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Electrostatics

From Wikipedia, the free encyclopedia
Study of still or slow electric charges
A tabby cat covered in packing peanuts.
Foam peanuts clinging to a cat's fur due tostatic electricity. The cat's fur becomes charged due to thetriboelectric effect. The electric field of the charged fur causes polarization of the molecules of the foam due toelectrostatic induction, resulting in a slight attraction of the light plastic pieces to the fur.[1][2][3][4] This effect is also the cause ofstatic cling in clothes.
Electromagnetism
Solenoid

Electrostatics is a branch ofphysics that studies slow-moving or stationaryelectric charges on macroscopic objects wherequantum effects can be neglected. Under these circumstances the electric field, electric potential, and the charge density are related without complications from magnetic effects.

Sinceclassical antiquity, it has been known that some materials, such asamber, attract lightweight particles afterrubbing.[5] TheGreek wordḗlektron (ἤλεκτρον), meaning 'amber', was thus theroot of the wordelectricity. Electrostatic phenomena arise from theforces that electric charges exert on each other. Suchforces are described byCoulomb'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, andphotocopier andlaser printer operation.

Coulomb's law

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Main article:Coulomb's law

Coulomb's law states that:[6]

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.

Ifr{\displaystyle r} is the distance (inmeters) between two charges, then the force between two point chargesQ{\displaystyle Q} andq{\displaystyle q} is:

F=14πε0|Qq|r2,{\displaystyle F={1 \over 4\pi \varepsilon _{0}}{|Qq| \over r^{2}},}

whereε0 =8.8541878188(14)×10−12 F⋅m−1‍[7] is thevacuum permittivity.[8]

TheSI unit ofε0 is equivalentlyA2⋅s4 ⋅kg−1⋅m−3 orC2⋅N−1⋅m−2 orF⋅m−1.

Electric field

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Main article:Electric field
Theelectrostatic field(lines with arrows) of a nearby positive charge(+) causes the mobile charges in conductive objects to separate due toelectrostatic induction. Negative charges(blue) are attracted and move to the surface of the object facing the external charge. Positive charges(red) 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 theelectrostatic potential is constant.

The electric field,E{\displaystyle \mathbf {E} }, in units ofnewtons percoulomb orvolts per meter, is avector field that can be defined everywhere, except at the location of point charges (where it diverges to infinity).[9] It is defined as the electrostatic forceF{\displaystyle \mathbf {F} } on a hypothetical smalltest charge at the point due to Coulomb's law, divided by the chargeq{\displaystyle q}

E=Fq{\displaystyle \mathbf {E} ={\mathbf {F} \over q}}

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 ofn{\displaystyle n} particles of chargeqi{\displaystyle q_{i}}, located at pointsri{\displaystyle \mathbf {r} _{i}} (calledsource points) generates the electric field atr{\displaystyle \mathbf {r} } (called thefield point) of:[9]

E(r)=14πε0∑i=1nqir−ri^|r−ri|2=14πε0∑i=1nqir−ri|r−ri|3,{\displaystyle \mathbf {E} (\mathbf {r} )={1 \over 4\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}},}

wherer−ri{\textstyle \mathbf {r} -\mathbf {r} _{i}} is thedisplacement vector from asource pointri{\displaystyle \mathbf {r} _{i}} to thefield pointr{\displaystyle \mathbf {r} }, andr−ri^ =def r−ri|r−ri|{\textstyle {\hat {\mathbf {r-r_{i}} }}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\mathbf {r-r_{i}} }{|\mathbf {r-r_{i}} |}}} is theunit vector of the displacement vector that indicates the direction of the field due to the source at pointri{\displaystyle \mathbf {r_{i}} }. For a single point charge,q{\displaystyle q}, at the origin, the magnitude of this electric field isE=q/4πε0r2{\displaystyle E=q/4\pi \varepsilon _{0}r^{2}} 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 thesuperposition principle. The electric field produced by a distribution of charges is given by thevolume charge densityρ(r){\displaystyle \rho (\mathbf {r} )} and can be obtained by converting this sum into atriple integral:

E(r)=14πε0∭ρ(r′)r−r′|r−r′|3d3|r′|{\displaystyle \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} '|}

Gauss's law

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Main articles:Gauss's law andGaussian surface

Gauss's law[10][11] states that "the totalelectric flux through any closed surface in free space of any shape drawn in an electric field is proportional to the totalelectric charge enclosed by the surface." Many numerical problems can be solved by considering aGaussian surface around a body. Mathematically, Gauss's law takes the form of an integral equation:

ΦE=∮SE⋅dA=Qenclosedε0=∫Vρε0d3r,{\displaystyle \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,}

whered3r=dx dy dz{\displaystyle \mathrm {d} ^{3}r=\mathrm {d} x\ \mathrm {d} y\ \mathrm {d} z} is a volume element. If the charge is distributed over a surface or along a line, replaceρd3r{\displaystyle \rho \,\mathrm {d} ^{3}r} byσdA{\displaystyle \sigma \,\mathrm {d} A} orλdℓ{\displaystyle \lambda \,\mathrm {d} \ell }. Thedivergence theorem allows Gauss's law to be written in differential form:

∇⋅E=ρε0.{\displaystyle \nabla \cdot \mathbf {E} ={\rho \over \varepsilon _{0}}.}

where∇⋅{\displaystyle \nabla \cdot } is thedivergence operator.

Poisson and Laplace equations

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Main articles:Poisson's equation andLaplace'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 densityρ:

∇2ϕ=−ρε0.{\displaystyle {\nabla }^{2}\phi =-{\rho \over \varepsilon _{0}}.}

This relationship is a form ofPoisson's equation.[12] In the absence of unpaired electric charge, the equation becomesLaplace's equation:

∇2ϕ=0,{\displaystyle {\nabla }^{2}\phi =0,}

Electrostatic approximation

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Summary of electrostatic relations between electric potential, electric field and charge density. Here,r=x−x′{\displaystyle \mathbf {r} =\mathbf {x} -\mathbf {x'} }.

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".[13]

The validity of the electrostatic approximation rests on the assumption that the electric field isirrotational, or nearly so:

∇×E≈0.{\displaystyle \nabla \times \mathbf {E} \approx 0.}

FromFaraday's law, this assumption implies the absence or near-absence of time-varying magnetic fields:

∂B∂t≈0.{\displaystyle {\partial \mathbf {B} \over \partial t}\approx 0.}

In other words, electrostatics does not require the absence of magnetic fields or electric currents. Rather, if magnetic fields or electric currentsdo exist, they must not change with time, or in the worst-case, they must change with time onlyvery slowly. In some problems, both electrostatics andmagnetostatics 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-relativisticGalilean limits for electromagnetism.[14] In addition, conventional electrostatics ignore quantum effects which have to be added for a complete description.[9]: 2 

Electrostatic potential

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Main article:Electrostatic potential

As the electric field isirrotational, it is possible to express the electric field as thegradient of a scalar function,ϕ{\displaystyle \phi }, called theelectrostatic potential (also known as thevoltage). An electric field,E{\displaystyle E}, points from regions of high electric potential to regions of low electric potential, expressed mathematically as

E=−∇ϕ.{\displaystyle \mathbf {E} =-\nabla \phi .}

Thegradient theorem can be used to establish that the electrostatic potential is the amount ofwork per unit charge required to move a charge from pointa{\displaystyle a} to pointb{\displaystyle b} with the followingline integral:

−∫abE⋅dℓ=ϕ(b)−ϕ(a).{\displaystyle -\int _{a}^{b}{\mathbf {E} \cdot \mathrm {d} \mathbf {\ell } }=\phi (\mathbf {b} )-\phi (\mathbf {a} ).}

