View source for Joule

{{Short description|SI unit of energy}}
{{About|the unit of energy or work}}
{{Use British English|date=February 2022}}
{{distinguish|Juul}}
{{Infobox unit
| bgcolour      = 
| name          = Joule
| image         = Definición del Joule.svg
| caption       = Intuitive representation of the joule as the work of a motive force
| standard      = [[SI]]
| quantity      = [[energy]]
| symbol        = J
| namedafter    = [[James Prescott Joule]]
| units1        = [[SI base unit]]s
| inunits1      = [[kilogram|kg]]⋅[[metre|m]]<sup>2</sup>⋅[[second|s]]<sup>−2</sup>
| units2        = [[CGS unit]]s
| inunits2      = {{val|1|e=7}} [[erg]]
| units3        = [[#Watt-second|watt-second]]s
| inunits3      = {{val|1}}&nbsp;[[Watt|W]]⋅[[second|s]]
| units4        = [[kilowatt-hour]]s
| inunits4      = ≈{{val|2.78|e=-7|u=kW⋅h}}
| units5        = [[calorie|kilocalories (thermochemical)]]
| inunits5      = {{val|2.390|e=-4|u=kcal<sub>th</sub>}}
| units6        = [[British thermal unit|BTUs]]
| inunits6      = {{val|9.48|e=-4|u=BTU}}
| units7        = [[electronvolt]]s
| inunits7      = ≈{{val|6.24|e=18|u=eV}}
}}

The '''joule''' ({{IPAc-en|dʒ|uː|l}} {{respell|JOOL}}, or {{IPAc-en|dʒ|aʊ|l}} {{respell|JOWL}}; symbol: '''J''') is the unit of [[energy]] in the [[International System of Units]] (SI).<ref>{{SIbrochure8th|page=120}}</ref> In terms of [[SI base units]], one joule corresponds to one [[kilogram]]-[[square metre]] per [[square seconds|square second]] {{nowrap|(1 J {{=}} 1 kg⋅m<sup>2</sup>⋅s<sup>−2</sup>).}} One joule is equal to the amount of [[work (physics)|work]] done when a force of one [[newton (unit)|newton]] displaces a [[body (physics)|body]] through a distance of one [[metre]] in the direction of that force. It is also the energy dissipated as heat when an [[electric current]] of one [[ampere]] passes through a [[electrical resistance and conductance|resistance]] of one [[ohm]] for one second. It is named after the English physicist [[James Prescott Joule]] (1818–1889).<ref>[https://web.archive.org/web/20060413141420/http://education.yahoo.com/reference/dictionary/entry/joule American Heritage Dictionary of the English Language], Online Edition (2009). Houghton Mifflin Co., hosted by [https://web.archive.org/web/20010502171832/http://education.yahoo.com/ Yahoo! Education].</ref><ref>''The American Heritage Dictionary'', Second College Edition (1985). Boston: Houghton Mifflin Co., p. 691.</ref><ref>''McGraw-Hill Dictionary of Physics'', Fifth Edition (1997). McGraw-Hill, Inc., p. 224.</ref>

== Definition ==
In terms of [[SI base units]] and in terms of [[SI derived units with special names]], the joule is defined as<ref>{{Cite journal |date=2016-01-28 |title=NIST Guide to the SI, Chapter 4: The Two Classes of SI Units and the SI Prefixes |url=https://www.nist.gov/pml/special-publication-811/nist-guide-si-chapter-4-two-classes-si-units-and-si-prefixes |journal=NIST |language=en}}</ref>

