Editing Voltage

{{Short description|Difference in electric potential between two points in space}}
{{other uses}}
{{Redirect|Potential difference|other uses|Potential}}
{{More citations needed|date=February 2018}}
{{Infobox physical quantity
|bgcolour = {default}
|name = Voltage
|image = [[File:AA AAA AAAA A23 battery comparison-1.jpg|frameless]]
|caption = [[Battery (electricity)|Batteries]] are sources of voltage in many [[Electrical network|electric circuits]].
|unit = [[volt]]
|symbols = {{math|''V''}} , {{math|∆''V''}} , {{math|''U''}} , {{math|∆''U''}}
|dimension = <math>\mathsf{M} \mathsf{L}^2 \mathsf{T}^{-3} \mathsf{I}^{-1}</math>
| derivations = Voltage = [[Energy]] / [[electric charge|charge]]
|baseunits=kg⋅m<sup>2</sup>⋅s<sup>−3</sup>⋅A<sup>−1</sup>}}
{{Electromagnetism|Network}}

'''Voltage''', also known as ('''electrical''') '''potential difference''', '''electric pressure''', or '''electric tension''', is the difference in [[electric potential]] between two points.<ref>{{Cite book |last1=Cretì |first1=Anna |url=https://books.google.com/books?id=7IKWDwAAQBAJ&pg=PA18|title=Economics of Electricity: Markets, Competition and Rules |last2=Fontini |first2=Fulvio |date=2019-05-30 |publisher=Cambridge University Press |isbn=978-1-107-18565-4 |pages=18 |language=en}}</ref><ref>{{Cite book |last=Tregub |first=Stanislav |url=https://books.google.com/books?id=HCxHEAAAQBAJ&pg=PA26|title=Theory of Energy Harmony: Mechanism of Fundamental Interactions |date=2020-08-08 |publisher=Stanislav Tregub |isbn=978-5-6044739-2-4 |pages=26 |language=en}}</ref> In a [[Electrostatics|static]] [[electric field]], it corresponds to the [[Work (electrical)|work]] needed per unit of [[Electric charge|charge]] to move a positive [[Test particle#Electrostatics|test charge]] from the first point to the second point. In the [[SI unit|International System of Units]] (SI), the [[SI derived unit|derived unit]] for voltage is the ''[[volt]]'' (''V'').<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=31 |access-date=2 January 2024}}</ref><ref>{{Cite book |last1=Holloway |first1=Michael D. |url=https://books.google.com/books?id=Jx0OEAAAQBAJ&pg=PA1259|title=Dictionary of Industrial Terminology |last2=Holloway |first2=Emma |date=2020-12-09 |publisher=John Wiley & Sons |isbn=978-1-119-36410-8 |pages=1259 |language=en}}</ref><ref>{{Cite book |last1=Aslam |first1=Dr S. |url=https://books.google.com/books?id=4Wf2EAAAQBAJ&pg=PA17|title=Integrating Electrical Systems With Intelligent Computing |last2=Sharma |first2=Dr Pradosh Kumar |last3=Rahul |first3=Satyakam |last4=Saluja |first4=Dr Hitanshu |date=2024-01-26 |publisher=Academic Guru Publishing House |isbn=978-81-19843-91-6 |pages=17 |language=en}}</ref>

The voltage between points can be caused by the build-up of [[electric charge]] (e.g., a [[capacitor]]), and from an [[electromotive force]] (e.g., [[electromagnetic induction]] in a [[Electric generator|generator]]).<ref>Demetrius T. Paris and F. Kenneth Hurd, ''Basic Electromagnetic Theory'', McGraw-Hill, New York 1969, {{ISBN|0-07-048470-8}}, pp. 512, 546</ref><ref>P. Hammond, ''Electromagnetism for Engineers'', p. 135, Pergamon Press 1969 {{OCLC|854336}}.</ref> On a macroscopic scale, a potential difference can be caused by electrochemical processes (e.g., cells and batteries), the pressure-induced [[piezoelectric effect]], and the [[thermoelectric effect]]. Since it is the difference in electric potential, it is a physical [[Scalar (physics)|scalar]] [[quantity]].<ref>{{Cite book |last=Experts |first=Disha |url=https://books.google.com/books?id=1OMyDwAAQBAJ&pg=PA64|title=10 in One Study Package for CBSE Physics Class 12 with 5 Model Papers |date=2017-08-29 |publisher=Disha Publications |isbn=978-93-86323-72-9 |pages=64 |language=en}}</ref>

A [[voltmeter]] can be used to measure the voltage between two points in a system.<ref>{{Cite book |last=International |first=Petrogav |url=https://books.google.com/books?id=ZS7JDwAAQBAJ&pg=PA328|title=Production Course for Hiring on Offshore Oil and Gas Rigs |publisher=Petrogav International |pages=328 |language=en}}</ref> Often a common reference potential such as the [[ground (electricity)|ground]] of the system is used as one of the points. In this case, voltage is often mentioned at a point without completely mentioning the other measurement point. A voltage can be associated with either a source of energy or the loss, dissipation, or storage of energy.

