Editing Voltage (section)

==Definition==
The SI unit of work per unit charge is the [[joule]] per [[coulomb]], where 1 volt = 1 joule (of work) per 1 coulomb of charge.{{cn|date=March 2024}} The old SI definition for ''volt'' used [[Electric power|power]] and [[Electric current|current]]; starting in 1990, the [[quantum Hall effect|quantum Hall]] and [[Josephson effect]] were used,<ref>{{cite report |author=David B. Newell, Eite Tiesinga |date=August 2019 |title=The International System of Units (SI) |url=https://nvlpubs.nist.gov/nistpubs/SpecialPublications/NIST.SP.330-2019.pdf |publisher=National Institute of Standards and Technology |page=88 |access-date=2 January 2024}}</ref> and in 2019 [[physical constant]]s were given defined values for the definition of all SI units.

Voltage is denoted symbolically by <math>\Delta V</math>, simplified ''V'',<ref>IEV: [http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=121-11-25 electric potential] {{Webarchive|url=https://web.archive.org/web/20210428033941/http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=121-11-25 |date=2021-04-28 }}</ref> especially in [[English language|English]]-speaking countries. Internationally, the symbol ''U'' is standardized.<ref>IEV: [http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=121-11-27 voltage] {{Webarchive|url=https://web.archive.org/web/20160203221151/http://www.electropedia.org/iev/iev.nsf/display?openform&ievref=121-11-27 |date=2016-02-03 }}</ref> It is used, for instance, in the context of [[Ohm's law|Ohm's]] or [[Kirchhoff's circuit laws]].

The [[electrochemical potential]] is the voltage that can be directly measured with a voltmeter.<ref>{{Cite book |last=Fischer |first=Traugott |url=https://books.google.com/books?id=r8Xasw5l8wQC&pg=PA434|title=Materials Science for Engineering Students |date=2009-03-13 |publisher=Academic Press |isbn=978-0-08-092002-3 |pages=434 |language=en}}</ref><ref>{{Cite book |last=Pulfrey |first=David L. |url=https://books.google.com/books?id=y9dYENs2SVUC&pg=PA93|title=Understanding Modern Transistors and Diodes |date=2010-01-28 |publisher=Cambridge University Press |isbn=978-1-139-48467-1 |pages=93 |language=en}}</ref> The [[Galvani potential]] that exists in structures with junctions of dissimilar materials, is also work per charge but cannot be measured with a voltmeter in the external circuit (see {{Section link||Galvani potential vs. electrochemical potential}}).

Voltage is defined so that negatively charged objects are pulled towards higher voltages, while positively charged objects are pulled towards lower voltages.<ref>{{Cite book |last=Vadari |first=Mani |url=https://books.google.com/books?id=Q85Lz70wdw8C&pg=PA41|title=Electric System Operations: Evolving to the Modern Grid |date=2013 |publisher=Artech House |isbn=978-1-60807-549-2 |pages=41 |language=en}}</ref><ref>{{Cite book |last=Vadari |first=Subramanian |url=https://books.google.com/books?id=c73PDwAAQBAJ&pg=PA47|title=Electric System Operations: Evolving to the Modern Grid, Second Edition |date=2020-01-31 |publisher=Artech House |isbn=978-1-63081-689-6 |pages=47 |language=en}}</ref> Therefore, the [[conventional current]] in a wire or [[resistor]] always flows from higher voltage to lower voltage.

Historically, voltage has been referred to using terms like "tension" and "pressure". Even today, the term "tension" is still used, for example within the phrase "[[High voltage|high tension]]" (HT) which is commonly used in the contexts of automotive electronics and systems using thermionic valves ([[vacuum tube]]s).

=== Electrostatics ===
[[File:Opfindelsernes bog3 fig282.png|thumb|The electric field around the rod exerts a force on the charged pith ball, in an [[electroscope]]]]
[[File:Electrostatic definition of voltage.svg|thumb|In a static field, the work is independent of the path]]

{{Main articles|Electric potential#Electrostatics}}
In [[electrostatics]], the voltage increase from point <math>\mathbf{r}_A</math> to some point <math>\mathbf{r}_B</math> is given by the change in [[Electric potential#Electrostatics|electrostatic potential]] <math display="inline">V</math> from <math>\mathbf{r}_A</math> to <math>\mathbf{r}_B</math>. By definition,<ref name=":1">{{Cite book|last=Griffiths|first=David J.|title=Introduction to Electrodynamics|publisher=Prentice Hall|year=1999|isbn=013805326X|edition=3rd|pages=}}</ref>{{Rp|78}} this is:

:<math>\begin{align}
\Delta V_{AB} &= V(\mathbf{r}_B) - V(\mathbf{r}_A) \\
&= -\int_{\mathbf{r}_0}^{\mathbf{r}_B} \mathbf{E} \cdot \mathrm{d}\boldsymbol{\ell} - \left(-\int_{\mathbf{r}_0}^{\mathbf{r}_A} \mathbf{E} \cdot \mathrm{d}\boldsymbol{\ell} \right)\\
&= -\int_{\mathbf{r}_A}^{\mathbf{r}_B} \mathbf{E} \cdot \mathrm{d}\boldsymbol{\ell}
\end{align} </math>

where <math>\mathbf{E}</math> is the intensity of the electric field.

