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class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Flux as flow rate per unit area</span> </div> </a> <button aria-controls="toc-Flux_as_flow_rate_per_unit_area-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Flux as flow rate per unit area subsection</span> </button> <ul id="toc-Flux_as_flow_rate_per_unit_area-sublist" class="vector-toc-list"> <li id="toc-General_mathematical_definition_(transport)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_mathematical_definition_(transport)"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>General mathematical definition (transport)</span> </div> </a> <ul id="toc-General_mathematical_definition_(transport)-sublist" class="vector-toc-list"> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1.1</span> <span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Transport_fluxes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Transport_fluxes"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Transport fluxes</span> </div> </a> <ul id="toc-Transport_fluxes-sublist" class="vector-toc-list"> <li id="toc-Chemical_diffusion" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Chemical_diffusion"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Chemical diffusion</span> </div> </a> <ul id="toc-Chemical_diffusion-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Quantum_mechanics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_mechanics"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Quantum mechanics</span> </div> </a> <ul id="toc-Quantum_mechanics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Flux_as_a_surface_integral" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Flux_as_a_surface_integral"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Flux as a surface integral</span> </div> </a> <button aria-controls="toc-Flux_as_a_surface_integral-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Flux as a surface integral subsection</span> </button> <ul id="toc-Flux_as_a_surface_integral-sublist" class="vector-toc-list"> <li id="toc-General_mathematical_definition_(surface_integral)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_mathematical_definition_(surface_integral)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>General mathematical definition (surface integral)</span> </div> </a> <ul id="toc-General_mathematical_definition_(surface_integral)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Electromagnetism" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Electromagnetism"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Electromagnetism</span> </div> </a> <ul id="toc-Electromagnetism-sublist" class="vector-toc-list"> <li id="toc-Electric_flux" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Electric_flux"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Electric flux</span> </div> </a> <ul id="toc-Electric_flux-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Magnetic_flux" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Magnetic_flux"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.2</span> <span>Magnetic flux</span> </div> </a> <ul id="toc-Magnetic_flux-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Poynting_flux" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Poynting_flux"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.3</span> <span>Poynting flux</span> </div> </a> <ul id="toc-Poynting_flux-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-SI_radiometry_units" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#SI_radiometry_units"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>SI radiometry units</span> </div> </a> <ul id="toc-SI_radiometry_units-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Flux</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 42 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-42" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">42 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Fluks" title="Fluks – Afrikaans" lang="af" hreflang="af" data-title="Fluks" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%AF%D9%81%D9%82_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA_%D9%88%D9%81%D9%8A%D8%B2%D9%8A%D8%A7%D8%A1)" title="تدفق (رياضيات وفيزياء) – Arabic" lang="ar" hreflang="ar" data-title="تدفق (رياضيات وفيزياء)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AB%E0%A7%8D%E0%A6%B2%E0%A6%BE%E0%A6%95%E0%A7%8D%E0%A6%B8" title="ফ্লাক্স – Bangla" lang="bn" hreflang="bn" data-title="ফ্লাক্স" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%92%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%BB%D0%B0_%D1%83%D0%B9%C4%83%D0%BD_%D1%8E%D1%85%C4%83%D0%BC%C4%95" title="Векторла уйăн юхăмĕ – Chuvash" lang="cv" hreflang="cv" data-title="Векторла уйăн юхăмĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Flux" title="Flux – Danish" lang="da" hreflang="da" data-title="Flux" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Fluss_(Physik)" title="Fluss (Physik) – German" lang="de" hreflang="de" data-title="Fluss (Physik)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Voog" title="Voog – Estonian" lang="et" hreflang="et" data-title="Voog" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A1%CE%BF%CE%AE" title="Ροή – Greek" lang="el" hreflang="el" data-title="Ροή" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Flujo" title="Flujo – Spanish" lang="es" hreflang="es" data-title="Flujo" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Fluo" title="Fluo – Esperanto" lang="eo" hreflang="eo" data-title="Fluo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B4%D8%A7%D8%B1" title="شار – Persian" lang="fa" hreflang="fa" data-title="شار" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Flux_(physique)" title="Flux (physique) – French" lang="fr" hreflang="fr" data-title="Flux (physique)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Flosc_(fisic)" title="Flosc (fisic) – Irish" lang="ga" hreflang="ga" data-title="Flosc (fisic)" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Fluxo_(f%C3%ADsica)" title="Fluxo (física) – Galician" lang="gl" hreflang="gl" data-title="Fluxo (física)" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%84%A0%EC%86%8D" title="선속 – Korean" lang="ko" hreflang="ko" data-title="선속" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8E%D5%A5%D5%AF%D5%BF%D5%B8%D6%80%D5%A1%D5%AF%D5%A1%D5%B6_%D5%A4%D5%A1%D5%B7%D5%BF%D5%AB_%D5%B0%D5%B8%D5%BD%D6%84" title="Վեկտորական դաշտի հոսք – Armenian" lang="hy" hreflang="hy" data-title="Վեկտորական դաշտի հոսք" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%AD%E0%A4%BF%E0%A4%B5%E0%A4%BE%E0%A4%B9" title="अभिवाह – Hindi" lang="hi" hreflang="hi" data-title="अभिवाह" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Tok_polja" title="Tok polja – Croatian" lang="hr" hreflang="hr" data-title="Tok polja" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Fluks" title="Fluks – Indonesian" lang="id" hreflang="id" data-title="Fluks" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Flusso" title="Flusso – Italian" lang="it" hreflang="it" data-title="Flusso" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A9%D7%98%D7%A3" title="שטף – Hebrew" lang="he" hreflang="he" data-title="שטף" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Fluxus" title="Fluxus – Hungarian" lang="hu" hreflang="hu" data-title="Fluxus" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%AA%E0%A5%8D%E0%A4%B0%E0%A4%B5%E0%A4%BE%E0%A4%B9_(%E0%A4%AD%E0%A5%8C%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A5%80)" title="प्रवाह (भौतिकी) – Marathi" lang="mr" hreflang="mr" data-title="प्रवाह (भौतिकी)" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Flux_(wis-_en_natuurkunde)" title="Flux (wis- en natuurkunde) – Dutch" lang="nl" hreflang="nl" data-title="Flux (wis- en natuurkunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%B5%81%E6%9D%9F" title="流束 – Japanese" lang="ja" hreflang="ja" data-title="流束" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Fluks" title="Fluks – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Fluks" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Fluks" title="Fluks – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Fluks" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Strumie%C5%84_pola" title="Strumień pola – Polish" lang="pl" hreflang="pl" data-title="Strumień pola" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Fluxo_(f%C3%ADsica)" title="Fluxo (física) – Portuguese" lang="pt" hreflang="pt" data-title="Fluxo (física)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Flux" title="Flux – Romanian" lang="ro" hreflang="ro" data-title="Flux" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D1%82%D0%BE%D0%BA_%D0%B2%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%BD%D0%BE%D0%B3%D0%BE_%D0%BF%D0%BE%D0%BB%D1%8F" title="Поток векторного поля – Russian" lang="ru" hreflang="ru" data-title="Поток векторного поля" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Fluksi" title="Fluksi – Albanian" lang="sq" hreflang="sq" data-title="Fluksi" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Flux" title="Flux – Simple English" lang="en-simple" hreflang="en-simple" data-title="Flux" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Tok_polja" title="Tok polja – Serbian" lang="sr" hreflang="sr" data-title="Tok polja" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Tok_polja" title="Tok polja – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Tok polja" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Vuo" title="Vuo – Finnish" lang="fi" hreflang="fi" data-title="Vuo" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Fl%C3%B6de" title="Flöde – Swedish" lang="sv" hreflang="sv" data-title="Flöde" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Ak%C4%B1" title="Akı – Turkish" lang="tr" hreflang="tr" data-title="Akı" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%BE%D1%82%D1%96%D0%BA_%D0%B2%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%BD%D0%BE%D0%B3%D0%BE_%D0%BF%D0%BE%D0%BB%D1%8F" title="Потік векторного поля – Ukrainian" lang="uk" hreflang="uk" data-title="Потік векторного поля" 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.hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about the concept of flux in natural science and mathematics. For other uses, see <a href="/wiki/Flux_(disambiguation)" class="mw-disambig" title="Flux (disambiguation)">Flux (disambiguation)</a>.</div> <p><b>Flux</b> describes any effect that appears to pass or travel (whether it actually moves or not) through a <a href="/wiki/Surface" title="Surface">surface</a> or substance. Flux is a concept in <a href="/wiki/Applied_mathematics" title="Applied mathematics">applied mathematics</a> and <a href="/wiki/Vector_calculus" title="Vector calculus">vector calculus</a> which has many applications in <a href="/wiki/Physics" title="Physics">physics</a>. For <a href="/wiki/Transport_phenomena" title="Transport phenomena">transport phenomena</a>, flux is a <a href="/wiki/Euclidean_vector" title="Euclidean vector">vector</a> quantity, describing the magnitude and direction of the flow of a substance or property. In <a href="/wiki/Vector_calculus" title="Vector calculus">vector calculus</a> flux is a <a href="/wiki/Scalar_(physics)" title="Scalar (physics)">scalar</a> quantity, defined as the <a href="/wiki/Surface_integral" title="Surface integral">surface integral</a> of the perpendicular component of a <a href="/wiki/Vector_field" title="Vector field">vector field</a> over a surface.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Terminology">Terminology</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Flux&action=edit&section=1" title="Edit section: Terminology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The word <i>flux</i> comes from <a href="/wiki/Latin" title="Latin">Latin</a>: <i>fluxus</i> means "flow", and <i>fluere</i> is "to flow".<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> As <i><a href="/wiki/Method_of_Fluxions" title="Method of Fluxions">fluxion</a></i>, this term was introduced into <a href="/wiki/Differential_calculus" title="Differential calculus">differential calculus</a> by <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a>. </p><p>The concept of <a href="/wiki/Heat_flux" title="Heat flux">heat flux</a> was a key contribution of <a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a>, in the analysis of heat transfer phenomena.