{|
|-
|style="vertical-align: top; padding-right: 2em;"|
<math display=block>\begin{alignat}{3}
\mathrm{J} \;
&=~ \mathrm{kg{\cdot}m^2{\cdot}s^{-2}} \\[0.7ex]
&=~ \mathrm{N{\cdot}m} \\[0.7ex]
&=~ \mathrm{Pa{\cdot}m^3} \\[0.7ex]
&=~ \mathrm{W{\cdot}s} \\[0.7ex]
&=~ \mathrm{C{\cdot}V} \\[0.7ex]
\end{alignat}</math>

|style="padding-left: 2em;"|
{| class="wikitable"
|-
! Symbol
! Name
|-
| J
| joule
|-
| kg
| [[kilogram]]
|-
| m
| [[metre]]
|-
| s
| [[second]]
|-
| N
| [[newton (unit)|newton]]
|-
| Pa
| [[pascal (unit)|pascal]]
|-
| W
| [[watt]]
|-
| C 
| [[coulomb]]
|-
| V 
| [[volt]]
|}

|}

One joule is also equivalent to any of the following:<ref>{{citation
|last1 = Halliday | first1 = David
|last2 = Resnick | first2 = Robert
|title = [[Fundamentals of Physics]] | edition = revised
|date = 1974
|publisher = Wiley |location = New York
|author-link1=David Halliday (physicist)
|author-link2=Robert Resnick
|pages = 516–517
|isbn = 0471344311
}}</ref>
* The work required to move an [[electric charge]] of one [[coulomb]] through an [[Voltage|electrical potential difference]] of one volt, or one coulomb-volt (C⋅V). This relationship can be used to define the volt.
* The work required to produce one watt of [[Power (physics)|power]] for one second, or one watt-second (W⋅s) (compare [[kilowatt-hour]], which is 3.6 megajoules). This relationship can be used to define the watt.

{{SI unit lowercase|James Prescott Joule|joule|J}}<ref>{{Cite web |title=What Is a Joule? - Chemistry Definition |url=https://www.thoughtco.com/definition-of-joule-604543 |access-date=2024-04-07 |website=ThoughtCo |language=en}}</ref>

== History ==
The [[CGS system]] had been declared official in 1881, at the first [[International Electrical Congress]].
The [[erg]] was adopted as its unit of energy in 1882. [[Carl Wilhelm Siemens|Wilhelm Siemens]], in his inauguration speech as chairman of the [[British Association for the Advancement of Science]] (23 August 1882) first proposed the ''joule'' as unit of [[heat]], to be derived from the electromagnetic units [[ampere]] and [[ohm]], in cgs units equivalent to {{val|e=7|u=erg}}.
The naming of the unit in honour of [[James Prescott Joule]] (1818–1889), at the time retired and aged 63, followed the recommendation of Siemens:

{{quote|Such a heat unit, if found acceptable, might with great propriety, I think, be called the Joule, after the man who has done so much to develop the dynamical theory of heat.<ref>{{cite conference | quote=The unit of heat has hitherto been taken variously as the heat required to raise a pound of water at the freezing-point through 1°&nbsp;Fahrenheit or Centigrade, or, again, the heat necessary to raise a kilogramme of water 1°&nbsp;Centigrade. The inconvenience of a unit so entirely arbitrary is sufficiently apparent to justify the introduction of one based on the electro-magnetic system, viz. the heat generated in one second by the current of an Ampère flowing through the resistance of an Ohm. In absolute measure its value is 10<sup>7</sup> C.G.S. units, and, assuming Joule's equivalent as 42,000,000, it is the heat necessary to raise 0.238&nbsp;grammes of water 1°&nbsp;Centigrade, or, approximately, the {{frac|1|1000}}th part of the arbitrary unit of a pound of water raised 1°&nbsp;Fahrenheit and the {{frac|1|4000}}th of the kilogramme of water raised 1°&nbsp;Centigrade. Such a heat unit, if found acceptable, might with great propriety, I think, be called the Joule, after the man who has done so much to develop the dynamical theory of heat. |first = Cal Wilhelm |last=Siemens |author-link=Carl Wilhelm Siemens |url=http://gallica.bnf.fr/ark:/12148/bpt6k781656 |title=Report of the Fifty-Second Meeting of the British Association for the Advancement of Science |pages=1–33 |quote-pages=6-7 |date= August 1882 |location=Southampton}}</ref>}}

