-1601-0|pages=416–417}}</ref> The electric potential and the magnetic vector potential together form a [[four vector|four-vector]], so that the two kinds of potential are mixed under [[Lorentz transformation]]s.

Practically, the electric potential is a [[continuous function]] 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]] (proportional to {{math|1 ⁄ ''r''}}, with {{math|''r''}} 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]], 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 {{math|ln(''r'')}}, with {{math|''r''}} the radial distance from the line of charge) is continuous everywhere except on the line of charge.

== Introduction ==
[[Classical mechanics]] explores concepts such as [[Force (physics)|force]], [[energy]], and [[potential]].<ref>{{cite book |last1=Young |first1=Hugh A. |last2=Freedman |first2=Roger D. |title=Sears and Zemansky's University Physics with Modern Physics |date=2012 |publisher=Addison-Wesley |location=Boston |page=754 |edition=13th}}</ref> Force and potential energy are directly related. A net force acting on any object will cause it to [[acceleration|accelerate]]. As an object moves in the direction of a force acting on it, its potential energy decreases. For example, the [[gravitational potential energy]] 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]].

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 [[gravity|gravitational field]] and an electric field (in the absence of time-varying magnetic fields). Such fields affect objects because of the intrinsic properties  (e.g., [[mass]] or charge) and positions of the objects.

An object may possess a property known as [[electric charge]]. Since an [[electric field]] 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]] 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,

<math display=block>|\mathbf{F}| = q |\mathbf{E}|.</math>

== Electrostatics ==
{{Main|Electrostatics}}{{Multiple image
| align = 
| direction = 
| total_width = 300
| image1 = VFPt plus thumb potential+contour.svg
| alt1 = 
| caption1 = 
| image2 = VFPt minus thumb potential+contour.svg
| caption2 = 
| footer = Electric potential of separate positive and negative point charges shown as color range from magenta (+), through yellow (0), to cyan (−). Circular contours are equipotential lines. Electric field lines leave the positive charge and enter the negative charge.
}}{{Multiple image
| align = 
| direction = 
| total_width = 300
| image1 = VFPt charges plus minus potential+contour.svg
| alt1 = 
| caption1 = 
| caption2 = 
| image2 = 
| footer = Electric potential in the vicinity of two opposite point charges.
}}

An electric potential at a point {{math|'''r'''}} in a static [[electric field]] {{math|'''E'''}} is given by the [[line integral]]

{{Equation box 1
|indent=:
|equation=<math>V_\mathbf{E} = - \int_{\mathcal{C} } \mathbf{E} \cdot \mathrm{d} \boldsymbol{\ell}\,</math>
|cellpadding
|border
|border colour = #50C478
|background colour = #ECFCC4}}

where {{mvar|C}} is an arbitrary path from some fixed reference point to {{math|'''r'''}}; it is uniquely determined up to a constant that is added or subtracted from the integral. In electrostatics, the [[Maxwell-Faraday equation]] reveals that the [[Curl (mathematics)|curl]] <math display="inline">\nabla\times\mathbf{E}</math> is zero, making the electric field [[Conservative vector field|conservative]]. Thus, the line integral above does not depend on the specific path {{mvar|C}} chosen but only on its endpoints, making <math display="inline">V_\mathbf{E}</math> well-defined everywhere. The [[gradient theorem]] then allows us to write:

{{Equation box 1
|indent=:
|equation=<math>\mathbf{E} = - \mathbf{\nabla} V_\mathbf{E}\,</math>
|cellpadding
|border
|border colour = #50C478
|background colour = #ECFCC4}}

This states that the electric field points "downhill" towards lower voltages. By [[Gauss's law]], the potential can also be found to satisfy [[Poisson's equation]]:
<math display="block">\mathbf{\nabla} \cdot \mathbf{E} = \mathbf{\nabla} \cdot \left (- \mathbf{\nabla} V_\mathbf{E} \right ) = -\nabla^2 V_\mathbf{E} = \rho / \varepsilon_0 </math>

where {{mvar|ρ}} is the total [[charge density]] and <math display="inline">\mathbf{\nabla}\cdot</math> denotes the [[divergence]].

