Editing Electric potential (section)

=== 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.