Editing Electric potential (section)

== Electrostatics ==
{{Main|Electrostatics}}{{Multiple image
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| 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.
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| footer = Electric potential in the vicinity of two opposite point charges.
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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>
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|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
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|equation=<math>\mathbf{E} = - \mathbf{\nabla} V_\mathbf{E}\,</math>
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|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.