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class="vector-toc-numb">2</span> <span>Electrostatics</span> </div> </a> <button aria-controls="toc-Electrostatics-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 Electrostatics subsection</span> </button> <ul id="toc-Electrostatics-sublist" class="vector-toc-list"> <li id="toc-Electric_potential_due_to_a_point_charge" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Electric_potential_due_to_a_point_charge"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Electric potential due to a point charge</span> </div> </a> <ul id="toc-Electric_potential_due_to_a_point_charge-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalization_to_electrodynamics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalization_to_electrodynamics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Generalization to electrodynamics</span> </div> </a> <button aria-controls="toc-Generalization_to_electrodynamics-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 Generalization to electrodynamics subsection</span> </button> <ul id="toc-Generalization_to_electrodynamics-sublist" class="vector-toc-list"> <li id="toc-Gauge_freedom" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gauge_freedom"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Gauge freedom</span> </div> </a> <ul id="toc-Gauge_freedom-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Units" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Units"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Units</span> </div> </a> <ul id="toc-Units-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Galvani_potential_versus_electrochemical_potential" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Galvani_potential_versus_electrochemical_potential"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Galvani potential versus electrochemical potential</span> </div> </a> <ul id="toc-Galvani_potential_versus_electrochemical_potential-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Common_formulas" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Common_formulas"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Common formulas</span> </div> </a> <ul id="toc-Common_formulas-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">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-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">9</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> </ul> 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</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">Electric potential</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 65 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-65" 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">65 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/Elektriese_potensiaal" title="Elektriese potensiaal – Afrikaans" lang="af" hreflang="af" data-title="Elektriese potensiaal" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8B%A8%E1%8A%A4%E1%88%8C%E1%8A%AD%E1%89%B5%E1%88%AA%E1%8A%AD_%E1%8A%A5%E1%88%9D%E1%89%85" title="የኤሌክትሪክ እምቅ – Amharic" lang="am" hreflang="am" data-title="የኤሌክትሪክ እምቅ" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%83%D9%85%D9%88%D9%86_%D9%83%D9%87%D8%B1%D8%A8%D8%A7%D8%A6%D9%8A" title="كمون كهربائي – Arabic" lang="ar" hreflang="ar" data-title="كمون كهربائي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Potencial_ll%C3%A9tricu" title="Potencial llétricu – Asturian" lang="ast" hreflang="ast" data-title="Potencial llétricu" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A4%E0%A6%A1%E0%A6%BC%E0%A6%BF%E0%A7%8E_%E0%A6%AC%E0%A6%BF%E0%A6%AD%E0%A6%AC" title="তড়িৎ বিভব – Bangla" lang="bn" hreflang="bn" data-title="তড়িৎ বিভব" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%AD%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D1%8B%D1%87%D0%BD%D1%8B_%D0%BF%D0%B0%D1%82%D1%8D%D0%BD%D1%86%D1%8B%D1%8F%D0%BB" title="Электрычны патэнцыял – Belarusian" lang="be" hreflang="be" data-title="Электрычны патэнцыял" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%AD%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D1%8B%D1%87%D0%BD%D1%8B_%D0%BF%D0%B0%D1%82%D1%8D%D0%BD%D1%86%D1%8B%D1%8F%D0%BB" title="Электрычны патэнцыял – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Электрычны патэнцыял" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%95%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%B8%D1%87%D0%B5%D0%BD_%D0%BF%D0%BE%D1%82%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB" title="Електричен