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> His seminal treatise <i>Théorie analytique de la chaleur</i> (<i>The Analytical Theory of Heat</i>),<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> defines <i>fluxion</i> as a central quantity and proceeds to derive the now well-known expressions of flux in terms of temperature differences across a slab, and then more generally in terms of temperature gradients or differentials of temperature, across other geometries. One could argue, based on the work of <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a>,<sup id="cite_ref-Maxwell_5-0" class="reference"><a href="#cite_note-Maxwell-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> that the transport definition precedes the <a href="/wiki/Magnetic_flux" title="Magnetic flux">definition of flux used in electromagnetism</a>. The specific quote from Maxwell is: </p> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>In the case of fluxes, we have to take the integral, over a surface, of the flux through every element of the surface. The result of this operation is called the <a href="/wiki/Surface_integral" title="Surface integral">surface integral</a> of the flux. It represents the quantity which passes through the surface. </p><div class="templatequotecite">— <cite>James Clerk Maxwell</cite></div></blockquote> <p>According to the transport definition, flux may be a single vector, or it may be a vector field / function of position. In the latter case flux can readily be integrated over a surface. By contrast, according to the electromagnetism definition, flux <i>is</i> the integral over a surface; it makes no sense to integrate a second-definition flux for one would be integrating over a surface twice. Thus, Maxwell's quote only makes sense if "flux" is being used according to the transport definition (and furthermore is a vector field rather than single vector). This is ironic because Maxwell was one of the major developers of what we now call "electric flux" and "magnetic flux" according to the electromagnetism definition. Their names in accordance with the quote (and transport definition) would be "surface integral of electric flux" and "surface integral of magnetic flux", in which case "electric flux" would instead be defined as "electric field" and "magnetic flux" defined as "magnetic field". This implies that Maxwell conceived of these fields as flows/fluxes of some sort. </p><p>Given a flux according to the electromagnetism definition, the corresponding <b>flux density</b>, if that term is used, refers to its derivative along the surface that was integrated. By the <a href="/wiki/Fundamental_theorem_of_calculus" title="Fundamental theorem of calculus">Fundamental theorem of calculus</a>, the corresponding <b>flux density</b> is a flux according to the transport definition. Given a <b>current</b> such as electric current—charge per time, <b>current density</b> would also be a flux according to the transport definition—charge per time per area. Due to the conflicting definitions of <i>flux</i>, and the interchangeability of <i>flux</i>, <i>flow</i>, and <i>current</i> in nontechnical English, all of the terms used in this paragraph are sometimes used interchangeably and ambiguously. Concrete fluxes in the rest of this article will be used in accordance to their broad acceptance in the literature, regardless of which definition of flux the term corresponds to. </p> <div class="mw-heading mw-heading2"><h2 id="Flux_as_flow_rate_per_unit_area">Flux as flow rate per unit area</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Flux&action=edit&section=2" title="Edit section: Flux as flow rate per unit area"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Transport_phenomena" title="Transport phenomena">transport phenomena</a> (<a href="/wiki/Heat_transfer" title="Heat transfer">heat transfer</a>, <a href="/wiki/Mass_transfer" title="Mass transfer">mass transfer</a> and <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a>), flux is defined as the <i>rate of flow of a property per unit area</i>, which has the <a href="/wiki/Dimensional_analysis" title="Dimensional analysis">dimensions</a> [quantity]·[time]<sup>−1</sup>·[area]<sup>−1</sup>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The area is of the surface the property is flowing "through" or "across". For example, the amount of water that flows through a cross section of a river each second divided by the area of that cross section, or the amount of sunlight energy that lands on a patch of ground each second divided by the area of the patch, are kinds of flux. </p> <div class="mw-heading mw-heading3"><h3 id="General_mathematical_definition_(transport)"><span id="General_mathematical_definition_.28transport.29"></span>General mathematical definition (transport)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Flux&action=edit&section=3" title="Edit section: General mathematical definition (transport)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:General_flux_diagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/General_flux_diagram.svg/330px-General_flux_diagram.svg.png" decoding="async" width="330" height="542" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/General_flux_diagram.svg/495px-General_flux_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/General_flux_diagram.svg/660px-General_flux_diagram.svg.png 2x" data-file-width="227" data-file-height="373" /></a><figcaption>The <a href="/wiki/Field_line" title="Field line">field lines</a> of a <a href="/wiki/Vector_field" title="Vector field">vector field</a> <span class="texhtml"><b>F</b></span> through surfaces with <a href="/wiki/Unit_vector" title="Unit vector">unit</a> normal <span class="texhtml"><b>n</b></span>, the angle from <span class="texhtml"><b>n</b></span> to <span class="texhtml"><b>F</b></span> is <span class="texhtml mvar" style="font-style:italic;">θ</span>. Flux is a measure of how much of the field passes through a given surface. <span class="texhtml"><b>F</b></span> is decomposed into components perpendicular (⊥) and parallel <span class="nowrap">( ‖ )</span> to <span class="texhtml"><b>n</b></span>. Only the parallel component contributes to flux because it is the maximum extent of the field passing through the surface at a point, the perpendicular component does not contribute. <br /><b>Top:</b> Three field lines through a plane surface, one normal to the surface, one parallel, and one intermediate. <br /><b>Bottom:</b> Field line through a <a href="/wiki/Curved_surface" class="mw-redirect" title="Curved surface">curved surface</a>, showing the setup of the unit normal and surface element to calculate flux.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Surface_integral_-_definition.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Surface_integral_-_definition.svg/330px-Surface_integral_-_definition.svg.png" decoding="async" width="330" height="149" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Surface_integral_-_definition.svg/495px-Surface_integral_-_definition.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bd/Surface_integral_-_definition.svg/660px-Surface_integral_-_definition.svg.png 2x" data-file-width="814" data-file-height="368" /></a><figcaption>To calculate the flux of a vector field <span class="texhtml"><b>F</b></span> <i>(red arrows)</i> through a surface <span class="texhtml mvar" style="font-style:italic;">S</span> the surface is divided into small patches <span class="texhtml mvar" style="font-style:italic;">dS</span>. The flux through each patch is equal to the normal (perpendicular) component of the field, the <a href="/wiki/Dot_product" title="Dot product">dot product</a> of <span class="texhtml"><b>F</b>(<b>x</b>)</span> with the unit normal vector <span class="texhtml"><b>n</b>(<b>x</b>)</span> <i>(blue arrows)</i> at the point <span class="texhtml"><b>x</b></span> multiplied by the area <span class="texhtml mvar" style="font-style:italic;">dS</span>. The sum of <span class="texhtml"><b>F</b> · <b>n</b>, <i>dS</i></span> for each patch on the surface is the flux through the surface.</figcaption></figure> <p>Here are 3 definitions in increasing order of complexity. Each is a special case of the following. In all cases the frequent symbol <i>j</i>, (or <i>J</i>) is used for flux, <i>q</i> for the <a href="/wiki/Physical_quantity" title="Physical quantity">physical quantity</a> that flows, <i>t</i> for time, and <i>A</i> for area. These identifiers will be written in bold when and only when they are vectors. </p><p>First, flux as a (single) scalar: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j={\frac {I}{A}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>I</mi> <mi>A</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j={\frac {I}{A}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79906be24ab8dcf6418997cf59df577514047f85" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-left: -0.027ex; width:7.309ex; height:5.343ex;" alt="{\displaystyle j={\frac {I}{A}},}" /></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=\lim _{\Delta t\to 0}{\frac {\Delta q}{\Delta t}}={\frac {\mathrm {d} q}{\mathrm {d} t}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>q</mi> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>q</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=\lim _{\Delta t\to 0}{\frac {\Delta q}{\Delta t}}={\frac {\mathrm {d} q}{\mathrm {d} t}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7252331d357d6613a21d777f65b91043ef976db" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.87ex; height:5.676ex;" alt="{\displaystyle I=\lim _{\Delta t\to 0}{\frac {\Delta q}{\Delta t}}={\frac {\mathrm {d} q}{\mathrm {d} t}}.}" /></span> In this case the surface in which flux is being measured is fixed and has area <i>A</i>. The surface is assumed to be flat, and the flow is assumed to be everywhere constant with respect to position and perpendicular to the surface. </p><p>Second, flux as a <a href="/wiki/Scalar_field" title="Scalar field">scalar field</a> defined along a surface, i.e. a function of points on the surface: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j(\mathbf {p} )={\frac {\partial I}{\partial A}}(\mathbf {p} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>I</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>A</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j(\mathbf {p} )={\frac {\partial I}{\partial A}}(\mathbf {p} ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c938afbd2e0eb005723ff54af65c849ef0b345c9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-left: -0.027ex; width:15.217ex; height:5.509ex;" alt="{\displaystyle j(\mathbf {p} )={\frac {\partial I}{\partial A}}(\mathbf {p} ),}" /></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(A,\mathbf {p} )={\frac {\mathrm {d} q}{\mathrm {d} t}}(A,\mathbf {p} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>q</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(A,\mathbf {p} )={\frac {\mathrm {d} q}{\mathrm {d} t}}(A,\mathbf {p} ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30e74fde1aa11d7452e2c29b97db1c4cf9c08575" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.259ex; height:5.509ex;" alt="{\displaystyle I(A,\mathbf {p} )={\frac {\mathrm {d} q}{\mathrm {d} t}}(A,\mathbf {p} ).}" /></span> As before, the surface is assumed to be flat, and the flow is assumed to be everywhere perpendicular to it. However the flow need not be constant. <i>q</i> is now a function of <b>p</b>, a point on the surface, and <i>A</i>, an area. Rather than measure the total flow through the surface, <i>q</i> measures the flow through the disk with area <i>A</i> centered at <i>p</i> along the surface. </p><p>Finally, flux as a <a href="/wiki/Vector_field" title="Vector field">vector field</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {j} (\mathbf {p} )={\frac {\partial \mathbf {I} }{\partial A}}(\mathbf {p} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>A</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {j} (\mathbf {p} )={\frac {\partial \mathbf {I} }{\partial A}}(\mathbf {p} ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f77da542e827989acdb1a4d3d73c01a404051f1d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-left: -0.164ex; width:15.212ex; height:5.509ex;" alt="{\displaystyle \mathbf {j} (\mathbf {p} )={\frac {\partial \mathbf {I} }{\partial A}}(\mathbf {p} ),}" /></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {I} (A,\mathbf {p} )={\underset {\mathbf {\hat {n}} }{\operatorname {arg\,max} }}\mathbf {\hat {n}} _{\mathbf {p} }{\frac {\mathrm {d} q}{\mathrm {d} t}}(A,\mathbf {p} ,\mathbf {\hat {n}} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">g</mi> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">n</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </munder> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">n</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>q</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">n</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {I} (A,\mathbf {p} )={\underset {\mathbf {\hat {n}} }{\operatorname {arg\,max} }}\mathbf {\hat {n}} _{\mathbf {p} }{\frac {\mathrm {d} q}{\mathrm {d} t}}(A,\mathbf {p} ,\mathbf {\hat {n}} ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/578fcf296c8064ae820e4809ce0a7f206f20d0d3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:33.337ex; height:6.343ex;" alt="{\displaystyle \mathbf {I} (A,\mathbf {p} )={\underset {\mathbf {\hat {n}} }{\operatorname {arg\,max} }}\mathbf {\hat {n}} _{\mathbf {p} }{\frac {\mathrm {d} q}{\mathrm {d} t}}(A,\mathbf {p} ,\mathbf {\hat {n}} ).