Editing Electrostatics

{{Short description|Study of still or slow electric charges}}
[[File:Cat demonstrating static cling with styrofoam peanuts.jpg|thumb|upright=1.3|alt=A tabby cat covered in packing peanuts.|[[Foam peanut]]s clinging to a cat's fur due to [[static electricity]]. The cat's fur becomes charged due to the [[triboelectricity|triboelectric effect]]. The electric field of the charged fur causes polarization of the molecules of the foam due to [[electrostatic induction]], resulting in a slight attraction of the light plastic pieces to the fur.<ref name="Ling">
{{cite book
 | last1  = Ling
 | first1 = Samuel J.
 | last2  = Moebs
 | first2 = William
 | last3  = Sanny
 | first3 = Jeff
 | title  = University Physics, Vol. 2
 | publisher = OpenStax
 | date   = 2019
 | url    = https://opentextbc.ca/universityphysicsv2openstax/chapter/conductors-insulators-and-charging-by-induction/
 | doi    = 
 | id     = 
 | isbn   = 9781947172210
}} Ch.30: Conductors, Insulators, and Charging by Induction</ref><ref name="Bloomfield">
{{cite book
 | last1  = Bloomfield
 | first1 = Louis A. 
 | title  = How Things Work: The Physics of Everyday Life
 | publisher = John Wiley and Sons
 | date   = 2015 
 | pages  = 270
 | url    = https://books.google.com/books?id=TLE7CwAAQBAJ&dq=polarization&pg=PA270
 | doi    = 
 | id     = 
 | isbn   = 9781119013846
}}</ref><ref name="PhysicsClassroom">
{{cite web
 | title = Polarization
 | work = Static Electricity – Lesson 1 – Basic Terminology and Concepts
 | publisher = The Physics Classroom
 | date = 2020
 | url = https://www.physicsclassroom.com/class/estatics/u8l1e.cfm
 | format = 
 | doi = 
 | accessdate = 18 June 2021}}</ref><ref name="Thompson">{{cite web
 | last = Thompson
 | first = Xochitl Zamora 
 | title = Charge It! All About Electrical Attraction and Repulsion
 | work = Teach Engineering: Stem curriculum for K-12
 | publisher = University of Colorado
 | date = 2004
 | url = https://www.teachengineering.org/activities/view/cub_electricity_lesson02_activity1
 | format = 
 | doi = 
 | accessdate = 18 June 2021
}}</ref> This effect is also the cause of [[static cling]] in clothes.]]
{{Electromagnetism|Electrostatics}}

'''Electrostatics''' is a branch of [[physics]] that studies slow-moving or stationary [[electric charge]]s.

Since [[classical antiquity|classical times]], it has been known that some materials, such as [[amber]], attract lightweight particles after [[triboelectric effect|rubbing]]. The [[Greek language|Greek]] word {{Transliteration|el|ḗlektron}} ({{lang|el|ἤλεκτρον}}), meaning 'amber', was thus the [[Root (linguistics)|root]] of the word ''[[electricity]]''. Electrostatic phenomena arise from the [[force]]s that electric charges exert on each other. Such [[forces]] are described by [[Coulomb's law]].

There are many examples of electrostatic phenomena, from those as simple as the attraction of plastic wrap to one's hand after it is removed from a package, to the apparently spontaneous explosion of grain silos, the damage of electronic components during manufacturing, and [[photocopier]] and [[laser printing|laser printer]] operation.

The electrostatic model accurately predicts electrical phenomena in "classical" cases where the velocities are low and the system is macroscopic so no quantum effects are involved. It also plays a role in quantum mechanics, where additional terms also need to be included.

== Coulomb's law  ==
{{Main|Coulomb's law}}
[[Coulomb's law]] states that:<ref>{{Cite book |last=J |first=Griffiths |url=http://dx.doi.org/10.1017/9781108333511.008 |title=Introduction to Electrodynamics |date=2017 |publisher=Cambridge University Press |isbn=978-1-108-33351-1 |pages=296–354 |doi=10.1017/9781108333511.008 |access-date=2023-08-11}}</ref>
{{blockquote|text=The magnitude of the electrostatic force of attraction or repulsion between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them.}}

The force is along the straight line joining them. If the two charges have the same sign, the electrostatic force between them is repulsive; if they have different signs, the force between them is attractive.