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

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Main articles:Electric potential energy andEnergy density

Atest particle's potential energy,UEsingle{\displaystyle U_{\mathrm {E} }^{\text{single}}}, can be calculated from aline integral of the work,qnE⋅dℓ{\displaystyle q_{n}\mathbf {E} \cdot \mathrm {d} \mathbf {\ell } }. We integrate from a point at infinity, and assume a collection ofN{\displaystyle N} particles of chargeQn{\displaystyle Q_{n}}, are already situated at the pointsri{\displaystyle \mathbf {r} _{i}}. This potential energy (inJoules) is:

UEsingle=qϕ(r)=q4πε0∑i=1NQi‖Ri‖{\displaystyle 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\|}}}

whereRi=r−ri{\displaystyle \mathbf {\mathcal {R_{i}}} =\mathbf {r} -\mathbf {r} _{i}} is the distance of each chargeQi{\displaystyle Q_{i}} from thetest chargeq{\displaystyle q}, which situated at the pointr{\displaystyle \mathbf {r} }, andϕ(r){\displaystyle \phi (\mathbf {r} )} is the electric potential that would be atr{\displaystyle \mathbf {r} } if thetest charge were not present. If only two charges are present, the potential energy isQ1Q2/(4πε0r){\displaystyle Q_{1}Q_{2}/(4\pi \varepsilon _{0}r)}. The totalelectric potential energy due a collection ofN charges is calculating by assembling these particlesone at a time:

UEtotal=14πε0∑j=1NQj∑i=1j−1Qirij=12∑i=1NQiϕi,{\displaystyle 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},}

where the following sum from,j = 1 toN, excludesi =j:

ϕi=14πε0∑j≠ij=1NQjrij.{\displaystyle \phi _{i}={\frac {1}{4\pi \varepsilon _{0}}}\sum _{\stackrel {j=1}{j\neq i}}^{N}{\frac {Q_{j}}{r_{ij}}}.}

This electric potential,ϕi{\displaystyle \phi _{i}} is what would be measured atri{\displaystyle \mathbf {r} _{i}} if the chargeQi{\displaystyle Q_{i}} 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∑(⋯)→∫(⋯)ρd3r{\textstyle \sum (\cdots )\rightarrow \int (\cdots )\rho \,\mathrm {d} ^{3}r}:

UEtotal=12∫ρ(r)ϕ(r)d3r=ε02∫|E|2d3r,{\displaystyle 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,}

This second expression forelectrostatic energy uses the fact that the electric field is the negativegradient of the electric potential, as well asvector calculus identities in a way that resemblesintegration by parts. These two integrals for electric field energy seem to indicate two mutually exclusive formulas for electrostatic energy density, namely12ρϕ{\textstyle {\frac {1}{2}}\rho \phi } and12ε0E2{\textstyle {\frac {1}{2}}\varepsilon _{0}E^{2}}; they yield equal values for the total electrostatic energy only if both are integrated over all space.

Electrostatic pressure

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Inside of an electricalconductor, there is no electric field.[15] The external electric field has been balanced by surface charges due to movement ofcharge carriers, either to or from other parts of the material, known aselectrostatic induction. The equation connecting the field just above a small patch of the surface and the surface charge isE⋅n^=σϵ0{\displaystyle \mathbf {E\cdot {\hat {n}}} ={\frac {\sigma }{\epsilon _{0}}}}where

The average electric field, half the external value,[16] also exerts a force (Coulomb's law) on the conductor patch where the forcef{\displaystyle \mathbf {f} } is given by

f=12ϵ0σ2n^{\displaystyle \mathbf {f} ={\frac {1}{2\epsilon _{0}}}\sigma ^{2}\mathbf {\hat {n}} }.