The concept of electric potential is closely linked with [[potential energy]]. A [[test charge]], {{math|''q''}}, has an [[electric potential energy]], {{math|''U''<sub>'''E'''</sub>}}, given by

<math display="block">U_ \mathbf{E} = q\,V.</math>

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 {{nowrap|<math display="inline">\nabla\times\mathbf{E}\neq\mathbf{0}</math>,}} i.e., in the case of a ''non-conservative electric field'' (caused by a changing [[magnetic field]]; see [[Maxwell's equations]]). The generalization of electric potential to this case is described in the section {{Section link||Generalization to electrodynamics}}.

=== Electric potential due to a point charge<!--'Coulomb potential' redirects here--> ===
{{See also|Coulomb's law}}
[[File:Electric potential varying charge.gif|right|thumb|150x150px|The electric potential created by a charge, ''Q'', is ''V'' = ''Q''/(4πε<sub>0</sub>''r''). Different values of ''Q'' yield different values of electric potential, ''V'', (shown in the image).]]

The electric potential arising from a point charge, {{math|''Q''}}, at a distance, {{math|''r''}}, from the location of {{math|''Q''}} is observed to be
<math display="block"> V_\mathbf{E} = \frac{1}{4 \pi \varepsilon_0} \frac{Q}{r}, </math>
where {{math|''ε''<sub>0</sub>}} is the [[permittivity of vacuum]]{{physconst|eps0|ref=only}}, {{math|''V''<sub>'''E'''</sub>}} is known as the '''Coulomb potential'''. Note that, in contrast to the magnitude of an [[electric field]] due to a point charge, the electric potential scales respective to the reciprocal of the radius, rather than the radius squared.

The electric potential at any location, {{math|'''r'''}}, 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 {{mvar|q<sub>i</sub>}} at points {{math|'''r'''<sub>''i''</sub>}} becomes

<math display="block"> V_\mathbf{E}(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0} \sum_{i=1}^n\frac{q_i}{|\mathbf{r}-\mathbf{r}_i|}\,</math>

where
*{{math|'''r'''}} is a point at which the potential is evaluated;
*{{math|'''r'''{{sub|''i''}}}} is a point at which there is a nonzero charge; and
*{{mvar|q{{sub|i}}}} is the charge at the point {{math|'''r'''{{sub|''i''}}}}.

And the potential of a continuous charge distribution {{math|''ρ''('''r''')}} becomes

<math display="block"> V_\mathbf{E}(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0} \int_R \frac{\rho(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \mathrm{d}^3 r'\,,</math>

where
*{{math|'''r'''}} is a point at which the potential is evaluated;
*{{mvar|R}} is a region containing all the points at which the charge density is nonzero;
*{{math|'''r'''{{'}}}} is a point inside {{mvar|R}}; and
*{{math|''ρ''('''r'''{{'}})}} is the charge density at the point {{math|'''r'''{{'}}}}.

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 [[Gaussian units|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 as a scalar potential {{math|''V''}} because the electric field is no longer [[conservative force|conservative]]: <math>\textstyle\int_C \mathbf{E}\cdot \mathrm{d}\boldsymbol{\ell}</math> is path-dependent because <math>\mathbf{\nabla} \times \mathbf{E} \neq \mathbf{0} </math> (due to the [[Maxwell-Faraday equation]]).

Instead, one can still define a scalar potential by also including the [[magnetic vector potential]] {{math|'''A'''}}. In particular, {{math|'''A'''}} is defined to satisfy:

<math display="block">\mathbf{B} = \mathbf{\nabla} \times \mathbf{A} </math>

where {{math|'''B'''}} is the [[magnetic field]]. By the [[fundamental theorem of vector calculus]], such an {{math|'''A'''}} can always be found, since [[Gauss's law for magnetism|the divergence of the magnetic field is always zero]] due to the absence of [[magnetic monopole]]s. Now, the quantity
<math display="block">\mathbf{F} = \mathbf{E} + \frac{\partial\mathbf{A}}{\partial t}</math>
''is'' a conservative field, since the curl of <math>\mathbf{E}</math> is canceled by the curl of <math>\frac{\partial\mathbf{A}}{\partial t}</math> according to the [[Maxwell–Faraday equation]]. One can therefore write
<math display="block">\mathbf{E} = -\mathbf{\nabla}V - \frac{\partial\mathbf{A}}{\partial t} ,</math>

where {{math|''V''}} is the scalar potential defined by the conservative field {{math|'''F'''}}.