If <math>r</math> is the distance (in [[meters]]) between two charges, then the force between two point charges <math>Q</math> and <math>q</math> is:
: <math>F = {1\over 4\pi\varepsilon_0}{|Qq|\over r^2},</math>
where ''ε''<sub>0</sub> = {{physconst|eps0}} is the [[vacuum permittivity]].<ref name="Sadiku">{{cite book |title=Elements of electromagnetics|year=2009|isbn=9780195387759|page=104|author=Matthew Sadiku|publisher=Oxford University Press }}</ref>

The [[SI unit]] of ''ε''<sub>0</sub> is equivalently [[ampere|A]]<sup>2</sup>⋅[[second|s]]<sup>4</sup> ⋅kg<sup>−1</sup>⋅m<sup>−3</sup> or [[coulomb|C]]<sup>2</sup>⋅[[newton (unit)|N]]<sup>−1</sup>⋅m<sup>−2</sup> or [[farad|F]]⋅m<sup>−1</sup>.

== Electric field ==
{{Main|Electric field}}
[[File:Electrostatic induction.svg|thumb|upright=2.0|The [[electrostatic field]] ''(lines with arrows)'' of a nearby positive charge ''<span style="color:red;">(+)</span>'' causes the mobile charges in conductive objects to separate due to [[electrostatic induction]]. Negative charges ''<span style="color:blue;">(blue)</span>'' are attracted and move to the surface of the object facing the external charge. Positive charges ''<span style="color:red;">(red)</span>'' are repelled and move to the surface facing away. These induced surface charges are exactly the right size and shape so their opposing electric field cancels the electric field of the external charge throughout the interior of the metal. Therefore, the electrostatic field everywhere inside a conductive object is zero, and the [[electrostatic potential]] is constant.]]

The electric field, <math>\mathbf E</math>, in units of [[newton (unit)|Newtons]] per [[Coulomb]] or [[volt]]s per meter, is a [[vector field]] that can be defined everywhere, except at the location of point charges (where it diverges to infinity).<ref name="Purcell">
{{cite book
 | last1=Purcell
 | first1=Edward M.
 | title=Electricity and Magnetism
 | publisher=Cambridge University Press
 | date=2013
 | pages=16–18
 | url=https://books.google.com/books?id=A2rS5vlSFq0C&q=%22electric+field%22&pg=PA16
 | isbn=978-1107014022
}}</ref> It is defined as the electrostatic force <math>\mathbf F</math> on a hypothetical small [[test charge]] at the point due to Coulomb's law, divided by the charge <math>q</math>
: <math>\mathbf E = {\mathbf F \over q}</math>

[[Field line|Electric field lines]] are useful for visualizing the electric field. Field lines begin on positive charge and terminate on negative charge. They are parallel to the direction of the electric field at each point, and the density of these field lines is a measure of the magnitude of the electric field at any given point.

A collection of <math>n</math> particles of charge <math>q_i</math>, located at points <math>\mathbf r_i</math> (called ''source points'') generates the electric field at <math>\mathbf r</math> (called the ''field point'') of:<ref name="Purcell"/>
: <math> \mathbf E(\mathbf r) = {1\over4\pi\varepsilon_0} \sum_{i=1}^n q_i {\hat\mathbf {r-r_i}\over {|\mathbf {r-r_i}|}^2} = {1\over 4\pi\varepsilon_0} \sum_{i=1}^n q_i {\mathbf {r-r_i}\over {|\mathbf {r-r_i}|}^3},</math>
where <math display="inline">\mathbf r-\mathbf r_i</math> is the [[displacement vector]] from a ''source point'' <math>\mathbf r_i</math> to the ''field point'' <math>\mathbf r</math>, and <math display="inline"> \hat\mathbf {r-r_i} \ \stackrel{\mathrm{def}}{=}\ \frac{\mathbf {r-r_i}}{|\mathbf {r-r_i}|}</math> is the [[unit vector]] of the displacement vector that indicates the direction of the field due to the source at point <math>\mathbf{r_i}</math>. For a single point charge, <math>q</math>, at the origin, the magnitude of this electric field is 
<math>E = q/4\pi\varepsilon_0 r^2</math> and points away from that charge if it is positive. The fact that the force (and hence the field) can be calculated by summing over all the contributions due to individual source particles is an example of the [[superposition principle]]. The electric field produced by a distribution of charges is given by the [[volume charge density]] <math>\rho(\mathbf r)</math> and can be obtained by converting this sum into a [[triple integral]]:
: <math>\mathbf E(\mathbf r) = \frac{1}{4\pi\varepsilon_0} \iiint \, \rho(\mathbf r') {\mathbf {r-r'}\over {|\mathbf {r-r'}|}^3} \mathrm{d}^3|\mathbf r'|</math>