At the second International Electrical Congress, on 31 August 1889, the joule was officially adopted alongside the [[watt]] and the ''quadrant'' (later renamed to [[Henry (unit)|henry]]).<ref>Pat Naughtin: [http://www.metricationmatters.com/docs/MetricationTimeline.pdf ''A chronological history of the modern metric system''], metricationmatters.com, 2009.</ref>
Joule died in the same year, on 11 October 1889.
At the fourth congress (1893), the "international ampere" and "international ohm" were defined, with slight changes in the specifications for their measurement, with the "international joule" being the unit derived from them.<ref>{{cite book |author=<!--Staff writer(s); no by-line.--> |title= Proceedings of the International Electrical Congress|url=https://archive.org/details/proceedingsinte01chicgoog|location=New York |publisher=American Institute of Electrical Engineers |date=1894 }}</ref>

In 1935, the [[International Electrotechnical Commission]] (as the successor organisation of the International Electrical Congress) adopted the "[[Giovanni Giorgi|Giorgi]] system", which by virtue of assuming a defined value for the [[magnetic constant]] also implied a redefinition of the joule. The Giorgi system was approved by the [[International Committee for Weights and Measures]] in 1946. The joule was now no longer defined based on electromagnetic unit, but instead as the unit of [[Work (physics)|work]] performed by one unit of force (at the time not yet named [[Newton (unit)|newton]]) over the distance of 1 [[metre]]. The joule was explicitly intended as the unit of energy to be used in both electromagnetic and mechanical contexts.<ref>[http://www.bipm.org/en/CIPM/db/1946/2/ ''CIPM, 1946, Resolution 2, Definitions of electric units'']. ''bipm.org''.</ref> The ratification of the definition at the ninth [[General Conference on Weights and Measures]], in 1948, added the specification that the joule was also to be preferred as the unit of [[heat]] in the context of [[calorimetry]], thereby officially deprecating the use of the [[calorie]].<ref>[http://www.bipm.org/en/CGPM/db/9/3/ ''9th CGPM, Resolution 3: Triple point of water; thermodynamic scale with a single fixed point; unit of quantity of heat (joule).''], ''bipm.org.''</ref>  This is the definition declared in the modern [[International System of Units]] in 1960.<ref>{{SIbrochure|pp=165–166}}</ref>

The definition of the joule as J&nbsp;=&nbsp;kg⋅m<sup>2</sup>⋅s<sup>−2</sup> has remained unchanged since 1946, but the joule as a derived unit has inherited changes in the definitions of the   [[second]] (in 1960 and 1967), the [[metre]] (in 1983) and the [[kilogram]] ([[2019 revision of the SI|in 2019]]).<ref>{{Cite journal |date=2018-05-11 |title=SI Redefinition |url=https://www.nist.gov/si-redefinition |journal=NIST |language=en}}</ref>

== Practical examples ==
One joule represents (approximately):
* The typical energy released as heat by a person at rest every 1/60 s (~{{val|16.6667|ul=ms}}, [[basal metabolic rate]]); about {{Convert|1200|kcal|kJ|lk=on|abbr=on|order=flip}} / day.
* The amount of electricity required to run a {{val|1|ul=W}} device for {{val|1|ul=s}}.
* The energy required to accelerate a {{val|1|ul=kg}} mass at {{val|1|ul=m/s2}} through a distance of {{val|1|ul=m}}.
* The [[kinetic energy]] of a {{val|2|ul=kg}} [[mass]] travelling at {{val|1|ul=m/s}}, or a {{val|1|ul=kg}} mass travelling at {{val|1.41|ul=m/s}}.
* The energy required to lift an apple up 1 m, assuming the apple has a mass of 101.97 g.
* The [[heat]] required to raise the temperature of 0.239&nbsp;g of water from 0 °C to 1 °C.<ref>{{cite web |url = http://www.engineeringtoolbox.com/heat-units-d_664.html |title=Units of Heat – BTU, Calorie and Joule |website=Engineering Toolbox |access-date=2021-06-14 }}</ref>
* The [[kinetic energy]] of a {{val|50|u=kg}} human moving very slowly ({{convert|0.2|m/s|km/h|abbr=on|disp=or}}).
* The kinetic energy of a {{val|56|u=g}} tennis ball moving at {{convert|6|m/s|km/h|abbr=on}}.<ref>{{cite book |last1=Ristinen |first1=Robert A. |first2=Jack J. |last2=Kraushaar |title=Energy and the Environment |edition=2nd |location=Hoboken, NJ |publisher=John Wiley & Sons |year=2006 |isbn=0-471-73989-8 |url-access=registration |url=https://archive.org/details/energyenvironmen00rist }}</ref>
* The food energy (kcal) in slightly more than half of an ordinary-sized sugar crystal ({{val|0.102|ul=mg}}/crystal).