}" /></span> In this case, there is no fixed surface we are measuring over. <i>q</i> is a function of a point, an area, and a direction (given by a unit vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {n}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">n</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {n}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4eb84e133d15551d660800ec29b44783ff36e19d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.343ex;" alt="{\displaystyle \mathbf {\hat {n}} }" /></span>), and measures the flow through the disk of area A perpendicular to that unit vector. <i>I</i> is defined picking the unit vector that maximizes the flow around the point, because the true flow is maximized across the disk that is perpendicular to it. The unit vector thus uniquely maximizes the function when it points in the "true direction" of the flow. (Strictly speaking, this is an <a href="/wiki/Abuse_of_notation" title="Abuse of notation">abuse of notation</a> because the "arg max" cannot directly compare vectors; we take the vector with the biggest norm instead.) </p> <div class="mw-heading mw-heading4"><h4 id="Properties">Properties</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Flux&action=edit&section=4" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>These direct definitions, especially the last, are rather unwieldy. For example, the arg max construction is artificial from the perspective of empirical measurements, when with a <a href="/wiki/Weathervane" class="mw-redirect" title="Weathervane">weathervane</a> or similar one can easily deduce the direction of flux at a point. Rather than defining the vector flux directly, it is often more intuitive to state some properties about it. Furthermore, from these properties the flux can uniquely be determined anyway. </p><p>If the flux <b>j</b> passes through the area at an angle θ to the area normal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {n}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">n</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {n}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4eb84e133d15551d660800ec29b44783ff36e19d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.343ex;" alt="{\displaystyle \mathbf {\hat {n}} }" /></span>, then the <a href="/wiki/Dot_product" title="Dot product">dot product</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {j} \cdot \mathbf {\hat {n}} =j\cos \theta .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">n</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>j</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {j} \cdot \mathbf {\hat {n}} =j\cos \theta .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77aed941d8d4fc933c96411460dfb5cec7f691f1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.164ex; width:13.824ex; height:2.676ex;" alt="{\displaystyle \mathbf {j} \cdot \mathbf {\hat {n}} =j\cos \theta .}" /></span> That is, the component of flux passing through the surface (i.e. normal to it) is <i>j</i> cos <i>θ</i>, while the component of flux passing tangential to the area is <i>j</i> sin <i>θ</i>, but there is <i>no</i> flux actually passing <i>through</i> the area in the tangential direction. The <i>only</i> component of flux passing normal to the area is the cosine component. </p><p>For vector flux, the <a href="/wiki/Surface_integral" title="Surface integral">surface integral</a> of <b>j</b> over a <a href="/wiki/Surface_(mathematics)" title="Surface (mathematics)">surface</a> <i>S</i>, gives the proper flowing per unit of time through the surface: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\mathrm {d} q}{\mathrm {d} t}}=\iint _{S}\mathbf {j} \cdot \mathbf {\hat {n}} \,dA=\iint _{S}\mathbf {j} \cdot d\mathbf {A} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>q</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">n</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>A</mi> <mo>=</mo> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} q}{\mathrm {d} t}}=\iint _{S}\mathbf {j} \cdot \mathbf {\hat {n}} \,dA=\iint _{S}\mathbf {j} \cdot d\mathbf {A} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2745674f547f5a46a8710df22c42f53546c330bc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:31.496ex; height:5.843ex;" alt="{\displaystyle {\frac {\mathrm {d} q}{\mathrm {d} t}}=\iint _{S}\mathbf {j} \cdot \mathbf {\hat {n}} \,dA=\iint _{S}\mathbf {j} \cdot d\mathbf {A} ,}" /></span> where <b>A</b> (and its infinitesimal) is the <a href="/wiki/Vector_area" title="Vector area">vector area</a> –  combination <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} =A\mathbf {\hat {n}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">n</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} =A\mathbf {\hat {n}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/176a3541048716f82aa79f9bdbb137fb3c44219e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.346ex; height:2.343ex;" alt="{\displaystyle \mathbf {A} =A\mathbf {\hat {n}} }" /></span> of the magnitude of the area <i>A</i> through which the property passes and a <a href="/wiki/Unit_vector" title="Unit vector">unit vector</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {n}} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">n</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {n}} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4eb84e133d15551d660800ec29b44783ff36e19d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.343ex;" alt="{\displaystyle \mathbf {\hat {n}} }" /></span> normal to the area. Unlike in the second set of equations, the surface here need not be flat. </p><p>Finally, we can integrate again over the time duration <i>t</i><sub>1</sub> to <i>t</i><sub>2</sub>, getting the total amount of the property flowing through the surface in that time (<i>t</i><sub>2</sub> − <i>t</i><sub>1</sub>): <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=\int _{t_{1}}^{t_{2}}\iint _{S}\mathbf {j} \cdot d\mathbf {A} \,dt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">j</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=\int _{t_{1}}^{t_{2}}\iint _{S}\mathbf {j} \cdot d\mathbf {A} \,dt.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ea9508e91bffae317c1102564435ad7cac2ee18" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.695ex; height:6.509ex;" alt="{\displaystyle q=\int _{t_{1}}^{t_{2}}\iint _{S}\mathbf {j} \cdot d\mathbf {A} \,dt.}" /></span> </p> <div class="mw-heading mw-heading3"><h3 id="Transport_fluxes">Transport fluxes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Flux&action=edit&section=5" title="Edit section: Transport fluxes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Eight of the most common forms of flux from the transport phenomena literature are defined as follows: </p> <ol><li><a href="/wiki/Transport_phenomena#Momentum_transfer" title="Transport phenomena">Momentum flux</a>, the rate of transfer of <a href="/wiki/Momentum" title="Momentum">momentum</a> across a unit area (N·s·m<sup>−2</sup>·s<sup>−1</sup>). (<a href="/wiki/Newton%27s_law_of_viscosity" class="mw-redirect" title="Newton's law of viscosity">Newton's law of viscosity</a>)<sup id="cite_ref-Physics_P.M_7-0" class="reference"><a href="#cite_note-Physics_P.M-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Heat_flux" title="Heat flux">Heat flux</a>, the rate of <a href="/wiki/Heat" title="Heat">heat</a> flow across a unit area (J·m<sup>−2</sup>·s<sup>−1</sup>). (<a href="/wiki/Fourier%27s_law" class="mw-redirect" title="Fourier's law">Fourier's law of conduction</a>)<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> (This definition of heat flux fits Maxwell's original definition.)<sup id="cite_ref-Maxwell_5-1" class="reference"><a href="#cite_note-Maxwell-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Diffusion_flux" class="mw-redirect" title="Diffusion flux">Diffusion flux</a>, the rate of movement of molecules across a unit area (mol·m<sup>−2</sup>·s<sup>−1</sup>). (<a href="/wiki/Fick%27s_law_of_diffusion" class="mw-redirect" title="Fick's law of diffusion">Fick's law of diffusion</a>)<sup id="cite_ref-Physics_P.M_7-1" class="reference"><a href="#cite_note-Physics_P.M-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Volumetric_flux" title="Volumetric flux">Volumetric flux</a>, the rate of <a href="/wiki/Volume" title="Volume">volume</a> flow across a unit area (m<sup>3</sup>·m<sup>−2</sup>·s<sup>−1</sup>). (<a href="/wiki/Darcy%27s_law" title="Darcy's law">Darcy's law of groundwater flow</a>)</li> <li><a href="/wiki/Mass_flux" title="Mass flux">Mass flux</a>, the rate of <a href="/wiki/Mass" title="Mass">mass</a> flow across a unit area (kg·m<sup>−2</sup>·s<sup>−1</sup>). (Either an alternate form of Fick's law that includes the molecular mass, or an alternate form of Darcy's law that includes the density.)</li> <li><a href="/wiki/Radiative_flux" title="Radiative flux">Radiative flux</a>, the amount of energy transferred in the form of <a href="/wiki/Photons" class="mw-redirect" title="Photons">photons</a> at a certain distance from the source per unit area per second (J·m<sup>−2</sup>·s<sup>−1</sup>). Used in astronomy to determine the <a href="/wiki/Magnitude_(astronomy)" title="Magnitude (astronomy)">magnitude</a> and <a href="/wiki/Spectral_class" class="mw-redirect" title="Spectral class">spectral class</a> of a star. Also acts as a generalization of heat flux, which is equal to the radiative flux when restricted to the electromagnetic spectrum.</li> <li><a href="/wiki/Energy_flux" title="Energy flux">Energy flux</a>, the rate of transfer of <a href="/wiki/Energy" title="Energy">energy</a> through a unit area (J·m<sup>−2</sup>·s<sup>−1</sup>). The radiative flux and heat flux are specific cases of energy flux.</li> <li><a href="/w/index.php?title=Particle_flux&action=edit&redlink=1" class="new" title="Particle flux (page does not exist)">Particle flux</a>, the rate of transfer of particles through a unit area ([number of particles] m<sup>−2</sup>·s<sup>−1</sup>)</li></ol> <p>These fluxes are vectors at each point in space, and have a definite magnitude and direction. Also, one can take the <a href="/wiki/Divergence" title="Divergence">divergence</a> of any of these fluxes to determine the accumulation rate of the quantity in a control volume around a given point in space. For <a href="/wiki/Incompressible_flow" title="Incompressible flow">incompressible flow</a>, the divergence of the volume flux is zero. </p> <div class="mw-heading mw-heading4"><h4 id="Chemical_diffusion">Chemical diffusion</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Flux&action=edit&section=6" title="Edit section: Chemical diffusion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As mentioned above, chemical <a href="/wiki/Mass_flux#Molar_fluxes" title="Mass flux">molar flux</a> of a component A in an <a href="/wiki/Isothermal" class="mw-redirect" title="Isothermal">isothermal</a>, <a href="/wiki/Isobaric_process" title="Isobaric process">isobaric system</a> is defined in <a href="/wiki/Fick%27s_law_of_diffusion" class="mw-redirect" title="Fick's law of diffusion">Fick's law of diffusion</a> as: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {J} _{A}=-D_{AB}\nabla c_{A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>B</mi> </mrow> </msub> <mi mathvariant="normal">∇</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J} _{A}=-D_{AB}\nabla c_{A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c43c685a30b0f2c43a1b1859311d9f88808e415" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.796ex; height:2.509ex;" alt="{\displaystyle \mathbf {J} _{A}=-D_{AB}\nabla c_{A}}" /></span> where the <a href="/wiki/Nabla_symbol" title="Nabla symbol">nabla symbol</a> ∇ denotes the <a href="/wiki/Gradient" title="Gradient">gradient</a> operator, <i>D<sub>AB</sub></i> is the diffusion coefficient (m<sup>2</sup>·s<sup>−1</sup>) of component A diffusing through component B, <i>c<sub>A</sub></i> is the <a href="/wiki/Concentration" title="Concentration">concentration</a> (<a href="/wiki/Mole_(unit)" title="Mole (unit)">mol</a>/m<sup>3</sup>) of component A.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>This flux has units of mol·m<sup>−2</sup>·s<sup>−1</sup>, and fits Maxwell's original definition of flux.