=== Gauss's law ===
{{Main|Gauss's law|Gaussian surface}}

[[Gauss's law]]<ref>{{cite journal |title=Sur l'attraction des sphéroides elliptiques, par M. de La Grange. |url=http://dx.doi.org/10.1163/9789004460409_mor2-b29447057 |access-date=2023-08-11 |website=Mathematics General Collection|doi=10.1163/9789004460409_mor2-b29447057 |url-access=subscription }}</ref><ref>{{Citation |last=Gauss |first=Carl Friedrich |title=Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum, methodo nova tractata |date=1877 |url=http://dx.doi.org/10.1007/978-3-642-49319-5_8 |work=Werke |pages=279–286 |access-date=2023-08-11 |place=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |doi=10.1007/978-3-642-49319-5_8 |isbn=978-3-642-49320-1|url-access=subscription }}</ref> states that "the total [[electric flux]] through any closed surface in free space of any shape drawn in an electric field is proportional to the total [[electric charge]] enclosed by the surface." Many numerical problems can be solved by considering a [[Gaussian surface]] around a body. Mathematically, Gauss's law takes the form of an integral equation:
: <math>\Phi_E = \oint_S\mathbf E\cdot \mathrm{d}\mathbf A = {Q_\text{enclosed}\over\varepsilon_0} = \int_V{\rho\over\varepsilon_0}\mathrm{d}^3 r,</math>
where <math>\mathrm{d}^3 r =\mathrm{d}x \ \mathrm{d}y \ \mathrm{d}z</math> is a volume element. If the charge is distributed over a surface or along a line, replace <math>\rho\,\mathrm{d}^3r</math> by <math>\sigma \, \mathrm{d}A</math> or <math>\lambda \, \mathrm{d}\ell</math>. The [[divergence theorem]] allows Gauss's Law to be written in differential form:
: <math>\nabla\cdot\mathbf E = {\rho\over\varepsilon_0}.</math>
where <math>\nabla \cdot</math> is the [[divergence|divergence operator]].

=== Poisson and Laplace equations ===
{{Main|Poisson's equation|Laplace's equation}}
The definition of electrostatic potential, combined with the differential form of Gauss's law (above), provides a relationship between the potential Φ and the charge density ''ρ'':
: <math>{\nabla}^2 \phi = - {\rho\over\varepsilon_0}.</math>

This relationship is a form of [[Poisson's equation]].<ref>{{Cite book |last1=Poisson |first1=M |url=https://www.biodiversitylibrary.org/item/55214 |title=Mémoires de l'Académie (royale) des sciences de l'Institut (imperial) de France |last2=sciences (France) |first2=Académie royale des |date=1827 |volume=6 |location=Paris}}</ref> In the absence of unpaired electric charge, the equation becomes [[Laplace's equation]]:
: <math>{\nabla}^2 \phi = 0,</math>