потенциал – Bulgarian" lang="bg" hreflang="bg" data-title="Електричен потенциал" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/Elektrisches_Potenzi%C3%A4u" title="Elektrisches Potenziäu – Bavarian" lang="bar" hreflang="bar" data-title="Elektrisches Potenziäu" data-language-autonym="Boarisch" data-language-local-name="Bavarian" class="interlanguage-link-target"><span>Boarisch</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Elektri%C4%8Dni_potencijal" title="Električni potencijal – Bosnian" lang="bs" hreflang="bs" data-title="Električni potencijal" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Potencial_el%C3%A8ctric" title="Potencial elèctric – Catalan" lang="ca" hreflang="ca" data-title="Potencial elèctric" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%AD%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%BE%D1%81%D1%82%D0%B0%D1%82%D0%B8%D0%BA%C4%83%D0%BB%D0%BB%D0%B0_%D0%BF%D0%BE%D1%82%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB" title="Электростатикăлла потенциал – Chuvash" lang="cv" hreflang="cv" data-title="Электростатикăлла потенциал" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Elektrick%C3%BD_potenci%C3%A1l" title="Elektrický potenciál – Czech" lang="cs" hreflang="cs" data-title="Elektrický potenciál" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Elektrisk_potential" title="Elektrisk potential – Danish" lang="da" hreflang="da" data-title="Elektrisk potential" 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/Elektrisches_Potential" title="Elektrisches Potential – German" lang="de" hreflang="de" data-title="Elektrisches Potential" 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/Elektriv%C3%A4lja_potentsiaal" title="Elektrivälja potentsiaal – Estonian" lang="et" hreflang="et" data-title="Elektrivälja potentsiaal" 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%97%CE%BB%CE%B5%CE%BA%CF%84%CF%81%CE%B9%CE%BA%CF%8C_%CE%B4%CF%85%CE%BD%CE%B1%CE%BC%CE%B9%CE%BA%CF%8C" title="Ηλεκτρικό δυναμικό – Greek" lang="el" hreflang="el" data-title="Ηλεκτρικό δυναμικό" 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/Potencial_el%C3%A9ctrico" title="Potencial eléctrico – Spanish" lang="es" hreflang="es" data-title="Potencial eléctrico" 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/Elektra_potencialo" title="Elektra potencialo – Esperanto" lang="eo" hreflang="eo" data-title="Elektra potencialo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Potentzial_elektriko" title="Potentzial elektriko – Basque" lang="eu" hreflang="eu" data-title="Potentzial elektriko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%BE%D8%AA%D8%A7%D9%86%D8%B3%DB%8C%D9%84_%D8%A7%D9%84%DA%A9%D8%AA%D8%B1%DB%8C%DA%A9%DB%8C" title="پتانسیل الکتریکی – Persian" lang="fa" hreflang="fa" data-title="پتانسیل الکتریکی" 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/Potentiel_%C3%A9lectrique" title="Potentiel électrique – French" lang="fr" hreflang="fr" data-title="Potentiel électrique" 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/Poit%C3%A9inseal_leictreach" title="Poitéinseal leictreach – Irish" lang="ga" hreflang="ga" data-title="Poitéinseal leictreach" 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/Potencial_el%C3%A9ctrico" title="Potencial eléctrico – Galician" lang="gl" hreflang="gl" data-title="Potencial eléctrico" 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%A0%84%EC%9C%84" title="전위 – Korean" lang="ko" hreflang="ko" data-title="전위" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%AF%E0%A5%81%E0%A4%A4_%E0%A4%B5%E0%A4%BF%E0%A4%AD%E0%A4%B5" title="विद्युत विभव – Hindi" lang="hi" hreflang="hi" data-title="विद्युत विभव" 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/Elektri%C4%8Dni_potencijal" title="Električni potencijal – Croatian" lang="hr" hreflang="hr" data-title="Električni potencijal" 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/Potensial_listrik" title="Potensial listrik – Indonesian" lang="id" hreflang="id" data-title="Potensial listrik" 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/Potenziale_elettrico" title="Potenziale elettrico – Italian" lang="it" hreflang="it" data-title="Potenziale elettrico" 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%A4%D7%95%D7%98%D7%A0%D7%A6%D7%99%D7%90%D7%9C_%D7%97%D7%A9%D7%9E%D7%9C%D7%99" title="פוטנציאל חשמלי – Hebrew" lang="he" hreflang="he" data-title="פוטנציאל