<sup id="cite_ref-Maxwell_5-2" class="reference"><a href="#cite_note-Maxwell-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>For dilute gases, kinetic molecular theory relates the diffusion coefficient <i>D</i> to the particle density <i>n</i> = <i>N</i>/<i>V</i>, the molecular mass <i>m</i>, the collision <a href="/wiki/Cross_section_(physics)" title="Cross section (physics)">cross section</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" /></span>, and the <a href="/wiki/Thermodynamic_temperature" title="Thermodynamic temperature">absolute temperature</a> <i>T</i> by <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D={\frac {2}{3n\sigma }}{\sqrt {\frac {kT}{\pi m}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mn>3</mn> <mi>n</mi> <mi>σ</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mi>k</mi> <mi>T</mi> </mrow> <mrow> <mi>π</mi> <mi>m</mi> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D={\frac {2}{3n\sigma }}{\sqrt {\frac {kT}{\pi m}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa1c3e054c02105cb76de41ea96b9f336f8202fa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.278ex; height:6.176ex;" alt="{\displaystyle D={\frac {2}{3n\sigma }}{\sqrt {\frac {kT}{\pi m}}}}" /></span> where the second factor is the <a href="/wiki/Mean_free_path" title="Mean free path">mean free path</a> and the square root (with the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a> <i>k</i>) is the <a href="/wiki/Maxwell%E2%80%93Boltzmann_distribution#Typical_speeds" title="Maxwell–Boltzmann distribution">mean velocity</a> of the particles. </p><p>In turbulent flows, the transport by eddy motion can be expressed as a grossly increased diffusion coefficient. </p> <div class="mw-heading mw-heading3"><h3 id="Quantum_mechanics">Quantum mechanics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Flux&action=edit&section=7" title="Edit section: Quantum mechanics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Probability_current" title="Probability current">Probability current</a></div> <p>In <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, particles of mass <i>m</i> in the <a href="/wiki/Quantum_state" title="Quantum state">quantum state</a> <i>ψ</i>(<b>r</b>, <i>t</i>) have a <a href="/wiki/Probability_amplitude" title="Probability amplitude">probability density</a> defined as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =\psi ^{*}\psi =|\psi |^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>ψ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =\psi ^{*}\psi =|\psi |^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5815f37de18e0cde2c512bb8ded3165d2756e5ef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.987ex; height:3.343ex;" alt="{\displaystyle \rho =\psi ^{*}\psi =|\psi |^{2}.}" /></span> So the probability of finding a particle in a differential <a href="/wiki/Volume_element" title="Volume element">volume element</a> d<sup>3</sup><b>r</b> is <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dP=|\psi |^{2}\,d^{3}\mathbf {r} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dP=|\psi |^{2}\,d^{3}\mathbf {r} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/171a2669529517aceddd29b75081c06e7ceb3e94" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.329ex; height:3.343ex;" alt="{\displaystyle dP=|\psi |^{2}\,d^{3}\mathbf {r} .}" /></span> Then the number of particles passing perpendicularly through unit area of a <a href="/wiki/Cross_section_(geometry)" title="Cross section (geometry)">cross-section</a> per unit time is the probability flux; <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {J} ={\frac {i\hbar }{2m}}\left(\psi \nabla \psi ^{*}-\psi ^{*}\nabla \psi \right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>i</mi> <mi class="MJX-variant">ℏ</mi> </mrow> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>ψ</mi> <mi mathvariant="normal">∇</mi> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>−</mo> <msup> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi mathvariant="normal">∇</mi> <mi>ψ</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J} ={\frac {i\hbar }{2m}}\left(\psi \nabla \psi ^{*}-\psi ^{*}\nabla \psi \right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/701531854e0d41bd20a41afa49c23d9c20a5575b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.622ex; height:5.343ex;" alt="{\displaystyle \mathbf {J} ={\frac {i\hbar }{2m}}\left(\psi \nabla \psi ^{*}-\psi ^{*}\nabla \psi \right).}" /></span> This is sometimes referred to as the probability current or current density,<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> or probability flux density.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Flux_as_a_surface_integral">Flux as a surface integral</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Flux&action=edit&section=8" title="Edit section: Flux as a surface integral"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="General_mathematical_definition_(surface_integral)"><span id="General_mathematical_definition_.28surface_integral.29"></span>General mathematical definition (surface integral)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Flux&action=edit&section=9" title="Edit section: General mathematical definition (surface integral)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Flux_diagram.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/72/Flux_diagram.png/260px-Flux_diagram.png" decoding="async" width="260" height="379" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/7/72/Flux_diagram.png 1.5x" data-file-width="384" data-file-height="560" /></a><figcaption>The flux visualized. The rings show the surface boundaries. The red arrows stand for the flow of charges, fluid particles, subatomic particles, photons, etc. The number of arrows that pass through each ring is the flux.</figcaption></figure> <p>As a mathematical concept, flux is represented by the <a href="/wiki/Surface_integral#Surface_integrals_of_vector_fields" title="Surface integral">surface integral of a vector field</a>,<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{F}=\iint _{A}\mathbf {F} \cdot \mathrm {d} \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mo>=</mo> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{F}=\iint _{A}\mathbf {F} \cdot \mathrm {d} \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07244c51343ac81dcd4963831a1eaa3ddeef39d1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.284ex; height:5.676ex;" alt="{\displaystyle \Phi _{F}=\iint _{A}\mathbf {F} \cdot \mathrm {d} \mathbf {A} }" /></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{F}=\iint _{A}\mathbf {F} \cdot \mathbf {n} \,\mathrm {d} A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mo>=</mo> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{F}=\iint _{A}\mathbf {F} \cdot \mathbf {n} \,\mathrm {d} A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22e75dac029ff62839c2c1fa416595dda86580a0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.88ex; height:5.676ex;" alt="{\displaystyle \Phi _{F}=\iint _{A}\mathbf {F} \cdot \mathbf {n} \,\mathrm {d} A}" /></span> where <b>F</b> is a <a href="/wiki/Vector_field" title="Vector field">vector field</a>, and d<b>A</b> is the <a href="/wiki/Vector_area" title="Vector area">vector area</a> of the surface <i>A</i>, directed as the <a href="/wiki/Normal_(geometry)" title="Normal (geometry)">surface normal</a>. For the second, <b>n</b> is the outward pointed <a href="/wiki/Unit_normal_vector" class="mw-redirect" title="Unit normal vector">unit normal vector</a> to the surface. </p><p>The surface has to be <a href="/wiki/Orientability" title="Orientability">orientable</a>, i.e. two sides can be distinguished: the surface does not fold back onto itself. Also, the surface has to be actually oriented, i.e. we use a convention as to flowing which way is counted positive; flowing backward is then counted negative. </p><p>The surface normal is usually directed by the <a href="/wiki/Right-hand_rule" title="Right-hand rule">right-hand rule</a>. </p><p>Conversely, one can consider the flux the more fundamental quantity and call the vector field the flux density. </p><p>Often a vector field is drawn by curves (field lines) following the "flow"; the magnitude of the vector field is then the line density, and the flux through a surface is the number of lines. Lines originate from areas of positive <a href="/wiki/Divergence" title="Divergence">divergence</a> (sources) and end at areas of negative divergence (sinks). </p><p>See also the image at right: the number of red arrows passing through a unit area is the flux density, the <a href="/wiki/Curve" title="Curve">curve</a> encircling the red arrows denotes the boundary of the surface, and the orientation of the arrows with respect to the surface denotes the sign of the <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner product</a> of the vector field with the surface normals. </p><p>If the surface encloses a 3D region, usually the surface is oriented such that the <b>influx</b> is counted positive; the opposite is the <b>outflux</b>. </p><p>The <a href="/wiki/Divergence_theorem" title="Divergence theorem">divergence theorem</a> states that the net outflux through a closed surface, in other words the net outflux from a 3D region, is found by adding the local net outflow from each point in the region (which is expressed by the <a href="/wiki/Divergence" title="Divergence">divergence</a>). </p><p>If the surface is not closed, it has an oriented curve as boundary. <a href="/wiki/Stokes%27_theorem" title="Stokes' theorem">Stokes' theorem</a> states that the flux of the <a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">curl</a> of a vector field is the <a href="/wiki/Line_integral" title="Line integral">line integral</a> of the vector field over this boundary. This path integral is also called <a href="/wiki/Circulation_(fluid_dynamics)" class="mw-redirect" title="Circulation (fluid dynamics)">circulation</a>, especially in fluid dynamics. Thus the curl is the circulation density. </p><p>We can apply the flux and these theorems to many disciplines in which we see currents, forces, etc., applied through areas. </p> <div class="mw-heading mw-heading3"><h3 id="Electromagnetism">Electromagnetism</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Flux&action=edit&section=10" title="Edit section: Electromagnetism"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Electric_flux">Electric flux</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Flux&action=edit&section=11" title="Edit section: Electric flux"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An electric "charge", such as a single proton in space, has a magnitude defined in coulombs. Such a charge has an electric field surrounding it. In pictorial form, the electric field from a positive point charge can be visualized as a dot radiating <a href="/wiki/Field_line" title="Field line">electric field lines</a> (sometimes also called "lines of force"). Conceptually, electric flux can be thought of as "the number of field lines" passing through a given area. Mathematically, electric flux is the integral of the <a href="/wiki/Normal_(geometry)" title="Normal (geometry)">normal</a> component of the electric field over a given area. Hence, units of electric flux are, in the <a href="/wiki/MKS_system_of_units" class="mw-redirect" title="MKS system of units">MKS system</a>, <a href="/wiki/Newton_(unit)" title="Newton (unit)">newtons</a> per <a href="/wiki/Coulomb_(unit)" class="mw-redirect" title="Coulomb (unit)">coulomb</a> times meters squared, or N m<sup>2</sup>/C. (Electric flux density is the electric flux per unit area, and is a measure of strength of the <a href="/wiki/Normal_(geometry)" title="Normal (geometry)">normal</a> component of the electric field averaged over the area of integration. Its units are N/C, the same as the electric field in MKS units.) </p><p>Two forms of <a href="/wiki/Electric_flux" title="Electric flux">electric flux</a> are used, one for the <b>E</b>-field:<sup id="cite_ref-Electromagnetism_2008_13-0" class="reference"><a href="#cite_note-Electromagnetism_2008-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Electrodynamics_2007_14-0" class="reference"><a href="#cite_note-Electrodynamics_2007-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="nowrap mw-no-invert"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{E}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{E}=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9810529d253a8cc85469e17185424ea235655087" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.619ex; height:2.509ex;" alt="{\displaystyle \Phi _{E}=}" /></span> <span class="mw-default-size" typeof="mw:File"><span><img alt="\oiint" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/40px-OiintLaTeX.svg.png" decoding="async" width="25" height="44" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/60px-OiintLaTeX.svg.png 2x" data-file-width="204" data-file-height="354" /></span></span><span style="position:relative; right:8px; top:18px; margin-right:-8px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\scriptstyle A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="1"> <mi>A</mi> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\scriptstyle A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c1dabbf21acfa318113b7dad41854d5d46cbce1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.233ex; height:1.843ex;" alt="{\displaystyle {\scriptstyle A}}" /></span></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} \cdot {\rm {d}}\mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} \cdot {\rm {d}}\mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fe9ba292db0251820a51aa41e1116b4b5874c55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.748ex; height:2.176ex;" alt="{\displaystyle \mathbf {E} \cdot {\rm {d}}\mathbf {A} }" /></span></span></dd></dl> <p>and one for the <b>D</b>-field (called the <a href="/wiki/Electric_displacement" class="mw-redirect" title="Electric displacement">electric displacement</a>): </p> <dl><dd><span class="nowrap mw-no-invert"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{D}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msub> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{D}=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2082046ceec613cfec93d2b26571554a0cb70be3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.724ex; height:2.509ex;" alt="{\displaystyle \Phi _{D}=}" /></span> <span class="mw-default-size" typeof="mw:File"><span><img alt="\oiint" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/40px-OiintLaTeX.svg.png" decoding="async" width="25" height="44" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/60px-OiintLaTeX.svg.png 2x" data-file-width="204" data-file-height="354" /></span></span><span style="position:relative; right:8px; top:18px; margin-right:-8px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\scriptstyle A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="1"> <mi>A</mi> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\scriptstyle A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c1dabbf21acfa318113b7dad41854d5d46cbce1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.233ex; height:1.843ex;" alt="{\displaystyle {\scriptstyle A}}" /></span></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {D} \cdot {\rm {d}}\mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {D} \cdot {\rm {d}}\mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4275ec35f4581cba031522dd9b7b1c638cb3a709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.041ex; height:2.176ex;" alt="{\displaystyle \mathbf {D} \cdot {\rm {d}}\mathbf {A} }" /></span></span></dd></dl> <p>This quantity arises in <a href="/wiki/Gauss%27s_law" title="Gauss's law">Gauss's law</a> – which states that the flux of the <a href="/wiki/Electric_field" title="Electric field">electric field</a> <b>E</b> out of a <a href="/wiki/Closed_surface" class="mw-redirect" title="Closed surface">closed surface</a> is proportional to the <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a> <i>Q<sub>A</sub></i> enclosed in the surface (independent of how that charge is distributed), the integral form is: </p> <dl><dd><span class="nowrap mw-no-invert"> <span class="mw-default-size" typeof="mw:File"><span><img alt="\oiint" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/40px-OiintLaTeX.svg.png" decoding="async" width="25" height="44" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/60px-OiintLaTeX.svg.png 2x" data-file-width="204" data-file-height="354" /></span></span><span style="position:relative; right:8px; top:18px; margin-right:-8px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\scriptstyle A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="1"> <mi>A</mi> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\scriptstyle A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c1dabbf21acfa318113b7dad41854d5d46cbce1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.233ex; height:1.843ex;" alt="{\displaystyle {\scriptstyle A}}" /></span></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} \cdot {\rm {d}}\mathbf {A} ={\frac {Q_{A}}{\varepsilon _{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} \cdot {\rm {d}}\mathbf {A} ={\frac {Q_{A}}{\varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf851c5bb276a1e9f9b5ca95b0450e96c40d5b72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.986ex; height:5.676ex;" alt="{\displaystyle \mathbf {E} \cdot {\rm {d}}\mathbf {A} ={\frac {Q_{A}}{\varepsilon _{0}}}}" /></span></span></dd></dl> <p>where <i>ε</i><sub>0</sub> is the <a href="/wiki/Permittivity_of_free_space" class="mw-redirect" title="Permittivity of free space">permittivity of free space</a>. </p><p>If one considers the flux of the electric field vector, <b>E</b>, for a tube near a point charge in the field of the charge but not containing it with sides formed by lines tangent to the field, the flux for the sides is zero and there is an equal and opposite flux at both ends of the tube. This is a consequence of Gauss's Law applied to an inverse square field. The flux for any cross-sectional surface of the tube will be the same. The total flux for any surface surrounding a charge <i>q</i> is <i>q</i>/<i>ε</i><sub>0</sub>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> </p><p>In free space the <a href="/wiki/Electric_displacement" class="mw-redirect" title="Electric displacement">electric displacement</a> is given by the <a href="/wiki/Constitutive_relation" class="mw-redirect" title="Constitutive relation">constitutive relation</a> <b>D</b> = <i>ε</i><sub>0</sub> <b>E</b>, so for any bounding surface the <b>D</b>-field flux equals the charge <i>Q<sub>A</sub></i> within it. Here the expression "flux of" indicates a mathematical operation and, as can be seen, the result is not necessarily a "flow", since nothing actually flows along electric field lines. </p> <div class="mw-heading mw-heading4"><h4 id="Magnetic_flux">Magnetic flux</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Flux&action=edit&section=12" title="Edit section: Magnetic flux"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The magnetic flux density (<a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a>) having the unit Wb/m<sup>2</sup> (<a href="/wiki/Tesla_(unit)" title="Tesla (unit)">Tesla</a>) is denoted by <b>B</b>, and <a href="/wiki/Magnetic_flux" title="Magnetic flux">magnetic flux</a> is defined analogously:<sup id="cite_ref-Electromagnetism_2008_13-1" class="reference"><a href="#cite_note-Electromagnetism_2008-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Electrodynamics_2007_14-1" class="reference"><a href="#cite_note-Electrodynamics_2007-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{B}=\iint _{A}\mathbf {B} \cdot \mathrm {d} \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo>=</mo> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{B}=\iint _{A}\mathbf {B} \cdot \mathrm {d} \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a5b3f8ee9a150b5e3c4b0249d3f42a41fc85779" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.519ex; height:5.676ex;" alt="{\displaystyle \Phi _{B}=\iint _{A}\mathbf {B} \cdot \mathrm {d} \mathbf {A} }" /></span> with the same notation above. The quantity arises in <a href="/wiki/Faraday%27s_law_of_induction" title="Faraday's law of induction">Faraday's law of induction</a>, where the magnetic flux is time-dependent either because the boundary is time-dependent or magnetic field is time-dependent. In integral form: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {{\rm {d}}\Phi _{B}}{{\rm {d}}t}}=\oint _{\partial A}\mathbf {E} \cdot d{\boldsymbol {\ell }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi>A</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ℓ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {{\rm {d}}\Phi _{B}}{{\rm {d}}t}}=\oint _{\partial A}\mathbf {E} \cdot d{\boldsymbol {\ell }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b2818ad6a6ffc7f9d19e2b7e3e98938e37d475e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.023ex; height:5.843ex;" alt="{\displaystyle -{\frac {{\rm {d}}\Phi _{B}}{{\rm {d}}t}}=\oint _{\partial A}\mathbf {E} \cdot d{\boldsymbol {\ell }}}" /></span> where <i>d</i><b>ℓ</b> is an infinitesimal vector <a href="/wiki/Line_element" title="Line element">line element</a> of the <a href="/wiki/Closed_curve" class="mw-redirect" title="Closed curve">closed curve</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc71b05684faf3a9114d7a81735a4894efe75baa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.061ex; height:2.176ex;" alt="{\displaystyle \partial A}" /></span>, with <a href="/wiki/Magnitude_(vector)" class="mw-redirect" title="Magnitude (vector)">magnitude</a> equal to the length of the <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitesimal</a> line element, and <a href="/wiki/Direction_(geometry)" title="Direction (geometry)">direction</a> given by the tangent to the curve <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc71b05684faf3a9114d7a81735a4894efe75baa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.061ex; height:2.176ex;" alt="{\displaystyle \partial A}" /></span>, with the sign determined by the integration direction. </p><p>The time-rate of change of the magnetic flux through a loop of wire is minus the <a href="/wiki/Electromotive_force" title="Electromotive force">electromotive force</a> created in that wire. The direction is such that if current is allowed to pass through the wire, the electromotive force will cause a current which "opposes" the change in magnetic field by itself producing a magnetic field opposite to the change. This is the basis for <a href="/wiki/Inductor" title="Inductor">inductors</a> and many <a href="/wiki/Electric_generator" title="Electric generator">electric generators</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Poynting_flux">Poynting flux</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Flux&action=edit&section=13" title="Edit section: Poynting flux"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Using this definition, the flux of the <a href="/wiki/Poynting_vector" title="Poynting vector">Poynting vector</a> <b>S</b> over a specified surface is the rate at which electromagnetic energy flows through that surface, defined like before:<sup id="cite_ref-Electrodynamics_2007_14-2" class="reference"><a href="#cite_note-Electrodynamics_2007-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="nowrap mw-no-invert"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{S}=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{S}=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6056884a7958c7f8a4c73c5ba376fc7d7b8b943c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.424ex; height:2.509ex;" alt="{\displaystyle \Phi _{S}=}" /></span> <span class="mw-default-size" typeof="mw:File"><span><img alt="\oiint" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/40px-OiintLaTeX.svg.png" decoding="async" width="25" height="44" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/86/OiintLaTeX.svg/60px-OiintLaTeX.svg.png 2x" data-file-width="204" data-file-height="354" /></span></span><span style="position:relative; right:8px; top:18px; margin-right:-8px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\scriptstyle A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="1"> <mi>A</mi> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\scriptstyle A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c1dabbf21acfa318113b7dad41854d5d46cbce1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.233ex; height:1.843ex;" alt="{\displaystyle {\scriptstyle A}}" /></span></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {S} \cdot {\rm {d}}\mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {S} \cdot {\rm {d}}\mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c061aa77d5365339592dbde08c8fb46263591793" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.476ex; height:2.176ex;" alt="{\displaystyle \mathbf {S} \cdot {\rm {d}}\mathbf {A} }" /></span></span></dd></dl> <p>The flux of the <a href="/wiki/Poynting_vector" title="Poynting vector">Poynting vector</a> through a surface is the electromagnetic <a href="/wiki/Power_(physics)" title="Power (physics)">power</a>, or <a href="/wiki/Energy" title="Energy">energy</a> per unit <a href="/wiki/Time" title="Time">time</a>, passing through that surface. This is commonly used in analysis of <a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">electromagnetic radiation</a>, but has application to other electromagnetic systems as well. </p><p>Confusingly, the Poynting vector is sometimes called the <i>power flux</i>, which is an example of the first usage of flux, above.