==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.

==Volt==
{{main|Volt}}
The volt (symbol: {{math|'''V'''}}) is the [[SI derived unit|derived unit]] for [[electric potential]], voltage, and [[electromotive force]].<ref>{{Cite book |last1=Hanssen |first1=Steven |url=https://books.google.com/books?id=zG2dEAAAQBAJ&pg=PA3|title=Electrical Trade Principles 6e |last2=Hampson |first2=Jeffery |date=2022-09-12 |publisher=Cengage AU |isbn=978-0-17-045885-6 |pages=3 |language=en}}</ref><ref>{{Cite book |last=Cardarelli |first=Francois |url=https://books.google.com/books?id=sEHtBwAAQBAJ&pg=PA340|title=Scientific Unit Conversion: A Practical Guide to Metrication |date=2012-12-06 |publisher=Springer Science & Business Media |isbn=978-1-4471-3394-0 |pages=340 |language=en}}</ref> The volt is named in honour of the Italian physicist [[Alessandro Volta]]  (1745–1827), who invented the [[voltaic pile]], possibly the first chemical [[battery (electricity)|battery]].

==Hydraulic analogy==
{{Main|Hydraulic analogy}}

A simple analogy for an [[electric circuit]] is water flowing in a closed circuit of [[pipework]], driven by a mechanical [[pump]].{{cn|date=March 2024}} This can be called a "water circuit".  The potential difference between two points corresponds to the [[fluid pressure|pressure difference]] between two points. If the pump creates a pressure difference between two points, then water flowing from one point to the other will be able to do work, such as driving a [[turbine]]. Similarly, work can be done by an [[electric current]] driven by the potential difference provided by a [[electric battery|battery]]. For example, the voltage provided by a sufficiently-charged automobile battery can "push" a large current through the windings of an automobile's [[starter motor]]. If the pump is not working, it produces no pressure difference, and the turbine will not rotate. Likewise, if the automobile's battery is very weak or "dead" (or "flat"), then it will not turn the starter motor.

The hydraulic analogy is a useful way of understanding many electrical concepts. In such a system, the work done to move water is equal to the "[[pressure]] drop" (compare p.d.) multiplied by the [[volume]] of water moved. Similarly, in an electrical circuit, the work done to move electrons or other charge carriers is equal to "electrical pressure difference" multiplied by the quantity of electrical charges moved. In relation to "flow", the larger the "pressure difference" between two points (potential difference or water pressure difference), the greater the flow between them (electric current or water flow). (See "[[Electric power#Definition|electric power]]".)

== Applications ==
[[File:US Navy 110315-N-0278E-002 High-voltage electricians from Naval Facilities Engineering Command (NAVFAC) Hawaii reconfigure electrical circuitry and.jpg|thumb|upright|Working on [[high voltage|high-voltage]] power lines]]

Specifying a voltage measurement requires explicit or implicit specification of the points across which the voltage is measured.  When using a voltmeter to measure voltage, one electrical lead of the voltmeter must be connected to the first point, one to the second point.

A common use of the term "voltage" is in describing the voltage dropped across an electrical device (such as a resistor). The [[voltage drop]] across the device can be understood as the difference between measurements at each terminal of the device with respect to a common reference point (or [[ground (electricity)|ground]]). The voltage drop is the difference between the two readings. Two points in an electric circuit that are connected by an ideal conductor without resistance and not within a changing [[magnetic field]] have a voltage of zero. Any two points with the same potential may be connected by a conductor and no current will flow between them.

===Addition of voltages===
The voltage between ''A'' and ''C'' is the sum of the voltage between ''A'' and ''B'' and the voltage between ''B'' and ''C''. The various voltages in a circuit can be computed using [[Kirchhoff's circuit laws]].

When talking about [[alternating current]] (AC) there is a difference between instantaneous voltage and average voltage. Instantaneous voltages can be added for [[direct current]] (DC) and AC, but average voltages can be meaningfully added only when they apply to signals that all have the same frequency and phase.

==Measuring instruments==

[[File:9VBatteryWithMeter.jpg|thumb|[[Multimeter]] set to measure voltage]]
Instruments for measuring voltages include the [[voltmeter]], the [[Potentiometer (measuring instrument)|potentiometer]], and the [[oscilloscope]]. [[Voltmeter|Analog voltmeter]]s, such as moving-coil instruments, work by measuring the current through a fixed resistor, which, according to [[Ohm's law]], is proportional to the voltage across the resistor. The potentiometer works by balancing the unknown voltage against a known voltage in a [[bridge circuit]]. The cathode-ray oscilloscope works by amplifying the voltage and using it to deflect an [[electron]] beam from a straight path, so that the deflection of the beam is proportional to the voltage.