חשמלי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%94%E1%83%9A%E1%83%94%E1%83%A5%E1%83%A2%E1%83%A0%E1%83%A3%E1%83%9A%E1%83%98_%E1%83%9E%E1%83%9D%E1%83%A2%E1%83%94%E1%83%9C%E1%83%AA%E1%83%98%E1%83%90%E1%83%9A%E1%83%98" title="ელექტრული პოტენციალი – Georgian" lang="ka" hreflang="ka" data-title="ელექტრული პოტენციალი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Elektromos_potenci%C3%A1l" title="Elektromos potenciál – Hungarian" lang="hu" hreflang="hu" data-title="Elektromos potenciál" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%95%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%B8%D1%87%D0%B5%D0%BD_%D0%BF%D0%BE%D1%82%D0%B5%D0%BD%D1%86%D0%B8%D1%98%D0%B0%D0%BB" title="Електричен потенцијал – Macedonian" lang="mk" hreflang="mk" data-title="Електричен потенцијал" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Angovon-toerana_araka_aratra" title="Angovon-toerana araka aratra – Malagasy" lang="mg" hreflang="mg" data-title="Angovon-toerana araka aratra" data-language-autonym="Malagasy" data-language-local-name="Malagasy" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%AF%E0%A5%81%E0%A4%A4_%E0%A4%B5%E0%A4%BF%E0%A4%AD%E0%A4%B5" title="विद्युत विभव – Marathi" lang="mr" hreflang="mr" data-title="विद्युत विभव" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Keupayaan_elektrik" title="Keupayaan elektrik – Malay" lang="ms" hreflang="ms" data-title="Keupayaan elektrik" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A6%D0%B0%D1%85%D0%B8%D0%BB%D0%B3%D0%B0%D0%B0%D0%BD_%D0%BF%D0%BE%D1%82%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB" title="Цахилгаан потенциал – Mongolian" lang="mn" hreflang="mn" data-title="Цахилгаан потенциал" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Elektrische_potentiaal" title="Elektrische potentiaal – Dutch" lang="nl" hreflang="nl" data-title="Elektrische potentiaal" 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/%E9%9B%BB%E4%BD%8D" title="電位 – Japanese" lang="ja" hreflang="ja" data-title="電位" 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/Elektrisk_potensial" title="Elektrisk potensial – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Elektrisk potensial" 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/Elektrisk_potensial" title="Elektrisk potensial – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Elektrisk potensial" 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-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%AC%E0%AC%BF%E0%AC%A6%E0%AD%8D%E0%AD%9F%E0%AD%81%E0%AC%A4%E0%AD%8D_%E0%AC%AC%E0%AC%BF%E0%AC%AD%E0%AC%AC" title="ବିଦ୍ୟୁତ୍ ବିଭବ – Odia" lang="or" hreflang="or" data-title="ବିଦ୍ୟୁତ୍ ବିଭବ" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="Odia" class="interlanguage-link-target"><span>ଓଡ଼ିଆ</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Elektrostatik_potensial" title="Elektrostatik potensial – Uzbek" lang="uz" hreflang="uz" data-title="Elektrostatik potensial" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Potencja%C5%82_elektryczny" title="Potencjał elektryczny – Polish" lang="pl" hreflang="pl" data-title="Potencjał elektryczny" 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/Potencial_el%C3%A9trico" title="Potencial elétrico – Portuguese" lang="pt" hreflang="pt" data-title="Potencial elétrico" 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/Poten%C8%9Bial_electric" title="Potențial electric – Romanian" lang="ro" hreflang="ro" data-title="Potențial electric" 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%AD%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%BE%D1%81%D1%82%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B9_%D0%BF%D0%BE%D1%82%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB" title="Электростатический потенциал – Russian" lang="ru" hreflang="ru" data-title="Электростатический потенциал" 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/Mundshm%C3%ABria_elektrike" title="Mundshmëria elektrike – Albanian" lang="sq" hreflang="sq" data-title="Mundshmëria elektrike" 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/Electric_potential" title="Electric potential – Simple English" lang="en-simple" hreflang="en-simple" data-title="Electric potential" 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-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Elektrick%C3%BD_potenci%C3%A1l" title="Elektrický potenciál – Slovak" lang="sk" hreflang="sk" data-title="Elektrický potenciál" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Elektri%C4%8Dni_potencial" title="Električni potencial – Slovenian" lang="sl" hreflang="sl" data-title="Električni potencial" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%BE%DB%86%D8%AA%DB%95%D9%86%D8%B4%DA%B5%DB%8C_%DA%A9%D8%A7%D8%B1%DB%95%D8%A8%D8%A7%DB%8C%DB%8C" title="پۆتەنشڵی کارەبایی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="پۆتەنشڵی کارەبایی" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%95%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%B8_%D0%BF%D0%BE%D1%82%D0%B5%D0%BD%D1%86%D0%B8%D1%98%D0%B0%D0%BB" title="Електрични потенцијал – Serbian" lang="sr" hreflang="sr" data-title="Електрични потенцијал" 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/Elektri%C4%8Dni_potencijal" title="Električni potencijal – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Električni potencijal" 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/S%C3%A4hk%C3%B6potentiaali" title="Sähköpotentiaali – Finnish" lang="fi" hreflang="fi" data-title="Sähköpotentiaali" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Elektrikong_potensiyal" title="Elektrikong potensiyal – Tagalog" lang="tl" hreflang="tl" data-title="Elektrikong potensiyal" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%AD%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%BE%D1%81%D1%82%D0%B0%D1%82%D0%B8%D0%BA_%D0%BF%D0%BE%D1%82%D0%B5%D0%BD%D1%86%D0%B8%D0%B0%D0%BB" title="Электростатик потенциал – Tatar" lang="tt" hreflang="tt" data-title="Электростатик потенциал" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A8%E0%B8%B1%E0%B8%81%E0%B8%A2%E0%B9%8C%E0%B9%84%E0%B8%9F%E0%B8%9F%E0%B9%89%E0%B8%B2" title="ศักย์ไฟฟ้า – Thai" lang="th" hreflang="th" data-title="ศักย์ไฟฟ้า" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Elektrik_potansiyeli" title="Elektrik potansiyeli – Turkish" lang="tr" hreflang="tr" data-title="Elektrik potansiyeli" 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%95%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%BE%D1%81%D1%82%D0%B0%D1%82%D0%B8%D1%87%D0%BD%D0%B8%D0%B9_%D0%BF%D0%BE%D1%82%D0%B5%D0%BD%D1%86%D1%96%D0%B0%D0%BB" title="Електростатичний потенціал – Ukrainian" lang="uk" hreflang="uk" data-title="Електростатичний потенціал" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur badge-Q70893996 mw-list-item" title=""><a href="https://ur.wikipedia.org/wiki/%D8%A8%D8%B1%D9%82%DB%8C_%D8%AC%D9%8F%DB%81%D8%AF" title="برقی جُہد – Urdu" lang="ur" hreflang="ur" data-title="برقی جُہد" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90i%E1%BB%87n_th%E1%BA%BF" title="Điện thế – Vietnamese" lang="vi" hreflang="vi" data-title="Điện thế" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%94%B5%E5%8A%BF" title="电势 – Wu" lang="wuu" hreflang="wuu" data-title="电势" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%9B%BB%E5%8B%A2" title="電勢 – Cantonese" lang="yue" hreflang="yue" data-title="電勢" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%9B%BB%E5%8B%A2" title="電勢 – Chinese" lang="zh" hreflang="zh" data-title="電勢" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a 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searchaux" style="display:none">Line integral of the electric field</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .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">Not to be confused with <a href="/wiki/Voltage" title="Voltage">Voltage</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above">Electric potential</th></tr><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:VFPt_metal_balls_largesmall_potential%2Bcontour.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/VFPt_metal_balls_largesmall_potential%2Bcontour.svg/220px-VFPt_metal_balls_largesmall_potential%2Bcontour.svg.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/VFPt_metal_balls_largesmall_potential%2Bcontour.svg/330px-VFPt_metal_balls_largesmall_potential%2Bcontour.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/68/VFPt_metal_balls_largesmall_potential%2Bcontour.svg/440px-VFPt_metal_balls_largesmall_potential%2Bcontour.svg.png 2x" data-file-width="800" data-file-height="600" /></a></span><div class="infobox-caption">Electric potential around two oppositely charged conducting spheres. Purple represents the highest potential, yellow zero, and cyan the lowest potential. The electric <a href="/wiki/Field_line" title="Field line">field lines</a> are shown leaving perpendicularly to the surface of each sphere.