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> It has units of <a href="/wiki/Watt" title="Watt">watts</a> per <a href="/wiki/Square_metre" title="Square metre">square metre</a> (W/m<sup>2</sup>). </p> <div class="mw-heading mw-heading2"><h2 id="SI_radiometry_units">SI radiometry units</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Flux&action=edit&section=14" title="Edit section: SI radiometry units"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="anchor" id="table-si-radiometry-units"></span> </p> <table class="wikitable plainrowheaders"> <caption><style data-mw-deduplicate="TemplateStyles:r1045256916">.mw-parser-output .navbar-header{text-align:center;position:relative;white-space:nowrap}.mw-parser-output .navbar-header .navbar{position:absolute;right:0;top:0;margin:0 5px}</style><div class="navbar-header"><span style="font-size:130%;">SI radiometry units</span><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:SI_radiometry_units" title="Template:SI radiometry units"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:SI_radiometry_units" title="Template talk:SI radiometry units"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:SI_radiometry_units" title="Special:EditPage/Template:SI radiometry units"><abbr title="Edit this template">e</abbr></a></li></ul></div></div> </caption> <tbody><tr> <th scope="col" colspan="2">Quantity </th> <th scope="col" colspan="2">Unit </th> <th scope="col" rowspan="2">Dimension </th> <th scope="col" rowspan="2">Notes </th></tr> <tr> <th scope="col">Name </th> <th scope="col">Symbol<sup id="cite_ref-note-suffix-e_17-0" class="reference"><a href="#cite_note-note-suffix-e-17"><span class="cite-bracket">[</span>nb 1<span class="cite-bracket">]</span></a></sup> </th> <th scope="col">Name </th> <th scope="col">Symbol </th></tr> <tr> <th scope="row"><a href="/wiki/Radiant_energy" title="Radiant energy">Radiant energy</a> </th> <td><span class="texhtml"><i>Q</i><sub>e</sub></span><sup id="cite_ref-note-alternative-symbol-radiometric_18-0" class="reference"><a href="#cite_note-note-alternative-symbol-radiometric-18"><span class="cite-bracket">[</span>nb 2<span class="cite-bracket">]</span></a></sup> </td> <td><a href="/wiki/Joule" title="Joule">joule</a> </td> <td><a href="/wiki/Joule" title="Joule">J</a> </td> <td><b>M</b>⋅<b>L</b><sup>2</sup>⋅<b>T</b><sup>−2</sup> </td> <td>Energy of electromagnetic radiation. </td></tr> <tr> <th scope="row" style="white-space:nowrap"><a href="/wiki/Radiant_energy_density" title="Radiant energy density">Radiant energy density</a> </th> <td><span class="texhtml"><i>w</i><sub>e</sub></span> </td> <td>joule per cubic metre </td> <td>J/m<sup>3</sup> </td> <td><b>M</b>⋅<b>L</b><sup>−1</sup>⋅<b>T</b><sup>−2</sup> </td> <td>Radiant energy per unit volume. </td></tr> <tr> <th scope="row"><a href="/wiki/Radiant_flux" title="Radiant flux">Radiant flux</a> </th> <td><span class="texhtml">Φ<sub>e</sub></span><sup id="cite_ref-note-alternative-symbol-radiometric_18-1" class="reference"><a href="#cite_note-note-alternative-symbol-radiometric-18"><span class="cite-bracket">[</span>nb 2<span class="cite-bracket">]</span></a></sup> </td> <td><a href="/wiki/Watt" title="Watt">watt</a> </td> <td><a href="/wiki/Watt" title="Watt">W</a> = J/s </td> <td><b>M</b>⋅<b>L</b><sup>2</sup>⋅<b>T</b><sup>−3</sup> </td> <td>Radiant energy emitted, reflected, transmitted or received, per unit time. This is sometimes also called "radiant power", and called <a href="/wiki/Luminosity" title="Luminosity">luminosity</a> in astronomy. </td></tr> <tr> <th scope="row" rowspan="2"><a href="/wiki/Radiant_flux" title="Radiant flux">Spectral flux</a> </th> <td><span class="texhtml">Φ<sub>e,<i>ν</i></sub></span><sup id="cite_ref-note-suffix-nu_19-0" class="reference"><a href="#cite_note-note-suffix-nu-19"><span class="cite-bracket">[</span>nb 3<span class="cite-bracket">]</span></a></sup> </td> <td>watt per <a href="/wiki/Hertz" title="Hertz">hertz</a> </td> <td>W/<a href="/wiki/Hertz" title="Hertz">Hz</a> </td> <td><b>M</b>⋅<b>L</b><sup>2</sup>⋅<b>T</b><sup> −2</sup> </td> <td rowspan="2">Radiant flux per unit frequency or wavelength. The latter is commonly measured in W⋅nm<sup>−1</sup>. </td></tr> <tr> <td><span class="texhtml">Φ<sub>e,<i>λ</i></sub></span><sup id="cite_ref-note-suffix-lambda_20-0" class="reference"><a href="#cite_note-note-suffix-lambda-20"><span class="cite-bracket">[</span>nb 4<span class="cite-bracket">]</span></a></sup> </td> <td>watt per metre </td> <td>W/m </td> <td><b>M</b>⋅<b>L</b>⋅<b>T</b><sup>−3</sup> </td></tr> <tr> <th scope="row"><a href="/wiki/Radiant_intensity" title="Radiant intensity">Radiant intensity</a> </th> <td><span class="texhtml"><i>I</i><sub>e,Ω</sub></span><sup id="cite_ref-note-suffix-omega_21-0" class="reference"><a href="#cite_note-note-suffix-omega-21"><span class="cite-bracket">[</span>nb 5<span class="cite-bracket">]</span></a></sup> </td> <td>watt per <a href="/wiki/Steradian" title="Steradian">steradian</a> </td> <td>W/<a href="/wiki/Steradian" title="Steradian">sr</a> </td> <td><b>M</b>⋅<b>L</b><sup>2</sup>⋅<b>T</b><sup>−3</sup> </td> <td>Radiant flux emitted, reflected, transmitted or received, per unit solid angle. This is a <i>directional</i> quantity. </td></tr> <tr> <th scope="row" rowspan="2"><a href="/wiki/Radiant_intensity" title="Radiant intensity">Spectral intensity</a> </th> <td><span class="texhtml"><i>I</i><sub>e,Ω,<i>ν</i></sub></span><sup id="cite_ref-note-suffix-nu_19-1" class="reference"><a href="#cite_note-note-suffix-nu-19"><span class="cite-bracket">[</span>nb 3<span class="cite-bracket">]</span></a></sup> </td> <td>watt per steradian per hertz </td> <td>W⋅sr<sup>−1</sup>⋅Hz<sup>−1</sup> </td> <td><b>M</b>⋅<b>L</b><sup>2</sup>⋅<b>T</b><sup>−2</sup> </td> <td rowspan="2">Radiant intensity per unit frequency or wavelength. The latter is commonly measured in W⋅sr<sup>−1</sup>⋅nm<sup>−1</sup>. This is a <i>directional</i> quantity. </td></tr> <tr> <td><span class="texhtml"><i>I</i><sub>e,Ω,<i>λ</i></sub></span><sup id="cite_ref-note-suffix-lambda_20-1" class="reference"><a href="#cite_note-note-suffix-lambda-20"><span class="cite-bracket">[</span>nb 4<span class="cite-bracket">]</span></a></sup> </td> <td>watt per steradian per metre </td> <td>W⋅sr<sup>−1</sup>⋅m<sup>−1</sup> </td> <td><b>M</b>⋅<b>L</b>⋅<b>T</b><sup>−3</sup> </td></tr> <tr> <th scope="row"><a href="/wiki/Radiance" title="Radiance">Radiance</a> </th> <td><span class="texhtml"><i>L</i><sub>e,Ω</sub></span><sup id="cite_ref-note-suffix-omega_21-1" class="reference"><a href="#cite_note-note-suffix-omega-21"><span class="cite-bracket">[</span>nb 5<span class="cite-bracket">]</span></a></sup> </td> <td>watt per steradian per square metre </td> <td>W⋅sr<sup>−1</sup>⋅m<sup>−2</sup> </td> <td><b>M</b>⋅<b>T</b><sup>−3</sup> </td> <td>Radiant flux emitted, reflected, transmitted or received by a <i>surface</i>, per unit solid angle per unit projected area. This is a <i>directional</i> quantity. This is sometimes also confusingly called "intensity". </td></tr> <tr> <th scope="row" rowspan="2" style="white-space:nowrap"><a href="/wiki/Spectral_radiance" title="Spectral radiance">Spectral radiance</a><br />Specific intensity </th> <td><span class="texhtml"><i>L</i><sub>e,Ω,<i>ν</i></sub></span><sup id="cite_ref-note-suffix-nu_19-2" class="reference"><a href="#cite_note-note-suffix-nu-19"><span class="cite-bracket">[</span>nb 3<span class="cite-bracket">]</span></a></sup> </td> <td>watt per steradian per square metre per hertz </td> <td>W⋅sr<sup>−1</sup>⋅m<sup>−2</sup>⋅Hz<sup>−1</sup> </td> <td><b>M</b>⋅<b>T</b><sup>−2</sup> </td> <td rowspan="2">Radiance of a <i>surface</i> per unit frequency or wavelength. The latter is commonly measured in W⋅sr<sup>−1</sup>⋅m<sup>−2</sup>⋅nm<sup>−1</sup>. This is a <i>directional</i> quantity. This is sometimes also confusingly called "spectral intensity". </td></tr> <tr> <td><span class="texhtml"><i>L</i><sub>e,Ω,<i>λ</i></sub></span><sup id="cite_ref-note-suffix-lambda_20-2" class="reference"><a href="#cite_note-note-suffix-lambda-20"><span class="cite-bracket">[</span>nb 4<span class="cite-bracket">]</span></a></sup> </td> <td>watt per steradian per square metre, per metre </td> <td>W⋅sr<sup>−1</sup>⋅m<sup>−3</sup> </td> <td><b>M</b>⋅<b>L</b><sup>−1</sup>⋅<b>T</b><sup>−3</sup> </td></tr> <tr> <th scope="row"><a href="/wiki/Irradiance" title="Irradiance">Irradiance</a><br /><a href="/wiki/Flux_density" class="mw-redirect" title="Flux density">Flux density</a> </th> <td><span class="texhtml"><i>E</i><sub>e</sub></span><sup id="cite_ref-note-alternative-symbol-radiometric_18-2" class="reference"><a href="#cite_note-note-alternative-symbol-radiometric-18"><span class="cite-bracket">[</span>nb 2<span class="cite-bracket">]</span></a></sup> </td> <td>watt per square metre </td> <td>W/m<sup>2</sup> </td> <td><b>M</b>⋅<b>T</b><sup>−3</sup> </td> <td>Radiant flux <i>received</i> by a <i>surface</i> per unit area. This is sometimes also confusingly called "intensity". </td></tr> <tr> <th scope="row" rowspan="2" style="white-space:nowrap"><a href="/wiki/Irradiance" title="Irradiance">Spectral irradiance</a><br /><a href="/wiki/Spectral_flux_density" title="Spectral flux density">Spectral flux density</a> </th> <td><span class="texhtml"><i>E</i><sub>e,<i>ν</i></sub></span><sup id="cite_ref-note-suffix-nu_19-3" class="reference"><a href="#cite_note-note-suffix-nu-19"><span class="cite-bracket">[</span>nb 3<span class="cite-bracket">]</span></a></sup> </td> <td>watt per square metre per hertz </td> <td>W⋅m<sup>−2</sup>⋅Hz<sup>−1</sup> </td> <td><b>M</b>⋅<b>T</b><sup>−2</sup> </td> <td rowspan="2">Irradiance of a <i>surface</i> per unit frequency or wavelength. This is sometimes also confusingly called "spectral intensity". Non-SI units of spectral flux density include <a href="/wiki/Jansky" title="Jansky">jansky</a> (<span class="nowrap"><span data-sort-value="7000100000000000000♠"></span>1 Jy</span> = <span class="nowrap"><span data-sort-value="6974100000000000000♠"></span>10<sup>−26</sup> W⋅m<sup>−2</sup>⋅Hz<sup>−1</sup></span>) and <a href="/wiki/Solar_flux_unit" title="Solar flux unit">solar flux unit</a> (<span class="nowrap"><span data-sort-value="7000100000000000000♠"></span>1 sfu</span> = <span class="nowrap"><span data-sort-value="6978100000000000000♠"></span>10<sup>−22</sup> W⋅m<sup>−2</sup>⋅Hz<sup>−1</sup></span> = <span class="nowrap"><span data-sort-value="7004100000000000000♠"></span>10<sup>4</sup> Jy</span>). </td></tr> <tr> <td><span class="texhtml"><i>E</i><sub>e,<i>λ</i></sub></span><sup id="cite_ref-note-suffix-lambda_20-3" class="reference"><a href="#cite_note-note-suffix-lambda-20"><span class="cite-bracket">[</span>nb 4<span class="cite-bracket">]</span></a></sup> </td> <td>watt per square metre, per metre </td> <td>W/m<sup>3</sup> </td> <td><b>M</b>⋅<b>L</b><sup>−1</sup>⋅<b>T</b><sup>−3</sup> </td></tr> <tr> <th scope="row"><a href="/wiki/Radiosity_(radiometry)" title="Radiosity (radiometry)">Radiosity</a> </th> <td><span class="texhtml"><i>J</i><sub>e</sub></span><sup id="cite_ref-note-alternative-symbol-radiometric_18-3" class="reference"><a href="#cite_note-note-alternative-symbol-radiometric-18"><span class="cite-bracket">[</span>nb 2<span class="cite-bracket">]</span></a></sup> </td> <td>watt per square metre </td> <td>W/m<sup>2</sup> </td> <td><b>M</b>⋅<b>T</b><sup>−3</sup> </td> <td>Radiant flux <i>leaving</i> (emitted, reflected and transmitted by) a <i>surface</i> per unit area. This is sometimes also confusingly called "intensity". </td></tr> <tr> <th scope="row" rowspan="2"><a href="/wiki/Radiosity_(radiometry)" title="Radiosity (radiometry)">Spectral radiosity</a> </th> <td><span class="texhtml"><i>J</i><sub>e,<i>ν</i></sub></span><sup id="cite_ref-note-suffix-nu_19-4" class="reference"><a href="#cite_note-note-suffix-nu-19"><span class="cite-bracket">[</span>nb 3<span class="cite-bracket">]</span></a></sup> </td> <td>watt per square metre per hertz </td> <td>W⋅m<sup>−2</sup>⋅Hz<sup>−1</sup> </td> <td><b>M</b>⋅<b>T</b><sup>−2</sup> </td> <td rowspan="2">Radiosity of a <i>surface</i> per unit frequency or wavelength. The latter is commonly measured in W⋅m<sup>−2</sup>⋅nm<sup>−1</sup>. This is sometimes also confusingly called "spectral intensity". </td></tr> <tr> <td><span class="texhtml"><i>J</i><sub>e,<i>λ</i></sub></span><sup id="cite_ref-note-suffix-lambda_20-4" class="reference"><a href="#cite_note-note-suffix-lambda-20"><span class="cite-bracket">[</span>nb 4<span class="cite-bracket">]</span></a></sup> </td> <td>watt per square metre, per metre </td> <td>W/m<sup>3</sup> </td> <td><b>M</b>⋅<b>L</b><sup>−1</sup>⋅<b>T</b><sup>−3</sup> </td></tr> <tr> <th scope="row"><a href="/wiki/Radiant_exitance" title="Radiant exitance">Radiant exitance</a> </th> <td><span class="texhtml"><i>M</i><sub>e</sub></span><sup id="cite_ref-note-alternative-symbol-radiometric_18-4" class="reference"><a href="#cite_note-note-alternative-symbol-radiometric-18"><span class="cite-bracket">[</span>nb 2<span class="cite-bracket">]</span></a></sup> </td> <td>watt per square metre </td> <td>W/m<sup>2</sup> </td> <td><b>M</b>⋅<b>T</b><sup>−3</sup> </td> <td>Radiant flux <i>emitted</i> by a <i>surface</i> per unit area. This is the emitted component of radiosity. "Radiant emittance" is an old term for this quantity. This is sometimes also confusingly called "intensity". </td></tr> <tr> <th scope="row" rowspan="2"><a href="/wiki/Radiant_exitance" title="Radiant exitance">Spectral exitance</a> </th> <td><span class="texhtml"><i>M</i><sub>e,<i>ν</i></sub></span><sup id="cite_ref-note-suffix-nu_19-5" class="reference"><a href="#cite_note-note-suffix-nu-19"><span class="cite-bracket">[</span>nb 3<span class="cite-bracket">]</span></a></sup> </td> <td>watt per square metre per hertz </td> <td>W⋅m<sup>−2</sup>⋅Hz<sup>−1</sup> </td> <td><b>M</b>⋅<b>T</b><sup>−2</sup> </td> <td rowspan="2">Radiant