In terms of the field just outside the surface, the force is equivalent to a pressure given by:

P=ε02(E⋅n^)2,{\displaystyle P={\frac {\varepsilon _{0}}{2}}(\mathbf {E\cdot {\hat {n}}} )^{2},}

This pressure acts normal to the surface of the conductor, independent of whether: the mobile charges are electrons,holes ormobile protons; the sign of the surface charge; or the sign of the surface normal component of the electric field.[16] Note that there is a similar form forelectrostriction in adielectric.[17]

See also

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References

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  1. ^Ling, Samuel J.; Moebs, William; Sanny, Jeff (2019).University Physics, Vol. 2. OpenStax.ISBN 9781947172210. Ch.30: Conductors, Insulators, and Charging by Induction
  2. ^Bloomfield, Louis A. (2015).How Things Work: The Physics of Everyday Life. John Wiley and Sons. p. 270.ISBN 9781119013846.
  3. ^"Polarization".Static Electricity – Lesson 1 – Basic Terminology and Concepts. The Physics Classroom. 2020. Retrieved18 June 2021.
  4. ^Thompson, Xochitl Zamora (2004)."Charge It! All About Electrical Attraction and Repulsion".Teach Engineering: Stem curriculum for K-12. University of Colorado. Retrieved18 June 2021.
  5. ^Brockman, C.J. (October 1929). "The history of electricity before the discovery of the voltaic pile".Journal of Chemical Education.6 (10):1726–1732.doi:10.1021/ed006p1726.
  6. ^J, Griffiths (2017).Introduction to Electrodynamics. Cambridge University Press. pp. 296–354.doi:10.1017/9781108333511.008.ISBN 978-1-108-33351-1. Retrieved2023-08-11.
  7. ^"2022 CODATA Value: vacuum electric permittivity".The NIST Reference on Constants, Units, and Uncertainty.NIST. May 2024. Retrieved2024-05-18.
  8. ^Matthew Sadiku (2009).Elements of electromagnetics. Oxford University Press. p. 104.ISBN 9780195387759.
  9. ^abcPurcell, Edward M. (2013).Electricity and Magnetism. Cambridge University Press. pp. 16–18.ISBN 978-1107014022.
  10. ^"Sur l'attraction des sphéroides elliptiques, par M. de La Grange".Mathematics General Collection.doi:10.1163/9789004460409_mor2-b29447057. Retrieved2023-08-11.
  11. ^Gauss, Carl Friedrich (1978-01-15) [1877],"Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum, methodo nova tractata",Werke, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 279–286,doi:10.1007/978-3-642-49319-5_8,ISBN 978-3-642-49320-1, retrieved2023-08-11
  12. ^Poisson, M; sciences (France), Académie royale des (1827).Mémoires de l'Académie (royale) des sciences de l'Institut (imperial) de France. Vol. 6. Paris.
  13. ^Montgomery, David (1970). "Validity of the electrostatic approximation".Physics of Fluids.13 (5):1401–1403.Bibcode:1970PhFl...13.1401M.doi:10.1063/1.1693079.hdl:2060/19700015014.
  14. ^Heras, J. A. (2010). "The Galilean limits of Maxwell's equations".American Journal of Physics.78 (10):1048–1055.arXiv:1012.1068.Bibcode:2010AmJPh..78.1048H.doi:10.1119/1.3442798.S2CID 118443242.
  15. ^Purcell, Edward M.; David J. Morin (2013).Electricity and Magnetism. Cambridge Univ. Press. pp. 127–128.ISBN 978-1107014022.
  16. ^abGriffiths, David J. (2023-11-02).Introduction to Electrodynamics (5 ed.). Cambridge University Press. pp. §2.5.3.doi:10.1017/9781009397735.ISBN 978-1-009-39773-5.
  17. ^Sundar, V.; Newnham, R. E. (1992-10-01)."Electrostriction and polarization".Ferroelectrics.135 (1):431–446.doi:10.1080/00150199208230043.ISSN 0015-0193.

Further reading

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External links

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Look upelectrostatics in Wiktionary, the free dictionary.
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Electrostatics

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