</div></td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Common symbols</div></th><td class="infobox-data"><i>V</i>, <i>φ</i></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI unit</a></th><td class="infobox-data"><a href="/wiki/Volt" title="Volt">volt</a></td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Other units</div></th><td class="infobox-data"><a href="/wiki/Statvolt" title="Statvolt">statvolt</a></td></tr><tr><th scope="row" class="infobox-label">In <a href="/wiki/SI_base_unit" title="SI base unit"><span class="wrap">SI base units</span></a></th><td class="infobox-data">V = kg⋅m<sup>2</sup>⋅s<sup>−3</sup>⋅A<sup>−1</sup></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Intensive_and_extensive_properties" title="Intensive and extensive properties">Extensive</a>?</th><td class="infobox-data">yes</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Dimensional_analysis#Formulation" title="Dimensional analysis">Dimension</a></th><td class="infobox-data"><b>M</b> <b>L</b><sup>2</sup> <b>T</b><sup>−3</sup> <b>I</b><sup>−1</sup></td></tr></tbody></table> <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 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href="/wiki/Watt" title="Watt">Watt</a></li> <li><a href="/wiki/Waveguide_(radio_frequency)" title="Waveguide (radio frequency)">Waveguides</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Magnetic_circuit" title="Magnetic circuit">Magnetic circuit</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/AC_motor" title="AC motor">AC motor</a></li> <li><a href="/wiki/DC_motor" title="DC motor">DC motor</a></li> <li><a href="/wiki/Electric_machine" title="Electric machine">Electric machine</a></li> <li><a href="/wiki/Electric_motor" title="Electric motor">Electric motor</a></li> <li><a href="/wiki/Gyrator%E2%80%93capacitor_model" title="Gyrator–capacitor model">Gyrator–capacitor</a></li> <li><a href="/wiki/Induction_motor" title="Induction motor">Induction motor</a></li> <li><a href="/wiki/Linear_motor" title="Linear motor">Linear motor</a></li> <li><a href="/wiki/Magnetomotive_force" title="Magnetomotive force">Magnetomotive force</a></li> <li><a href="/wiki/Permeance" title="Permeance">Permeance</a></li> <li><a href="/wiki/Magnetic_complex_reluctance" title="Magnetic complex reluctance">Reluctance (complex)</a></li> <li><a href="/wiki/Magnetic_reluctance" title="Magnetic reluctance">Reluctance (real)</a></li> <li><a href="/wiki/Rotor_(electric)" title="Rotor (electric)">Rotor</a></li> <li><a href="/wiki/Stator" title="Stator">Stator</a></li> <li><a href="/wiki/Transformer" title="Transformer">Transformer</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a href="/wiki/Covariant_formulation_of_classical_electromagnetism" title="Covariant formulation of classical electromagnetism">Covariant formulation</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Electromagnetic_tensor" title="Electromagnetic tensor">Electromagnetic tensor</a></li> <li><a href="/wiki/Classical_electromagnetism_and_special_relativity" title="Classical electromagnetism and special relativity">Electromagnetism and special relativity</a></li> <li><a href="/wiki/Four-current" title="Four-current">Four-current</a></li> <li><a href="/wiki/Electromagnetic_four-potential" title="Electromagnetic four-potential">Four-potential</a></li> <li><a href="/wiki/Mathematical_descriptions_of_the_electromagnetic_field" title="Mathematical descriptions of the electromagnetic field">Mathematical descriptions</a></li> <li><a href="/wiki/Maxwell%27s_equations_in_curved_spacetime" title="Maxwell's equations in curved spacetime">Maxwell equations in curved spacetime</a></li> <li><a href="/wiki/Relativistic_electromagnetism" title="Relativistic electromagnetism">Relativistic electromagnetism</a></li> <li><a href="/wiki/Electromagnetic_stress%E2%80%93energy_tensor" title="Electromagnetic stress–energy tensor">Stress–energy tensor</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Andr%C3%A9-Marie_Amp%C3%A8re" title="André-Marie Ampère">Ampère</a></li> <li><a href="/wiki/Jean-Baptiste_Biot" title="Jean-Baptiste Biot">Biot</a></li> <li><a href="/wiki/Charles-Augustin_de_Coulomb" title="Charles-Augustin de Coulomb">Coulomb</a></li> <li><a href="/wiki/Humphry_Davy" title="Humphry Davy">Davy</a></li> <li><a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a></li> <li><a href="/wiki/Michael_Faraday" title="Michael Faraday">Faraday</a></li> <li><a href="/wiki/Hippolyte_Fizeau" title="Hippolyte Fizeau">Fizeau</a></li> <li><a href="/wiki/Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li> <li><a