exitance of a <i>surface</i> per unit frequency or wavelength. The latter is commonly measured in W⋅m<sup>−2</sup>⋅nm<sup>−1</sup>. "Spectral emittance" is an old term for this quantity. This is sometimes also confusingly called "spectral intensity". </td></tr> <tr> <td><span class="texhtml"><i>M</i><sub>e,<i>λ</i></sub></span><sup id="cite_ref-note-suffix-lambda_20-5" class="reference"><a href="#cite_note-note-suffix-lambda-20"><span class="cite-bracket">[</span>nb 4<span class="cite-bracket">]</span></a></sup> </td> <td>watt per square metre, per metre </td> <td>W/m<sup>3</sup> </td> <td><b>M</b>⋅<b>L</b><sup>−1</sup>⋅<b>T</b><sup>−3</sup> </td></tr> <tr> <th scope="row"><a href="/wiki/Radiant_exposure" title="Radiant exposure">Radiant exposure</a> </th> <td><span class="texhtml"><i>H</i><sub>e</sub></span> </td> <td>joule per square metre </td> <td>J/m<sup>2</sup> </td> <td><b>M</b>⋅<b>T</b><sup>−2</sup> </td> <td>Radiant energy received by a <i>surface</i> per unit area, or equivalently irradiance of a <i>surface</i> integrated over time of irradiation. This is sometimes also called "radiant fluence". </td></tr> <tr> <th scope="row" rowspan="2"><a href="/wiki/Radiant_exposure" title="Radiant exposure">Spectral exposure</a> </th> <td><span class="texhtml"><i>H</i><sub>e,<i>ν</i></sub></span><sup id="cite_ref-note-suffix-nu_19-6" class="reference"><a href="#cite_note-note-suffix-nu-19"><span class="cite-bracket">[</span>nb 3<span class="cite-bracket">]</span></a></sup> </td> <td>joule per square metre per hertz </td> <td>J⋅m<sup>−2</sup>⋅Hz<sup>−1</sup> </td> <td><b>M</b>⋅<b>T</b><sup>−1</sup> </td> <td rowspan="2">Radiant exposure of a <i>surface</i> per unit frequency or wavelength. The latter is commonly measured in J⋅m<sup>−2</sup>⋅nm<sup>−1</sup>. This is sometimes also called "spectral fluence". </td></tr> <tr> <td><span class="texhtml"><i>H</i><sub>e,<i>λ</i></sub></span><sup id="cite_ref-note-suffix-lambda_20-6" class="reference"><a href="#cite_note-note-suffix-lambda-20"><span class="cite-bracket">[</span>nb 4<span class="cite-bracket">]</span></a></sup> </td> <td>joule per square metre, per metre </td> <td>J/m<sup>3</sup> </td> <td><b>M</b>⋅<b>L</b><sup>−1</sup>⋅<b>T</b><sup>−2</sup> </td></tr> <tr> <td style="text-align:center;" colspan="6">See also: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><div class="hlist inline"><ul><li><a href="/wiki/SI" class="mw-redirect" title="SI">SI</a></li><li><a href="/wiki/Radiometry" title="Radiometry">Radiometry</a></li><li><a href="/wiki/Photometry_(optics)" title="Photometry (optics)">Photometry</a></li></ul></div> </td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-note-suffix-e-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-note-suffix-e_17-0">^</a></b></span> <span class="reference-text"><a href="/wiki/Standards_organization" title="Standards organization">Standards organizations</a> recommend that radiometric <a href="/wiki/Physical_quantity" title="Physical quantity">quantities</a> should be denoted with suffix "e" (for "energetic") to avoid confusion with photometric or <a href="/wiki/Photon" title="Photon">photon</a> quantities.</span> </li> <li id="cite_note-note-alternative-symbol-radiometric-18"><span class="mw-cite-backlink">^ <a href="#cite_ref-note-alternative-symbol-radiometric_18-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-note-alternative-symbol-radiometric_18-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-note-alternative-symbol-radiometric_18-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-note-alternative-symbol-radiometric_18-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-note-alternative-symbol-radiometric_18-4"><sup><i><b>e</b></i></sup></a></span> <span class="reference-text">Alternative symbols sometimes seen: <span class="texhtml mvar" style="font-style:italic;">W</span> or <span class="texhtml mvar" style="font-style:italic;">E</span> for radiant energy, <span class="texhtml mvar" style="font-style:italic;">P</span> or <span class="texhtml mvar" style="font-style:italic;">F</span> for radiant flux, <span class="texhtml mvar" style="font-style:italic;">I</span> for irradiance, <span class="texhtml mvar" style="font-style:italic;">W</span> for radiant exitance.</span> </li> <li id="cite_note-note-suffix-nu-19"><span class="mw-cite-backlink">^ <a href="#cite_ref-note-suffix-nu_19-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-note-suffix-nu_19-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-note-suffix-nu_19-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-note-suffix-nu_19-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-note-suffix-nu_19-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-note-suffix-nu_19-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-note-suffix-nu_19-6"><sup><i><b>g</b></i></sup></a></span> <span class="reference-text">Spectral quantities given per unit <a href="/wiki/Frequency" title="Frequency">frequency</a> are denoted with suffix "<i><a href="/wiki/%CE%9D" class="mw-redirect" title="Ν">ν</a></i>" (Greek letter <a href="/wiki/Nu_(letter)" title="Nu (letter)">nu</a>, not to be confused with a letter "v", indicating a photometric quantity.)</span> </li> <li id="cite_note-note-suffix-lambda-20"><span class="mw-cite-backlink">^ <a href="#cite_ref-note-suffix-lambda_20-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-note-suffix-lambda_20-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-note-suffix-lambda_20-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-note-suffix-lambda_20-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-note-suffix-lambda_20-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-note-suffix-lambda_20-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-note-suffix-lambda_20-6"><sup><i><b>g</b></i></sup></a></span> <span class="reference-text">Spectral quantities given per unit <a href="/wiki/Wavelength" title="Wavelength">wavelength</a> are denoted with suffix "<i><a href="/wiki/%CE%9B" class="mw-redirect" title="Λ">λ</a></i>".</span> </li> <li id="cite_note-note-suffix-omega-21"><span class="mw-cite-backlink">^ <a href="#cite_ref-note-suffix-omega_21-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-note-suffix-omega_21-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">Directional quantities are denoted with suffix "<a href="/wiki/%CE%A9" class="mw-redirect" title="Ω">Ω</a>".</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Flux&action=edit&section=15" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1266661725">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output .portalbox-image{display:table-cell;padding:0.2em;vertical-align:middle;text-align:center}.mw-parser-output .portalbox-link{display:table-cell;padding:0.2em 0.2em 0.2em 0.3em;vertical-align:middle}@media(min-width:720px){.mw-parser-output .portalleft{margin:0.5em 1em 0.5em 0}.mw-parser-output .portalright{clear:right;float:right;margin:0.5em 0 0.5em 1em}}</style><ul role="navigation" aria-label="Portals" class="noprint portalbox portalborder portalright"> <li class="portalbox-entry"><span class="portalbox-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span></span><span class="portalbox-link"><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></span></li></ul> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 28em;"> <ul><li><a href="/wiki/AB_magnitude" title="AB magnitude">AB magnitude</a></li> <li><a href="/wiki/Explosively_pumped_flux_compression_generator" title="Explosively pumped flux compression generator">Explosively pumped flux compression generator</a></li> <li><a href="/wiki/Eddy_covariance" title="Eddy covariance">Eddy covariance</a> flux (aka, eddy correlation, eddy flux)</li> <li><a href="/wiki/Fast_Flux_Test_Facility" title="Fast Flux Test Facility">Fast Flux Test Facility</a></li> <li><a href="/wiki/Fluence" class="mw-redirect" title="Fluence">Fluence</a> (flux of the first sort for particle beams)</li> <li><a href="/wiki/Fluid_dynamics" title="Fluid dynamics">Fluid dynamics</a></li> <li><a href="/wiki/Flux_footprint" title="Flux footprint">Flux footprint</a></li> <li><a href="/wiki/Flux_pinning" title="Flux pinning">Flux pinning</a></li> <li><a href="/wiki/Flux_quantization" class="mw-redirect" title="Flux quantization">Flux quantization</a></li> <li><a href="/wiki/Gauss%27s_law" title="Gauss's law">Gauss's law</a></li> <li><a href="/wiki/Inverse-square_law" title="Inverse-square law">Inverse-square law</a></li> <li><a href="/wiki/Jansky" title="Jansky">Jansky</a> (non SI unit of spectral flux density)</li> <li><a href="/wiki/Latent_heat_flux" class="mw-redirect" title="Latent heat flux">Latent heat flux</a></li> <li><a href="/wiki/Luminous_flux" title="Luminous flux">Luminous flux</a></li> <li><a href="/wiki/Magnetic_flux" title="Magnetic flux">Magnetic flux</a></li> <li><a href="/wiki/Magnetic_flux_quantum" title="Magnetic flux quantum">Magnetic flux quantum</a></li> <li><a href="/wiki/Neutron_flux" title="Neutron flux">Neutron flux</a></li> <li><a href="/wiki/Poynting_flux" class="mw-redirect" title="Poynting flux">Poynting flux</a></li> <li><a href="/wiki/Poynting_theorem" class="mw-redirect" title="Poynting theorem">Poynting theorem</a></li> <li><a href="/wiki/Radiant_flux" title="Radiant flux">Radiant flux</a></li> <li><a href="/wiki/Rapid_single_flux_quantum" title="Rapid single flux quantum">Rapid single flux quantum</a></li> <li><a href="/wiki/Sound_energy_flux" class="mw-redirect" title="Sound energy flux">Sound energy flux</a></li> <li><a href="/wiki/Volumetric_flux" title="Volumetric flux">Volumetric flux</a> (flux of the first sort for fluids)</li> <li><a href="/wiki/Volumetric_flow_rate" title="Volumetric flow rate">Volumetric flow rate</a> (flux of the second sort for fluids)</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Flux&action=edit&section=16" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Purcell, p. 22–26</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFWeekley1967" class="citation book cs1">Weekley, Ernest (1967). <i>An Etymological Dictionary of Modern English</i>. Courier Dover Publications. p. 581. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-21873-2" title="Special:BookSources/0-486-21873-2"><bdi>0-486-21873-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Etymological+Dictionary+of+Modern+English&rft.pages=581&rft.pub=Courier+Dover+Publications&rft.date=1967&rft.isbn=0-486-21873-2&rft.aulast=Weekley&rft.aufirst=Ernest&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFlux" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHerivel1975" class="citation book cs1">Herivel, John (1975). <i>Joseph Fourier: the man and the physicist</i>. Oxford: Clarendon Press. pp. <span class="nowrap">181–</span>191. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-858149-1" title="Special:BookSources/0-19-858149-1"><bdi>0-19-858149-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Joseph+Fourier%3A+the+man+and+the+physicist&rft.place=Oxford&rft.pages=%3Cspan+class%3D%22nowrap%22%3E181-%3C%2Fspan%3E191&rft.pub=Clarendon+Press&rft.date=1975&rft.isbn=0-19-858149-1&rft.aulast=Herivel&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFlux" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFFourier1822" class="citation book cs1 cs1-prop-foreign-lang-source">Fourier, Joseph (1822). <a rel="nofollow" class="external text" href="https://archive.org/details/bub_gb_TDQJAAAAIAAJ"><i>Théorie analytique de la chaleur</i></a> (in French). Paris: Firmin Didot Père et Fils. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/2688081">2688081</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Th%C3%A9orie+analytique+de+la+chaleur&rft.place=Paris&rft.pub=Firmin+Didot+P%C3%A8re+et+Fils&rft.date=1822&rft_id=info%3Aoclcnum%2F2688081&rft.aulast=Fourier&rft.aufirst=Joseph&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fbub_gb_TDQJAAAAIAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFlux" class="Z3988"></span></span> </li> <li id="cite_note-Maxwell-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Maxwell_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Maxwell_5-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Maxwell_5-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMaxwell1892" class="citation book cs1"><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell, James Clerk</a> (1892). <i>Treatise on Electricity and Magnetism</i>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-60636-8" title="Special:BookSources/0-486-60636-8"><bdi>0-486-60636-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Treatise+on+Electricity+and+Magnetism&rft.date=1892&rft.isbn=0-486-60636-8&rft.aulast=Maxwell&rft.aufirst=James+Clerk&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFlux" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBirdStewart,_Warren_E.Lightfoot,_Edwin_N.1960" class="citation book cs1"><a href="/wiki/Robert_Byron_Bird" title="Robert Byron Bird">Bird, R. Byron</a>; Stewart, Warren E.; <a href="/wiki/Edwin_N._Lightfoot" title="Edwin N. Lightfoot">Lightfoot, Edwin N.