== Multiples ==
{{for|additional examples|Orders of magnitude (energy)}}
{{SI multiples
|unit=joule
|symbol=J
|note=Common multiples are in bold face
|m=|k=|mc=|M=|n=|G=|p=|T=
}}

; {{vanchor|zeptojoule}} : {{convert|160|zJ|eV|disp=x|&nbsp;is about&nbsp;|sigfig=1|lk=out|abbr=off}}.{{paragraph}} The minimal energy needed to change a bit of data in computation at around room temperature – approximately {{val|2.75|u=zJ}} – is given by the [[Landauer limit]].{{Citation needed|date=January 2024}}
; {{vanchor|nanojoule}} : {{val|160|u=nanojoule}} is about the [[kinetic energy]] of a flying mosquito.<ref>{{cite web|url=http://public.web.cern.ch/Public/en/Science/Glossary-en.php|title=Physics – CERN|website=public.web.cern.ch|url-status=dead|archive-url=https://web.archive.org/web/20121213173112/https://public.web.cern.ch/public/en/Science/Glossary-en.php|archive-date=2012-12-13}}</ref>
; {{vanchor|microjoule}} : The [[Large Hadron Collider]] (LHC) produces collisions of the microjoule order (7 TeV) per particle.{{Citation needed|date=January 2024}}
; {{vanchor|kilojoule}} : Nutritional food labels in most countries express energy in kilojoules (kJ).<ref name="Cal vs kJ">{{cite web |title=You Say Calorie, We Say Kilojoule: Who's Right? |url=https://www.coca-colacompany.com/au/news/you-say-calorie--we-say-kilojoule-who-s-right- |archive-url=https://web.archive.org/web/20230515172626/https://www.coca-colacompany.com/au/news/you-say-calorie--we-say-kilojoule-who-s-right- |access-date=2 May 2017|archive-date=2023-05-15 }}</ref>{{paragraph}} One square metre of the [[Earth]] receives about {{val|1.4|u=kilojoules}} of [[sunlight#Solar constant|solar radiation]] every second in full daylight.<ref name=TSI>{{cite web |title=Construction of a Composite Total Solar Irradiance (TSI) Time Series from 1978 to present |url=http://www.pmodwrc.ch/pmod.php?topic=tsi/composite/SolarConstant |access-date=2005-10-05 |archive-url=https://web.archive.org/web/20110830221302/http://www.pmodwrc.ch/pmod.php?topic=tsi%2Fcomposite%2FSolarConstant |archive-date=2011-08-30 }}</ref> A human in a sprint has approximately 3&nbsp;kJ of kinetic energy,<ref>{{sfrac|1|2}} × 70&nbsp;kg × (10&nbsp;m/s)<sup>2</sup> = 3500&nbsp;J</ref> while a cheetah in a {{val|122|ul=km/h}} (76&nbsp;mph) sprint has approximately 20&nbsp;kJ.<ref>{{sfrac|1|2}} × 35&nbsp;kg × (35&nbsp;m/s)<sup>2</sup> = {{val|21,400|u=J}}</ref> {{convert|1|Wh|kJ|disp=x|, of electricity or any other form of energy, is&nbsp;|spell=In|lk=in}}.
; {{vanchor|megajoule}} : The megajoule is approximately the kinetic energy of a one megagram (tonne) vehicle moving at {{val|161|ul=km/h}} (100&nbsp;mph).{{Citation needed|date=January 2024}}{{paragraph}} The energy required to heat {{val|10|u=liters}} of liquid water at constant pressure from {{convert|0|C}} to {{convert|100|C}} is approximately {{val|4.2|ul=MJ}}.{{Citation needed|date=January 2024}}{{paragraph}} {{convert|1|kWh|MJ|disp=x|, of electricity or any other form of energy, is&nbsp;|spell=In|lk=in}}.
; {{vanchor|gigajoule}} : {{val|6|ul=gigajoule}} is about the [[chemical energy]] of combusting {{convert|1|oilbbl|L|0}} of [[petroleum]].<ref>{{cite web|url=https://www.eia.gov/energyexplained/index.cfm?page=about_energy_units|title=Energy Units – Energy Explained, Your Guide To Understanding Energy – Energy Information Administration|website=www.eia.gov}}</ref> 2&nbsp;GJ is about the [[Planck units|Planck energy]] unit.  {{convert|1|MWh|GJ|disp=x|, of electricity or any other form of energy, is&nbsp;|spell=In|lk=in}}.