==Typical voltages==
{{main|Volt#Common voltages|Orders of magnitude (voltage)|Mains electricity#Choice of voltage}}
A common voltage for [[Battery (electricity)|flashlight batteries]] is 1.5&nbsp;volts (DC).
A common voltage for [[Automotive battery|automobile batteries]] is 12&nbsp;volts (DC).

Common voltages supplied by power companies to consumers are 110 to 120 volts (AC) and 220 to 240&nbsp;volts (AC). The voltage in [[electric power transmission]] lines used to distribute electricity from power stations can be several hundred times greater than consumer voltages, typically 110 to 1200&nbsp;kV (AC).

The voltage used in [[overhead line]]s to power railway locomotives is between 12&nbsp;kV and 50&nbsp;kV (AC) or between 0.75&nbsp;kV and 3&nbsp;kV (DC).

==Galvani potential vs. electrochemical potential==
{{main|Galvani potential|Electrochemical potential|Fermi level}}
Inside a conductive material, the energy of an electron is affected not only by the average electric potential but also by the specific thermal and atomic environment that it is in.
When a [[voltmeter]] is connected between two different types of metal, it measures not the electrostatic potential difference, but instead something else that is affected by thermodynamics.<ref>{{cite book|url=https://books.google.com/books?id=09QI-assq1cC&pg=PA22 |title=Fundamentals of electrochemistry|first= Vladimir Sergeevich|last= Bagotskii|page=22|isbn=978-0-471-70058-6|year=2006|publisher=John Wiley & Sons }}</ref>
The quantity measured by a voltmeter is the negative of the difference of the [[electrochemical potential]] of electrons ([[Fermi level]]) divided by the electron charge and commonly referred to as the voltage difference, while the pure unadjusted [[electrostatic potential]] (not measurable with a voltmeter) is sometimes called [[Galvani potential]].
The terms "voltage" and "electric potential" are ambiguous in that, in practice, they can refer to ''either'' of these in different contexts.

== History ==
The term ''electromotive force'' was first used by Volta in a letter to [[Giovanni Aldini]] in 1798, and first appeared in a published paper in 1801 in ''[[Annales de chimie et de physique]]''.<ref name=Varney/>{{rp|408}}  Volta meant by this a force that was not an [[electrostatic]] force, specifically, an [[electrochemical]] force.<ref name=Varney>Robert N. Varney, Leon H. Fisher, [https://aapt.scitation.org/doi/abs/10.1119/1.12115 "Electromotive force: Volta's forgotten concept"] {{Webarchive|url=https://web.archive.org/web/20210416012226/https://aapt.scitation.org/doi/abs/10.1119/1.12115 |date=2021-04-16 }}, ''American Journal of Physics'', vol. 48, iss. 5, pp. 405–408, May 1980.</ref>{{rp|405}}  The term was taken up by [[Michael Faraday]] in connection with [[electromagnetic induction]] in the 1820s.  However, a clear definition of voltage and method of measuring it had not been developed at this time.<ref>C. J. Brockman, [https://pubs.acs.org/doi/abs/10.1021/ed005p549?journalCode=jceda8 "The origin of voltaic electricity: The contact vs. chemical theory before the concept of E. M. F. was developed"] {{Webarchive|url=https://web.archive.org/web/20220717034921/https://pubs.acs.org/doi/abs/10.1021/ed005p549?journalCode=jceda8 |date=2022-07-17 }}, ''Journal of Chemical Education'', vol. 5, no. 5, pp. 549–555, May 1928</ref>{{rp|554}} Volta distinguished electromotive force (emf) from ''tension'' (potential difference): the observed potential difference at the terminals of an electrochemical cell when it was open circuit must exactly balance the emf of the cell so that no current flowed.<ref name=Varney/>{{rp|405}}

==See also==
{{Portal|Electronics}}
{{div col start}}
* [[Electric shock]]
* [[Mains electricity by country]] (list of countries with mains voltage and frequency)
* [[Open-circuit voltage]]
* [[Phantom voltage]]
{{div col end}}

==References==
<references />

== Footnotes ==
<references group="note" />

==External links==
{{Wiktionary}}
* [http://www.sengpielaudio.com/calculator-ohm.htm Electrical voltage ''V'', current ''I'', resistivity ''R'', impedance ''Z'', wattage ''P'']

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[[Category:Voltage| ]]
[[Category:Electrical systems]]
[[Category:Electromagnetic quantities]]