== Electrostatic approximation ==
[[File:Electrostatics_relation_triangle.svg|thumb|Summary of electrostatic relations between electric potential, electric field and charge density. Here, <math>\mathbf r = \mathbf x - \mathbf{x'}</math>.]]
If the electric field in a system can be assumed to result from static charges, that is, a system that exhibits no significant time-varying magnetic fields, the system is justifiably analyzed using only the principles of electrostatics. This is called the "electrostatic approximation".<ref>{{cite journal |last1=Montgomery|first1=David|title=Validity of the electrostatic approximation|journal=[[Physics of Fluids]]|date=1970|volume=13|pages=1401–1403|doi=10.1063/1.1693079|hdl=2060/19700015014|hdl-access=free}}</ref>

The validity of the electrostatic approximation rests on the assumption that the electric field is [[irrotational]], or nearly so:
: <math>\nabla\times\mathbf E \approx 0.</math>

From [[Faraday's law of induction|Faraday's law]], this assumption implies the absence or near-absence of time-varying magnetic fields:
: <math>{\partial\mathbf B\over\partial t} \approx 0.</math>

In other words, electrostatics does not require the absence of magnetic fields or electric currents. Rather, if magnetic fields or electric currents ''do'' exist, they must not change with time, or in the worst-case, they must change with time only ''very slowly''. In some problems, both electrostatics and [[magnetostatics]] may be required for accurate predictions, but the coupling between the two can still be ignored. Electrostatics and magnetostatics can both be seen as non-relativistic [[Galilean electromagnetism|Galilean limits]] for electromagnetism.<ref>{{cite journal |last1=Heras|first1=J. A.|title=The Galilean limits of Maxwell's equations|journal=[[American Journal of Physics]]|date=2010|volume=78|issue=10|pages=1048–1055|doi=10.1119/1.3442798|arxiv=1012.1068 |bibcode=2010AmJPh..78.1048H|s2cid=118443242}}</ref> In addition, conventional electrostatics ignore quantum effects which have to be added for a complete description.<ref name="Purcell"/>{{rp|2}}

=== Electrostatic potential ===
{{Main | Electrostatic potential}}
As the electric field is [[irrotational]], it is possible to express the electric field as the [[gradient]] of a scalar function, <math>\phi</math>, called the [[electrostatic potential]] (also known as the [[voltage]]). An electric field, <math>E</math>, points from regions of high electric potential to regions of low electric potential, expressed mathematically as
: <math>\mathbf E = -\nabla\phi.</math>

The [[gradient theorem]] can be used to establish that the electrostatic potential     is the amount of [[work (physics)|work]] per unit charge required to move a charge from point <math>a</math> to point <math>b</math> with the following [[line integral]]:
: <math> -\int_a^b {\mathbf E\cdot \mathrm{d}\mathbf \ell} = \phi (\mathbf b) -\phi(\mathbf a).</math>

From these equations, we see that the electric potential is constant in any region for which the electric field vanishes (such as occurs inside a conducting object).

=== Electrostatic energy ===
{{Main|Electric potential energy|Energy density}}

A [[test particle]]'s potential energy, <math>U_\mathrm{E}^{\text{single}}</math>, can be calculated from a [[line integral]] of the work, <math>q_n\mathbf E\cdot\mathrm d\mathbf\ell</math>. We integrate from a point at infinity, and assume a collection of <math>N</math> particles of charge <math>Q_n</math>, are already situated at the points <math>\mathbf r_i</math>. This potential energy (in [[Joule]]s) is:
: <math> U_\mathrm{E}^{\text{single}}=q\phi(\mathbf r)=\frac{q }{4\pi \varepsilon_0}\sum_{i=1}^N \frac{Q_i}{\left \|\mathcal{\mathbf R_i} \right \|}</math>
where <math>\mathbf\mathcal {R_i} = \mathbf r - \mathbf r_i</math> is the distance of each charge <math>Q_i</math> from the [[test charge]] <math>q</math>, which situated at the point <math>\mathbf r</math>, and <math>\phi(\mathbf r)</math> is the electric potential that would be at <math>\mathbf r</math> if the [[test charge]] were not present. If only two charges are present, the potential energy is <math>Q_1 Q_2/(4\pi\varepsilon_0 r)</math>. The total [[electric potential energy]] due a collection of ''N'' charges is calculating by assembling these particles [[Electric Potential Energy|one at a time]]:
: <math>U_\mathrm{E}^{\text{total}} = \frac{1 }{4\pi\varepsilon _0}\sum_{j=1}^N Q_j \sum_{i=1}^{j-1} \frac{Q_i}{r_{ij}}= \frac{1}{2}\sum_{i=1}^N Q_i\phi_i ,</math>
where the following sum from, ''j'' = 1 to ''N'', excludes ''i'' = ''j'':
: <math>\phi_i = \frac{1}{4\pi \varepsilon _0} \sum_{\stackrel{j=1}{j \ne i}}^N \frac{Q_j}{r_{ij}}.</math>