; {{vanchor|terajoule}} : The terajoule is about {{val|0.278|ul=GWh}} (which is often used in energy tables). About {{val|63|ul=TJ}} of energy was released by [[Little Boy]].<ref name="hironaga">{{cite web |url = http://www.mbe.doe.gov/me70/manhattan/publications/LANLHiroshimaNagasakiYields.pdf |title=Report LA-8819: The yields of the Hiroshima and Nagasaki nuclear explosions |last=Malik |first=John |date=September 1985 |publisher=[[Los Alamos National Laboratory]]|access-date=18 March 2015 |archive-url = https://web.archive.org/web/20091011030043/http://www.mbe.doe.gov/me70/manhattan/publications/LANLHiroshimaNagasakiYields.pdf |archive-date=11 October 2009 }}</ref> The [[International Space Station]], with a mass of approximately {{val|450|ul=megagrams}} and orbital velocity of {{val|7700|ul=m/s}},<ref name="iss">{{cite web |url = http://www.spaceflight.esa.int/users/downloads/factsheets/fs001_12_iss.pdf |title=International Space Station Final Configuration |publisher=[[European Space Agency]] |access-date=18 March 2015 |archive-url = https://web.archive.org/web/20110721012349/http://www.spaceflight.esa.int/users/downloads/factsheets/fs001_12_iss.pdf |archive-date=21 July 2011 }}</ref> has a [[kinetic energy]] of roughly {{val|13|u=TJ}}. In 2017, [[Hurricane Irma]] was estimated to have a peak wind energy of {{val|112|u=TJ}}.<ref>{{cite news|url=https://www.washingtonpost.com/graphics/2017/national/how-big-is-hurricane-irma/|title=Analysis – How Big Is Hurricane Irma?|newspaper=Washington Post|author1=Bonnie Berkowitz|author2=Laris Karklis|author3=Reuben Fischer-Baum|author4=Chiqui Esteban|date=11 September 2017|access-date=2 November 2017}}</ref><ref>{{Cite news |last=Rathbone |first=John-Paul |last2=Fontanella-Khan |first2=James |last3=Rovnick |first3=Naomi |date=11 September 2017 |title=A weakened Irma unleashes more damage on Florida coast |url=https://www.ft.com/content/2c58ce3e-9621-11e7-b83c-9588e51488a0 |url-status=live |archive-url=https://archive.today/20240804212211/https://webcache.googleusercontent.com/search?q=cache:https://www.ft.com/content/2c58ce3e-9621-11e7-b83c-9588e51488a0 |archive-date=4 August 2024 |access-date=11 September 2017 |work=[[Financial Times]] |location=New York (Rathbone), Miami (Fontanella-Khan), London (Rovnick) |issn=0307-1766}}</ref>  {{convert|1|GWh|TJ|disp=x|, of electricity or any other form of energy, is&nbsp;|spell=In|lk=in}}.
; {{vanchor|petajoule}} : {{val|210|u=petajoule}} is about {{val|50|ul=megatons}} of TNT, which is the amount of energy released by the [[Tsar Bomba]], the largest man-made explosion ever. {{convert|1|TWh|PJ|disp=x|, of electricity or any other form of energy, is&nbsp;|spell=In|lk=in}}.
; {{vanchor|exajoule}} : The [[2011 Tōhoku earthquake and tsunami]] in Japan had {{val|1.41|u=EJ}} of energy according to its rating of 9.0 on the [[moment magnitude scale]]. Yearly [[Energy in the United States|U.S. energy consumption]] amounts to roughly {{val|94|u=EJ}}, and the world final energy consumption was {{val|439|u=EJ}} in 2021.<ref>{{cite report |date=2022 |title=World Energy Outlook 2022 |url=https://www.iea.org/reports/world-energy-outlook-2022 |publisher=International Energy Agency |page=239 |access-date=7 September 2023 |quote=}}</ref>  One [[petawatt-hour]] of electricity, or any other form of energy, is {{convert|1000|TWh|EJ|disp=out}}.
; {{vanchor|zettajoule}} : The zettajoule is somewhat more than the amount of energy required to heat the [[Baltic Sea]] by 1&nbsp;°C, assuming properties similar to [[Properties of water|those of pure water]].<ref name="Volumes of the World's Oceans">{{cite web |title=Volumes of the World's Oceans from ETOPO1 |url=https://ngdc.noaa.gov/mgg/global/etopo1_ocean_volumes.html |website=noaa.gov |date=19 August 2020 |publisher=National Oceanic and Atmospheric Administration |access-date=8 March 2022}}</ref> Human annual [[world energy consumption]] is approximately {{val|0.5|u=ZJ}}.  The energy to raise the temperature of Earth's atmosphere 1&nbsp;°C is approximately {{val|2.2|u=ZJ}}.{{Citation needed|date=January 2024}}
; {{vanchor|yottajoule}} : The yottajoule is a little less than the amount of energy required to heat the [[Indian Ocean]] by 1&nbsp;°C, assuming properties similar to those of pure water.<ref name="Volumes of the World's Oceans" /> The thermal output of the [[Sun]] is approximately {{val|400|u=YJ}} per second.<ref name="pve">{{cite web |title=The Sun |url=https://www.pveducation.org/pvcdrom/properties-of-sunlight/the-sun |website=pveducation.org |access-date=22 November 2024}}</ref>