href="/wiki/Oliver_Heaviside" title="Oliver Heaviside">Heaviside</a></li> <li><a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">Helmholtz</a></li> <li><a href="/wiki/Joseph_Henry" title="Joseph Henry">Henry</a></li> <li><a href="/wiki/Heinrich_Hertz" title="Heinrich Hertz">Hertz</a></li> <li><a href="/wiki/John_Hopkinson" title="John Hopkinson">Hopkinson</a></li> <li><a href="/wiki/Oleg_D._Jefimenko" title="Oleg D. Jefimenko">Jefimenko</a></li> <li><a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">Joule</a></li> <li><a href="/wiki/Lord_Kelvin" title="Lord Kelvin">Kelvin</a></li> <li><a href="/wiki/Gustav_Kirchhoff" title="Gustav Kirchhoff">Kirchhoff</a></li> <li><a href="/wiki/Joseph_Larmor" title="Joseph Larmor">Larmor</a></li> <li><a href="/wiki/Emil_Lenz" title="Emil Lenz">Lenz</a></li> <li><a href="/wiki/Alfred-Marie_Li%C3%A9nard" title="Alfred-Marie Liénard">Liénard</a></li> <li><a href="/wiki/Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li> <li><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a></li> <li><a href="/wiki/Franz_Ernst_Neumann" title="Franz Ernst Neumann">Neumann</a></li> <li><a href="/wiki/Georg_Ohm" title="Georg Ohm">Ohm</a></li> <li><a href="/wiki/Hans_Christian_%C3%98rsted" title="Hans Christian Ørsted">Ørsted</a></li> <li><a href="/wiki/Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Poisson</a></li> <li><a href="/wiki/John_Henry_Poynting" title="John Henry Poynting">Poynting</a></li> <li><a href="/wiki/William_Ritchie_(physicist)" title="William Ritchie (physicist)">Ritchie</a></li> <li><a href="/wiki/F%C3%A9lix_Savart" title="Félix Savart">Savart</a></li> <li><a href="/wiki/George_Singer" title="George Singer">Singer</a></li> <li><a href="/wiki/Charles_Proteus_Steinmetz" title="Charles Proteus Steinmetz">Steinmetz</a></li> <li><a href="/wiki/Nikola_Tesla" title="Nikola Tesla">Tesla</a></li> <li><a href="/wiki/J._J._Thomson" title="J. J. Thomson">Thomson</a></li> <li><a href="/wiki/Alessandro_Volta" title="Alessandro Volta">Volta</a></li> <li><a href="/wiki/Wilhelm_Eduard_Weber" title="Wilhelm Eduard Weber">Weber</a></li> <li><a href="/wiki/Emil_Wiechert" title="Emil Wiechert">Wiechert</a></li></ul></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><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:Electromagnetism" title="Template:Electromagnetism"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Electromagnetism" title="Template talk:Electromagnetism"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Electromagnetism" title="Special:EditPage/Template:Electromagnetism"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Electric potential</b> (also called the <i>electric field potential</i>, potential drop, the <b>electrostatic potential</b>) is defined as the amount of <a href="/wiki/Work_(physics)" title="Work (physics)">work</a>/<a href="/wiki/Energy" title="Energy">energy</a> needed per unit of <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a> to move the charge from a reference point to a specific point in an electric field. More precisely, the electric potential is the energy per unit charge for a <a href="/wiki/Test_charge" class="mw-redirect" title="Test charge">test charge</a> that is so small that the disturbance of the field under consideration is negligible. The motion across the field is supposed to proceed with negligible acceleration, so as to avoid the test charge acquiring kinetic energy or producing radiation. By definition, the electric potential at the reference point is zero units. Typically, the reference point is <a href="/wiki/Earth_(electricity)" class="mw-redirect" title="Earth (electricity)">earth</a> or a point at <a href="/wiki/Infinity" title="Infinity">infinity</a>, although any point can be used. </p><p>In classical <a href="/wiki/Electrostatics" title="Electrostatics">electrostatics</a>, the electrostatic field is a vector quantity expressed as the gradient of the electrostatic potential, which is a <a href="/wiki/Scalar_(physics)" title="Scalar (physics)">scalar</a> quantity denoted by <span class="texhtml"><i>V</i></span> or occasionally <span class="texhtml"><i>φ</i></span>,<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> equal to the <a href="/wiki/Electric_potential_energy" title="Electric potential energy">electric potential energy</a> of any <a href="/wiki/Charged_particle" title="Charged particle">charged particle</a> at any location (measured in <a href="/wiki/Joule" title="Joule">joules</a>) divided