The electrostatic potential is simply the special case of this definition where {{math|'''A'''}} is time-invariant.  On the other hand, for time-varying fields,
<math display="block">-\int_a^b \mathbf{E} \cdot \mathrm{d}\boldsymbol{\ell} \neq V_{(b)} - V_{(a)} </math>
unlike electrostatics.

=== Gauge freedom ===
{{Main articles|Gauge fixing}}
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, {{math|𝜓}}, we can perform the following [[Gauge Transformation|gauge transformation]] to find a new set of potentials that produce exactly the same electric and magnetic fields:<ref>{{Cite book|last=Griffiths|first=David J.|title=Introduction to Electrodynamics|publisher=Prentice Hall|year=1999|isbn=013805326X|edition=3rd|pages=420}}</ref>

<math display="block">\begin{align}
V^\prime &= V - \frac{\partial\psi}{\partial t} \\
\mathbf{A}^\prime &= \mathbf{A} + \nabla\psi
\end{align}</math>

Given different choices of gauge, the electric potential could have quite different properties. In the [[Coulomb Gauge|Coulomb gauge]], the electric potential is given by [[Poisson's equation]]

<math display="block">\nabla^2 V=-\frac{\rho}{\varepsilon_0} </math>

just like in electrostatics. However, in the [[Lorenz gauge condition|Lorenz gauge]], the electric potential is a [[retarded potential]] that propagates at the speed of light and is the solution to an [[Inhomogeneous electromagnetic wave equation#A and %CF%86 potential fields|inhomogeneous wave equation]]:

<math display="block">\nabla^2 V - \frac{1}{c^2}\frac{\partial^2 V}{\partial t^2} = -\frac{\rho}{\varepsilon_0} </math>

==Units==
The [[SI derived unit]] of electric potential is the [[volt]] (in honor of [[Alessandro Volta]]), denoted as V, which is why the electric potential difference between two points in space is known as a [[voltage]]. Older units are rarely used today.  Variants of the [[centimetre–gram–second system of units]] included a number of different units for electric potential, including the [[Conversion of units#Electromotive force, electric potential difference|abvolt]] and the [[statvolt]].

==Galvani potential versus electrochemical potential==
{{Main|Galvani potential|Electrochemical potential|Fermi level}}
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]] is connected between two different types of metal, it measures the '''potential difference''' corrected for the different atomic environments.<ref>{{cite book|url=https://books.google.com/books?id=09QI-assq1cC&pg=PA22 |title=Fundamentals of electrochemistry|first= Vladimir Sergeevich|last= Bagotskii | name-list-style = vanc |page=22|isbn=978-0-471-70058-6|year=2006|publisher=John Wiley & Sons }}</ref> The quantity measured by a voltmeter is called [[electrochemical potential]] or [[fermi level]], while the pure unadjusted electric potential, {{math|''V''}}, is sometimes called the [[Galvani potential]], {{mvar|ϕ}}. The terms "voltage" and "electric potential" are a bit ambiguous but one may refer to {{em|either}} of these in different contexts.

== Common formulas==
{| class="wikitable"
|+
!Charge configuration
!Figure
! colspan="2" |Electric potential
|-
|Infinite wire
| [[File:Charged infinite wire problem.svg|frameless|235x235px]]
| colspan="2" |<math>V = -\frac{\lambda}{2\pi\varepsilon_0}\ln x,</math> 