== Conversions ==
{{Main|Conversion of units of energy}}
1 joule is equal to (approximately unless otherwise stated):
* {{convert|1e0|J|erg|disp=out}} (exactly)
* {{convert|1|J|eV|disp=out|sigfig=6}}
* {{convert|1e0|J|BTU|disp=out|sigfig=6}}
* {{convert|1|J|ftlb|disp=out|sigfig=6}} (foot-pound)
* {{convert|1|J|ftpdl|disp=out|sigfig=6}} (foot-poundal)

Units with exact equivalents in joules include:
* 1 thermochemical [[calorie]] = 4.184{{nbsp}}J<ref name=FAO>[http://www.fao.org/docrep/meeting/009/ae906e/ae906e17.htm The adoption of joules as units of energy], FAO/WHO Ad Hoc Committee of Experts on Energy and Protein, 1971. A report on the changeover from calories to joules in nutrition.</ref>
* 1 International Table calorie = 4.1868{{nbsp}}J<ref>{{cite web|last=Feynman|first=Richard|author-link=Richard Feynman|title=Physical Units|url=http://www.numericana.com/answer/feynman.htm|work=Feynman's Lectures on Physics|year=1963|access-date=2014-03-07}}</ref>
* 1{{nbsp}}W⋅h = {{convert|1|Wh|J kJ|disp=out}}
* 1{{nbsp}}kW⋅h = {{convert|1|kWh|e6J MJ|disp=out}}
* 1{{nbsp}}W⋅s = {{val|1|u=J}}
* 1{{nbsp}}[[ton TNT]] = {{convert|1|tonTNT|GJ|3|disp=out}}
* 1{{nbsp}}[[foe (unit)|foe]] = {{val|e=44|u=J}}<ref>{{cite web |url=http://library.lanl.gov/cgi-bin/getfile?25-14.pdf |title=Neutrinos and Supernovae |author1=Marc Herant |author2=Stirling A. Colgate |author3=Willy Benz |author4=Chris Fryer |date=October 25, 1997 |work=Los Alamos Sciences |publisher=[[Los Alamos National Laboratory]] |access-date=2008-04-23 |archive-url=https://web.archive.org/web/20090114233021/http://library.lanl.gov/cgi-bin/getfile?25-14.pdf |archive-date=2009-01-14}}</ref>