</a> (1960). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/transportphenome00bird"><i>Transport Phenomena</i></a></span>. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-07392-X" title="Special:BookSources/0-471-07392-X"><bdi>0-471-07392-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Transport+Phenomena&rft.pub=Wiley&rft.date=1960&rft.isbn=0-471-07392-X&rft.aulast=Bird&rft.aufirst=R.+Byron&rft.au=Stewart%2C+Warren+E.&rft.au=Lightfoot%2C+Edwin+N.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftransportphenome00bird&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFlux" class="Z3988"></span></span> </li> <li id="cite_note-Physics_P.M-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-Physics_P.M_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Physics_P.M_7-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFP.M._WhelanM.J._Hodgeson1978" class="citation book cs1">P.M. Whelan; M.J. Hodgeson (1978). <i>Essential Principles of Physics</i> (2nd ed.). John Murray. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7195-3382-1" title="Special:BookSources/0-7195-3382-1"><bdi>0-7195-3382-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Essential+Principles+of+Physics&rft.edition=2nd&rft.pub=John+Murray&rft.date=1978&rft.isbn=0-7195-3382-1&rft.au=P.M.+Whelan&rft.au=M.J.+Hodgeson&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFlux" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFCarslawJaeger,_J.C.1959" class="citation book cs1">Carslaw, H.S.; Jaeger, J.C. (1959). <i>Conduction of Heat in Solids</i> (Second ed.). Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-853303-9" title="Special:BookSources/0-19-853303-9"><bdi>0-19-853303-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Conduction+of+Heat+in+Solids&rft.edition=Second&rft.pub=Oxford+University+Press&rft.date=1959&rft.isbn=0-19-853303-9&rft.aulast=Carslaw&rft.aufirst=H.S.&rft.au=Jaeger%2C+J.C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFlux" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWeltyWicks,_Wilson_and_Rorrer2001" class="citation book cs1">Welty; Wicks, Wilson and Rorrer (2001). <i>Fundamentals of Momentum, Heat, and Mass Transfer</i> (4th ed.). Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-38149-7" title="Special:BookSources/0-471-38149-7"><bdi>0-471-38149-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Momentum%2C+Heat%2C+and+Mass+Transfer&rft.edition=4th&rft.pub=Wiley&rft.date=2001&rft.isbn=0-471-38149-7&rft.au=Welty&rft.au=Wicks%2C+Wilson+and+Rorrer&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFlux" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFD._McMahon2008" class="citation book cs1">D. McMahon (2008). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/quantumfieldtheo0000mcma"><i>Quantum Mechanics Demystified</i></a></span> (2nd ed.). Mc Graw Hill. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-145546-6" title="Special:BookSources/978-0-07-145546-6"><bdi>978-0-07-145546-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantum+Mechanics+Demystified&rft.edition=2nd&rft.pub=Mc+Graw+Hill&rft.date=2008&rft.isbn=978-0-07-145546-6&rft.au=D.+McMahon&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fquantumfieldtheo0000mcma&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFlux" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSakurai,_J._J.1967" class="citation book cs1">Sakurai, J. J. (1967). <i>Advanced Quantum Mechanics</i>. Addison Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-06710-2" title="Special:BookSources/0-201-06710-2"><bdi>0-201-06710-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Quantum+Mechanics&rft.pub=Addison+Wesley&rft.date=1967&rft.isbn=0-201-06710-2&rft.au=Sakurai%2C+J.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFlux" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMurray_R._SpiegelS._LipcshutzD._Spellman2009" class="citation book cs1">Murray R. Spiegel; S. Lipcshutz; D. Spellman (2009). <a rel="nofollow" class="external text" href="https://archive.org/details/vectoranalysis0000unse_t6w7"><i>Vector Analysis</i></a>. Schaum's Outlines (2nd ed.). McGraw Hill. p. 100. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-07-161545-7" title="Special:BookSources/978-0-07-161545-7"><bdi>978-0-07-161545-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Vector+Analysis&rft.series=Schaum%27s+Outlines&rft.pages=100&rft.edition=2nd&rft.pub=McGraw+Hill&rft.date=2009&rft.isbn=978-0-07-161545-7&rft.au=Murray+R.+Spiegel&rft.au=S.+Lipcshutz&rft.au=D.+Spellman&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fvectoranalysis0000unse_t6w7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFlux" class="Z3988"></span></span> </li> <li id="cite_note-Electromagnetism_2008-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-Electromagnetism_2008_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Electromagnetism_2008_13-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFI.S._GrantW.R._Phillips2008" class="citation book cs1">I.S. Grant; W.R. Phillips (2008). <i>Electromagnetism</i>. Manchester Physics (2nd ed.). <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-92712-9" title="Special:BookSources/978-0-471-92712-9"><bdi>978-0-471-92712-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electromagnetism&rft.series=Manchester+Physics&rft.edition=2nd&rft.pub=John+Wiley+%26+Sons&rft.date=2008&rft.isbn=978-0-471-92712-9&rft.au=I.S.+Grant&rft.au=W.R.+Phillips&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFlux" class="Z3988"></span></span> </li> <li id="cite_note-Electrodynamics_2007-14"><span class="mw-cite-backlink">^ <a href="#cite_ref-Electrodynamics_2007_14-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Electrodynamics_2007_14-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Electrodynamics_2007_14-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFD.J._Griffiths2007" class="citation book cs1">D.J. Griffiths (2007). <i>Introduction to Electrodynamics</i> (3rd ed.). Pearson Education, <a href="/wiki/Dorling_Kindersley" class="mw-redirect" title="Dorling Kindersley">Dorling Kindersley</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-7758-293-2" title="Special:BookSources/978-81-7758-293-2"><bdi>978-81-7758-293-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Electrodynamics&rft.edition=3rd&rft.pub=Pearson+Education%2C+Dorling+Kindersley&rft.date=2007&rft.isbn=978-81-7758-293-2&rft.au=D.J.+Griffiths&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFlux" class="Z3988"></span></span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/II_04.html#Ch4-S5-p7">The Feynman Lectures on Physics Vol. II Ch. 4: Electrostatics</a></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFWangsness1986" class="citation book cs1">Wangsness, Roald K. (1986). <i>Electromagnetic Fields</i> (2nd ed.). Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-81186-6" title="Special:BookSources/0-471-81186-6"><bdi>0-471-81186-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electromagnetic+Fields&rft.edition=2nd&rft.pub=Wiley&rft.date=1986&rft.isbn=0-471-81186-6&rft.aulast=Wangsness&rft.aufirst=Roald+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFlux" class="Z3988"></span> p.357</span> </li> </ol></div></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBrowne2010" class="citation book cs1">Browne, Michael (2010). <i>Physics for Engineering and Science, 2nd Edition</i>. Schaum Outlines. New York, Toronto: <a href="/wiki/McGraw-Hill_Education" class="mw-redirect" title="McGraw-Hill Education">McGraw-Hill Publishing</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-0716-1399-6" title="Special:BookSources/978-0-0716-1399-6"><bdi>978-0-0716-1399-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics+for+Engineering+and+Science%2C+2nd+Edition.&rft.place=New+York%2C+Toronto&rft.series=Schaum+Outlines&rft.pub=McGraw-Hill+Publishing&rft.date=2010&rft.isbn=978-0-0716-1399-6&rft.aulast=Browne&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFlux" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFPurcell2013" class="citation book cs1">Purcell, Edward (2013). <i>Electricity and Magnetism, 3rd Edition</i>. Cambridge, UK: <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978110-7014022" title="Special:BookSources/978110-7014022"><bdi>978110-7014022</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electricity+and+Magnetism%2C+3rd+Edition&rft.place=Cambridge%2C+UK&rft.pub=Cambridge+University+Press&rft.date=2013&rft.isbn=978110-7014022&rft.aulast=Purcell&rft.aufirst=Edward&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFlux" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Flux&action=edit&section=17" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFStauffer,_P.H.2006" class="citation journal cs1">Stauffer, P.H. (2006). <a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1745-6584.2006.00197.x">"Flux Flummoxed: A Proposal for Consistent Usage"</a>. <i>Ground Water</i>. <b>44</b> (2): <span class="nowrap">125–</span>128. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006GrWat..44..125S">2006GrWat..44..125S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1745-6584.2006.00197.x">10.1111/j.1745-6584.2006.00197.x</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16556188">16556188</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:21812226">21812226</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ground+Water&rft.atitle=Flux+Flummoxed%3A+A+Proposal+for+Consistent+Usage&rft.volume=44&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E125-%3C%2Fspan%3E128&rft.date=2006&rft_id=info%3Adoi%2F10.1111%2Fj.1745-6584.2006.00197.x&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A21812226%23id-name%3DS2CID&rft_id=info%3Apmid%2F16556188&rft_id=info%3Abibcode%2F2006GrWat..44..125S&rft.au=Stauffer%2C+P.H.&rft_id=https%3A%2F%2Fdoi.org%2F10.1111%252Fj.1745-6584.2006.00197.x&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFlux" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Flux&action=edit&section=18" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wiktionary-logo-en-v2.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/20px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png 1.5x" data-file-width="512" data-file-height="512" /></a></span> The dictionary definition of <a href="https://en.wiktionary.org/wiki/Special:Search/flux" class="extiw" title="wiktionary:Special:Search/flux"><i>flux</i></a> at Wiktionary</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Flux&oldid=1277640585">https://en.wikipedia.org/w/index.php?title=Flux&oldid=1277640585</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Physical_quantities" title="Category:Physical quantities">Physical quantities</a></li><li><a href="/wiki/Category:Vector_calculus" title="Category:Vector calculus">Vector calculus</a></li><li><a href="/wiki/Category:Rates" title="Category:Rates">Rates</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_French-language_sources_(fr)" title="Category:CS1 French-language sources (fr)">CS1 French-language sources (fr)</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 25 February 2025, at 21:22<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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vector-button-flush-left" > <input type="checkbox" id="vector-sticky-header-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-sticky-header-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-sticky-header-toc-label" for="vector-sticky-header-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-sticky-header-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div class="vector-sticky-header-context-bar-primary" aria-hidden="true" ><span class="mw-page-title-main">Flux</span></div> </div> </div> <div class="vector-sticky-header-end" aria-hidden="true"> <div class="vector-sticky-header-icons"> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-talk-sticky-header" tabindex="-1" data-event-name="talk-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbles mw-ui-icon-wikimedia-speechBubbles"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-subject-sticky-header" tabindex="-1" data-event-name="subject-sticky-header"><span class="vector-icon mw-ui-icon-article mw-ui-icon-wikimedia-article"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-history-sticky-header" tabindex="-1" data-event-name="history-sticky-header"><span class="vector-icon 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data-event-name="ve-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-edit mw-ui-icon-wikimedia-wikimedia-edit"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-editLock mw-ui-icon-wikimedia-wikimedia-editLock"></span> <span></span> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-language mw-ui-icon-wikimedia-wikimedia-language"></span> <span>42 languages</span> </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet 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