In this case, the voltage increase from point A to point B is equal to the work done per unit charge, against the electric field, to move the charge from A to B without causing any acceleration.<ref name=":1" />{{Rp|90-91}} Mathematically, this is expressed as the [[line integral]] of the [[electric field]] along that path. In electrostatics, this line integral is independent of the path taken.<ref name=":1" />{{Rp|91}}

Under this definition, any circuit where there are time-varying magnetic fields, such as [[Alternating current|AC circuits]], will not have a well-defined voltage between nodes in the circuit, since the electric force is not a [[conservative force]] in those cases.<ref group="note" name=":0">This follows from the [[Maxwell-Faraday equation]]:

<math>\nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}</math>

If there are changing magnetic fields in some [[Simply connected space|simply connected]] region, then the [[Curl (mathematics)|curl]] of the electric field in that region is non-zero, and as a result the electric field is not conservative. For more, see {{Section link|Conservative force|Mathematical description}}.</ref> However, at lower frequencies when the electric and magnetic fields are not rapidly changing, this can be neglected (see [[Electrostatics#Electrostatic approximation|electrostatic approximation]]).

=== Electrodynamics ===
{{Main articles|Electric potential#Generalization to electrodynamics}}
The electric potential can be generalized to electrodynamics, so that differences in electric potential between points are well-defined even in the presence of time-varying fields. However, unlike in electrostatics, the electric field can no longer be expressed only in terms of the electric potential.<ref name=":1" />{{Rp|417}} Furthermore, the potential is no longer uniquely determined up to a constant, and can take significantly different forms depending on the choice of [[Gauge fixing|gauge]].<ref group="note">For example, in the [[Lorenz gauge condition|Lorenz gauge]], the electric potential is a [[retarded potential]], which propagates at the [[speed of light]]; whereas in the [[Coulomb Gauge|Coulomb gauge]], the potential changes instantaneously when the source charge distribution changes.</ref><ref name=":1" />{{Rp|419-422}}

In this general case, some authors<ref>{{Cite book|last1=Moon|first1=Parry|url=https://books.google.com/books?id=lijEAgAAQBAJ&pg=PA126|title=Foundations of Electrodynamics|last2=Spencer|first2=Domina Eberle|publisher=Dover Publications|year=2013|isbn=978-0-486-49703-7|pages=126|access-date=2021-11-19|archive-date=2022-03-19|archive-url=https://web.archive.org/web/20220319091311/https://books.google.com/books?id=lijEAgAAQBAJ&pg=PA126|url-status=live}}</ref> use the word "voltage" to refer to the line integral of the electric field, rather than to differences in electric potential. In this case, the voltage rise along some path <math>\mathcal{P}</math> from <math>\mathbf{r}_A</math> to <math>\mathbf{r}_B</math> is given by:
:<math>\Delta V_{AB} = -\int_\mathcal{P} \mathbf{E} \cdot \mathrm{d}\boldsymbol{\ell}
</math>
However, in this case the "voltage" between two points depends on the path taken.

=== Circuit theory ===
In [[Network analysis (electrical circuits)|circuit analysis]] and [[electrical engineering]], [[lumped element model]]s are used to represent and analyze circuits. These elements are idealized and self-contained circuit elements used to model physical components.<ref name=":2">{{Cite web|last=A. Agarwal & J. Lang|date=2007|title=Course materials for 6.002 Circuits and Electronics|url=https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-002-circuits-and-electronics-spring-2007/video-lectures/6002_l1.pdf|access-date=4 December 2018|website=MIT OpenCourseWare|archive-date=9 April 2016|archive-url=https://web.archive.org/web/20160409071008/http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-002-circuits-and-electronics-spring-2007/video-lectures/6002_l1.pdf|url-status=live}}</ref>

When using a lumped element model, it is assumed that the effects of changing magnetic fields produced by the circuit are suitably contained to each element.<ref name=":2" /> Under these assumptions, the electric field in the region exterior to each component is conservative, and voltages between nodes in the circuit are well-defined, where<ref name=":2" />

:<math>\Delta V_{AB} = -\int_{\mathbf{r}_A}^{\mathbf{r}_B} \mathbf{E} \cdot \mathrm{d}\boldsymbol{\ell} </math>

as long as the path of integration does not pass through the inside of any component. The above is the same formula used in electrostatics. This integral, with the path of integration being along the test leads, is what a voltmeter will actually measure.<ref>{{Cite journal|last=Bossavit|first=Alain|date=January 2008|title=What do voltmeters measure?|journal=COMPEL - the International Journal for Computation and Mathematics in Electrical and Electronic Engineering|volume=27|pages=9–16|doi=10.1108/03321640810836582|via=ResearchGate}}</ref><ref group="note">This statement makes a few assumptions about the nature of the voltmeter (these are discussed in the cited paper). One of these assumptions is that the current drawn by the voltmeter is negligible.</ref>

If uncontained magnetic fields throughout the circuit are not negligible, then their effects can be modelled by adding [[mutual inductance]] elements. In the case of a physical inductor though, the ideal lumped representation is often accurate. This is because the external fields of inductors are generally negligible, especially if the inductor has a closed [[Magnetic circuit|magnetic path]]. If external fields are negligible, we find that

:<math>\Delta V_{AB} = -\int_\mathrm{exterior}\mathbf{E}\cdot \mathrm{d}\boldsymbol{\ell}=L\frac{dI}{dt}
</math>

is path-independent, and there is a well-defined voltage across the inductor's terminals.<ref>{{Cite web|last1=Feynman|first1=Richard|last2=Leighton|first2=Robert B.|last3=Sands|first3=Matthew|title=The Feynman Lectures on Physics Vol. II Ch. 22: AC Circuits|url=https://feynmanlectures.caltech.edu/II_22.html|access-date=2021-10-09|website=Caltech}}</ref> This is the reason that measurements with a voltmeter across an inductor are often reasonably independent of the placement of the test leads.