== Newton-metre and torque ==

{{Main|Newton-metre}}
In [[mechanics]], the concept of [[force]] (in some direction) has a close analogue in the concept of [[torque]] (about some angle):{{Citation needed|date=January 2024}}

{| class="wikitable"
|-
! Linear   !! Angular
|-
| Force    || Torque
|-
| [[Mass]] || [[Moment of inertia]]
|-
| Displacement
| Angle
|}

A result of this similarity is that the SI unit for torque is the [[newton-metre]], which works out [[algebra]]ically to have the same [[dimensional analysis|dimension]]s as the joule, but they are not interchangeable. The [[General Conference on Weights and Measures]] has given the unit of [[energy]] the name ''joule'', but has not given the unit of torque any special name, hence it is simply the newton-metre (N⋅m) – a compound name derived from its constituent parts.<ref name="BIPM2" /> The use of newton-metres for torque but joules for energy is helpful to avoid misunderstandings and miscommunication.<ref name="BIPM2">{{cite web |url = http://www.bipm.org/en/si/si_brochure/chapter2/2-2/2-2-2.html |title=Units with special names and symbols; units that incorporate special names and symbols |publisher=[[International Bureau of Weights and Measures]] |access-date=18 March 2015 |archive-url = https://web.archive.org/web/20090628084157/http://www.bipm.org/en/si/si_brochure/chapter2/2-2/2-2-2.html |archive-date=28 June 2009 |quote = A derived unit can often be expressed in different ways by combining base units with derived units having special names. Joule, for example, may formally be written newton metre, or kilogram metre squared per second squared. This, however, is an algebraic freedom to be governed by common sense physical considerations; in a given situation some forms may be more helpful than others. In practice, with certain quantities, preference is given to the use of certain special unit names, or combinations of unit names, to facilitate the distinction between different quantities having the same dimension.}}</ref>

The distinction may be seen also in the fact that energy is a [[scalar (physics)|scalar]] quantity – the [[dot product]] of a force [[Euclidean vector|vector]] and a displacement vector. By contrast, torque is a vector – the [[cross product]] of a force vector and a distance vector. Torque and energy are related to one another by the equation{{Citation needed|date=January 2024}}
<math display="block">E = \tau \theta\, ,</math>

where ''E'' is energy, ''τ'' is (the [[magnitude (mathematics)#Euclidean vector space|vector magnitude]] of) torque, and ''θ'' is the angle swept (in [[radian]]s). Since plane angles are dimensionless, it follows that torque and energy have the same dimensions.{{Citation needed|date=January 2024}}

== Watt-second ==
A '''watt-second''' (symbol '''W&nbsp;s''' or '''W⋅s''') is a [[derived unit]] of [[energy]] equivalent to the joule.<ref>{{SIbrochure8th|pages=39–40, 53}}</ref> The watt-second is the energy equivalent to the power of one [[watt]] sustained for one [[second]].  While the watt-second is equivalent to the joule in both units and meaning, there are some contexts in which the term "watt-second" is used instead of "joule", such as in the rating of photographic [[Flash (photography)|electronic flash units]]. <ref>{{cite web|url=http://www.imaginginfo.com/print/Studio-Photography/What-Is-A-Watt-Second/3$1043|title=What Is A Watt Second?|access-date=2018-10-23|archive-date=2017-06-02|archive-url=https://web.archive.org/web/20170602221233/http://www.imaginginfo.com/print/Studio-Photography/What-Is-A-Watt-Second/3$1043|url-status=dead}}</ref>

== References ==
{{reflist|1=30em|liststyle=decimal}}

== External links ==
* {{wiktionary inline|joule}}
{{Spoken Wikipedia|date=2024-03-06|Joule.ogg}}

{{SI units}}
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[[Category:James Prescott Joule]]
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