This electric potential, <math>\phi_i</math> is what would be measured at <math>\mathbf r_i</math> if the charge <math>Q_i</math> were missing. This formula obviously excludes the (infinite) energy that would be required to assemble each point charge from a disperse cloud of charge. The sum over charges can be converted into an integral over charge density using the prescription <math display="inline">\sum (\cdots) \rightarrow \int(\cdots)\rho \, \mathrm d^3r</math>:
: <math>U_\mathrm{E}^{\text{total}} = \frac{1}{2} \int\rho(\mathbf r)\phi(\mathbf r) \, \mathrm{d}^3 r = \frac{\varepsilon_0 }{2} \int \left|{\mathbf{E}}\right|^2 \, \mathrm{d}^3 r,</math>

This second expression for [[electrostatic energy]] uses the fact that the electric field is the negative [[gradient]] of the electric potential, as well as [[vector calculus identities]] in a way that resembles [[integration by parts]]. These two integrals for electric field energy seem to indicate two mutually exclusive formulas for electrostatic energy density, namely <math display="inline">\frac{1}{2}\rho\phi</math> and <math display="inline">\frac{1}{2}\varepsilon_0 E^2</math>; they yield equal values for the total electrostatic energy only if both are integrated over all space.

=== Electrostatic pressure ===
On a [[electrical conductor|conductor]], a surface charge will experience a force in the presence of an [[electric field]]. This force is the average of the discontinuous electric field at the surface charge. This average in terms of the field just outside the surface amounts to:
: <math> P = \frac{ \varepsilon_0 }{2} E^2,</math>

This pressure tends to draw the conductor into the field, regardless of the sign of the surface charge.

== See also ==
* {{annotated link|Electromagnetism}}
* [[Electrostatic generator]], machines that create static electricity.
* [[Electrostatic induction]], separation of charges due to electric fields.
* [[Permittivity]] and [[relative permittivity]], the electric polarizability of materials.
* [[Elementary charge|Quantization of charge]], the charge units carried by electrons or protons.
* [[Static electricity]], stationary charge accumulated on a material.
* [[Triboelectric effect]], separation of charges due to sliding or contact.

== References ==
{{reflist}}

== Further reading ==
* {{cite book |author1=Hermann A. Haus |author2=James R. Melcher |title=Electromagnetic Fields and Energy |location=Englewood Cliffs, NJ |publisher=Prentice-Hall |year=1989 |isbn=0-13-249020-X}}
* {{cite book |author1=Halliday, David |author2=Robert Resnick |author3=Kenneth S. Krane |title=Physics |url=https://archive.org/details/isbn_9780471559184 |url-access=registration |location=New York |publisher=John Wiley & Sons |year=1992 |isbn=0-471-80457-6}}
* {{cite book |author=Griffiths, David J. |title=Introduction to Electrodynamics |location=Upper Saddle River, NJ |publisher=Prentice Hall |year=1999 |isbn=0-13-805326-X |url-access=registration |url=https://archive.org/details/introductiontoel00grif_0 |author-link=David J. Griffiths}}

== External links ==
{{EB1911 poster|Electrostatics}}
* {{Commons category-inline}}
{{Wiktionary}}
* [https://feynmanlectures.caltech.edu/II_04.html The Feynman Lectures on Physics Vol. II Ch. 4: Electrostatics]
* [http://physics.gmu.edu/~joe/PHYS685/Topic1.pdf Introduction to Electrostatics]: Point charges can be treated as a distribution using the [[Dirac delta function]]
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