View source for Physics:Electrostatics

{{Short description|Study of stationary or slow-moving electric charges}}
{{Electromagnetism|Electrostatics}}
[[File:Cat demonstrating static cling with styrofoam peanuts upscayled 4x.jpg|thumb|upright=1.4|alt=A tabby cat covered in packing peanuts.|Foam peanuts clinging to a cat's fur due to static electricity. The electric field of the charged fur causes polarization of the molecules of the foam due to [[Physics:Electrostatic induction|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 [[Physics:Static cling|static cling]] in clothes.]]

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

Since [[History:Classical antiquity|classical times]], it has been known that some materials, such as [[Earth:Amber|amber]], attract lightweight particles after [[Physics:Triboelectric effect|rubbing]]. The [[Social:Greek language|Greek]] word for amber, {{lang|el|ἤλεκτρον}} ({{transl|el|ḗlektron}}), was thus the source of the word '[[Physics:Electricity|electricity]]'. Electrostatic phenomena arise from the [[Physics:Force|force]]s that electric charges exert on each other. Such forces are described by [[Physics:Coulomb's law|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 [[Engineering:Photocopier|photocopier]] and [[Engineering: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 article|Physics:Coulomb's law}}
[[Physics: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 |work= |publisher=Cambridge University Press |isbn=978-1-108-33351-1 |pages=296–354 |access-date=2023-08-11}}</ref>

'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 (in [[Newton (unit)|newton]]s) between two point charges <math>q</math> and <math>Q</math> (in [[Coulomb|coulomb]]s) is:
:<math>F = \frac{1}{4\pi \varepsilon_0}\frac{qQ}{r^2}= k_\text{e}\frac{qQ}{r^2}\, ,</math>

where ''ε''<sub>0</sub> is the [[Physics:Vacuum permittivity|vacuum permittivity]], or permittivity of free space:<ref name="Sadiku">{{cite book |title=Elements of electromagnetics|year=2009|isbn=9780195387759|page=104|author=Matthew Sadiku}}</ref>

:<math>\varepsilon_0 \approx  \mathrm{8.854\ 187\ 817 \times 10^{-12} ~C^2{\cdot}N^{-1}{\cdot}m^{-2}}.</math>

The [[International System of Units|SI]] units<ref>{{Cite journal |date=2010-04-12 |title=SI Units |url=https://www.nist.gov/pml/owm/metric-si/si-units |journal=NIST |language=en}}</ref> of ''ε''<sub>0</sub> are 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>. The [[Physics:Coulomb constant|Coulomb constant]] is:

:<math> k_\text{e} = \frac{1}{4\pi\varepsilon_0}\approx \mathrm{8.987\ 551\ 792 \times 10^9 ~N{\cdot}m^2{\cdot}C^{-2}}.</math>

These [[Physics:Physical constant|physical constant]]s (''ε''<sub>0</sub>, ''k''<sub>e</sub>, ''e'') are currently defined so that ''e'' is exactly defined, and ''ε''<sub>0</sub> and ''k''<sub>e</sub> are measured quantities.

==Electric field==
{{Main|Physics: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>\vec{E}</math>, in units of [[Newton (unit)|Newtons]] per [[Coulomb]] or [[Volt|volt]]s per meter, is a [[Vector field|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>\vec{F}\,</math> in newtons on a hypothetical small [[Physics:Test charge|test charge]] at the point due to Coulomb's Law, divided by the magnitude of the charge <math>q\,</math> in coulombs
:<math>\vec{E} = {\vec{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.

Consider a collection of <math>N</math> particles of charge <math>Q_i</math>, located at points <math>\vec r_i</math> (called ''source points''), the electric field at <math>\vec r</math> (called the ''field point'') is:<ref name="Purcell"/>

:<math> \vec{E}(\vec r)
=\frac{1}{4\pi \varepsilon _0}\sum_{i=1}^N \frac{\widehat\mathcal R_i Q_i}{\left \|\mathcal\vec R_i \right \|^2} ,</math>

where <math>\vec\mathcal R_i = \vec r - \vec r_i ,</math> is the displacement vector from a ''source point'' <math>\vec r_i</math> to the ''field point'' <math>\vec r</math>, and <math> \widehat\mathcal R_i = \vec\mathcal R_i / \left \|\vec\mathcal R_i \right \|</math>
is a [[Unit vector|unit vector]] that indicates the direction of the field. For a single point charge at the origin, the magnitude of this electric field is <math>E =k_\text{e}Q/\mathcal 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|superposition principle]]. The electric field produced by a distribution of charges is given by the volume [[Physics:Charge density|charge density]] <math>\rho (\vec r)</math> and can be obtained by converting this sum into a triple integral:

:<math>\vec{E}(\vec r)= \frac {1}{4 \pi \varepsilon_0} \iiint \frac {\vec r - \vec r \,'}{\left \| \vec r - \vec r \,' \right \|^3} \rho (\vec r \,') \, \mathrm{d}^3 r\,'</math>

===Gauss' law===

[[Physics:Gauss's law|Gauss's law]]<ref>{{Cite web |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}}</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 |isbn=978-3-642-49320-1}}</ref> states that "the total [[Physics:Electric flux|electric flux]] through any closed surface in free space of any shape drawn in an electric field is proportional to the total [[Physics:Electric charge|electric charge]] enclosed by the surface." Many numerical problems can be solved by considering a


a [[Gaussian surface]] around a body. 
Mathematically, Gauss's law takes the form of an integral equation:

:<math>\oint_S\vec{E} \cdot\mathrm{d}\vec{A} = \frac{1}{\varepsilon_0}\,Q_\text{enclosed} = \int_V{\rho\over\varepsilon_0}\cdot \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|divergence theorem]] allows Gauss's Law to be written in differential form:

:<math>\vec{\nabla}\cdot\vec{E} = {\rho\over\varepsilon_0}.</math>

where <math>\vec{\nabla} \cdot</math> is the [[Divergence|divergence operator]].

===Poisson and Laplace equations===
{{Main|Physics: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 [[Physics:Poisson's equation|Poisson's equation]].<ref>{{Cite book |last=Poisson |first=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==
The validity of the electrostatic approximation rests on the assumption that the electric field is irrotational:
:<math>\vec{\nabla}\times\vec{E} = 0.</math>

From [[Physics:Faraday's law of induction|Faraday's law]], this assumption implies the absence or near-absence of time-varying magnetic fields:
:<math>{\partial\vec{B}\over\partial t} = 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 [[Physics:Magnetostatics|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 [[Physics:Galilean electromagnetism|Galilean limits]] for electromagnetism.<ref>{{cite journal |last1=Heras|first1=J. A.|title=The Galilean limits of Maxwell's equations|journal=[[Physics:American Journal of Physics|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===
As the electric field is irrotational, it is possible to express the electric field as the [[Gradient|gradient]] of a scalar function, <math>\phi</math>, called the electrostatic potential (also known as the [[Physics:Voltage|voltage]]). An electric field, <math>E</math>, points from regions of high electric potential to regions of low electric potential, expressed mathematically as
:<math>\vec{E} = -\vec{\nabla}\phi.</math>

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

A [[Physics:Test particle|test particle]]'s potential energy, <math>U_\mathrm{E}^{\text{single}}</math>, can be calculated from a [[Line integral|line integral]] of the work, <math>q_n\vec E\cdot\mathrm d\vec\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>\vec r_i</math>. This potential energy (in [[Joule]]s) is:

:<math> U_\mathrm{E}^{\text{single}}=q\phi(\vec r)=\frac{q }{4\pi \varepsilon_0}\sum_{i=1}^N \frac{Q_i}{\left \|\mathcal{\vec R_i} \right \|}</math>

where <math>\vec\mathcal {R_i} = \vec r - \vec r_i</math> is the distance of each charge <math>Q_i</math> from the [[Physics:Test charge|test charge]] <math>q</math>, which situated at the point <math>\vec r</math>, and <math>\phi(\vec r)</math> is the electric potential that would be at <math>\vec r</math> if the [[Physics:Test charge|test charge]] were not present. If only two charges are present, the potential energy is <math>k_\text{e}Q_1Q_2/r</math>. The total [[Physics:Electric potential energy|electric potential energy]] due a collection of ''N'' charges is calculating by assembling these particles 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>\vec 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(\vec{r})\phi(\vec{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|gradient]] of the electric potential, as well as [[Vector calculus identities|vector calculus identities]] in a way that resembles [[Integration by parts|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 [[Physics:Electrical conductor|conductor]], a surface charge will experience a force in the presence of an [[Physics:Electric field|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|Physics:Electromagnetism}}
*[[Physics:Electrostatic generator|Electrostatic generator]], machines that create static electricity.
*[[Physics:Electrostatic induction|Electrostatic induction]], separation of charges due to electric fields.
*[[Physics:Permittivity|Permittivity]] and [[Chemistry:Relative permittivity|relative permittivity]], the electric polarizability of materials.
*[[Physics:Elementary charge|Quantisation of charge]], the charge units carried by electrons or protons.
*[[Physics:Static electricity|Static electricity]], stationary charge accumulated on a material.
*[[Physics:Triboelectric effect|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 |}}

==External links==
{{EB1911 Poster|Electrostatics}}
*[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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