where <math>\lambda</math> is uniform linear charge density.
|-
|Infinitely large surface
|[[File:Charged infinite plane problem.svg|frameless|235x235px]]
| colspan="2" |<math>V = -\frac{\sigma x}{2\varepsilon_0},</math>
where <math>\sigma</math> is uniform surface charge density.
|-
|Infinitely long cylindrical volume
|[[File:Charged infinite cylinder problem.svg|frameless|235x235px]]
| colspan="2" |<math>V = -\frac{\lambda}{2\pi\varepsilon_0}\ln x, </math>
where <math>\lambda</math> is uniform linear charge density.
|-
|Spherical volume
|[[File:Charged solid sphere problem.svg|frameless|235x235px]]
|<math>V = \frac{Q}{4\pi\varepsilon_0x}, </math> 
outside the sphere, where <math>Q</math> is the total charge uniformly distributed in the volume.
|<math>V = \frac{Q(3R^2-r^2)}{8\pi\varepsilon_0R^3},</math>
inside the sphere, where <math>Q</math> is the total charge uniformly distributed in the volume.
|-
|Spherical surface
|[[File:Charged spherical surface problem.svg|frameless|235x235px]]
|<math>V = \frac{Q}{4\pi\varepsilon_0x},</math> 
outside the sphere, where <math>Q</math> is the total charge uniformly distributed on the surface.
|<math>V = \frac{Q}{4\pi\varepsilon_0R},</math> 
inside the sphere for uniform charge distribution.
|-
|Charged Ring
|[[File:Charged ring problem.svg|frameless|235x235px]]
| colspan="2" | <math>V = \frac{Q}{4\pi\varepsilon_0\sqrt{R^2+x^2}},</math> 
on the axis, where <math>Q</math> is the total charge uniformly distributed on the ring.
|-
|Charged Disc
|[[File:Charged disc problem.svg|frameless|235x235px]]
| colspan="2" | <math>V = \frac{\sigma}{2\varepsilon_0} \left[\sqrt{x^2+R^2} - x\right],</math>
on the axis, where <math>\sigma</math> is the uniform surface charge density.
|-
|Electric Dipole
|[[File:Electric dipole - axial and equatorial problem.svg|frameless|235x235px]]
| <math>V = 0,</math>

on the equatorial plane.
| <math>V = \frac{p}{4\pi\varepsilon_0x^2},</math>

on the axis (given that <math>x \gg d</math>), where <math>x</math> can also be negative to indicate position at the opposite direction on the axis, and <math>p</math> is the magnitude of [[electric dipole moment]].
|}

==See also==
* [[Absolute electrode potential]]
* [[Electrochemical potential]]
* [[Electrode potential]]

==References==
{{Reflist}}

== Further reading ==
{{Commons category|Electric potential}}
{{Refbegin}}
* {{cite book|last1=Politzer|first1=Peter|last2=Truhlar|first2=Donald G.| name-list-style = vanc | title=Chemical Applications of Atomic and Molecular Electrostatic Potentials: Reactivity, Structure, Scattering, and Energetics of Organic, Inorganic, and Biological Systems |date=1981|publisher=Springer US|location=Boston, MA|isbn=978-1-4757-9634-6}}
* {{cite book|last1=Sen|first1=Kalidas|last2=Murray|first2=Jane S. | name-list-style = vanc | title=Molecular Electrostatic Potentials: Concepts and Applications|date=1996|publisher=Elsevier|location=Amsterdam|isbn=978-0-444-82353-3}}
* {{cite book | last = Griffiths | first = David J. | name-list-style = vanc | title = Introduction to Electrodynamics | edition = 3rd. | publisher = Prentice Hall | year = 1999 | isbn = 0-13-805326-X | url-access = registration | url = https://archive.org/details/introductiontoel00grif_0 }} 
* {{cite book|last=Jackson|first=John David | name-list-style = vanc |title=Classical Electrodynamics|publisher = John Wiley & Sons, Inc. |year=1999|location=USA|edition=3rd.|isbn=978-0-471-30932-1}}
* {{cite book | last = Wangsness | first = Roald K. | name-list-style = vanc |title=Electromagnetic Fields| publisher=Wiley |year=1986|edition=2nd., Revised, illustrated|isbn=978-0-471-81186-2}}
{{Refend}}
{{Authority control}}

[[Category:Potentials]]
[[Category:Electrostatics]]
[[Category:Voltage]]
[[Category:Electromagnetic quantities]]