by the <a href="/wiki/Electric_charge" title="Electric charge">charge</a> of that particle (measured in <a href="/wiki/Coulomb" title="Coulomb">coulombs</a>). By dividing out the charge on the particle a quotient is obtained that is a property of the electric field itself. In short, an electric potential is the <a href="/wiki/Electric_potential_energy" title="Electric potential energy">electric potential energy</a> per unit charge. </p><p>This value can be calculated in either a static (time-invariant) or a dynamic (time-varying) <a href="/wiki/Electric_field" title="Electric field">electric field</a> at a specific time with the unit joules per coulomb (J⋅C<sup>−1</sup>) or <a href="/wiki/Volt" title="Volt">volt</a> (V). The electric potential at infinity is assumed to be zero. </p><p>In <a href="/wiki/Electrodynamics" class="mw-redirect" title="Electrodynamics">electrodynamics</a>, when time-varying fields are present, the electric field cannot be expressed only as a <a href="/wiki/Scalar_potential" title="Scalar potential">scalar potential</a>. Instead, the electric field can be expressed as both the scalar electric potential and the <a href="/wiki/Magnetic_vector_potential" title="Magnetic vector potential">magnetic vector potential</a>.<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> The electric potential and the magnetic vector potential together form a <a href="/wiki/Four_vector" class="mw-redirect" title="Four vector">four-vector</a>, so that the two kinds of potential are mixed under <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformations</a>. </p><p>Practically, the electric potential is a <a href="/wiki/Continuous_function" title="Continuous function">continuous function</a> in all space, because a spatial derivative of a discontinuous electric potential yields an electric field of impossibly infinite magnitude. Notably, the electric potential due to an idealized <a href="/wiki/Point_charge" class="mw-redirect" title="Point charge">point charge</a> (proportional to <span class="texhtml">1 ⁄ <i>r</i></span>, with <span class="texhtml"><i>r</i></span> the distance from the point charge) is continuous in all space except at the location of the point charge. Though electric field is not continuous across an idealized <a href="/wiki/Surface_charge" title="Surface charge">surface charge</a>, it is not infinite at any point. Therefore, the electric potential is continuous across an idealized surface charge. Additionally, an idealized line of charge has electric potential (proportional to <span class="texhtml">ln(<i>r</i>)</span>, with <span class="texhtml"><i>r</i></span> the radial distance from the line of charge) is continuous everywhere except on the line of charge. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Introduction">Introduction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electric_potential&action=edit&section=1" title="Edit section: Introduction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Classical_mechanics" title="Classical mechanics">Classical mechanics</a> explores concepts such as <a href="/wiki/Force_(physics)" class="mw-redirect" title="Force (physics)">force</a>, <a href="/wiki/Energy" title="Energy">energy</a>, and <a href="/wiki/Potential" title="Potential">potential</a>.<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> Force and potential energy are directly related. A net force acting on any object will cause it to <a href="/wiki/Acceleration" title="Acceleration">accelerate</a>. As an object moves in the direction of a force acting on it, its potential energy decreases. For example, the <a href="/wiki/Gravitational_potential_energy" class="mw-redirect" title="Gravitational potential energy">gravitational potential energy</a> of a cannonball at the top of a hill is greater than at the base of the hill. As it rolls downhill, its potential energy decreases and is being translated to motion – <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a>. </p><p>It is possible to define the potential of certain force fields so that the potential energy of an object in that field depends only on the position of the object with respect to the field. Two such force fields are a <a href="/wiki/Gravity" title="Gravity">gravitational field</a> and an electric field (in the absence of time-varying magnetic fields). Such fields affect objects because of the intrinsic properties (e.g., <a href="/wiki/Mass" title="Mass">mass</a> or charge) and positions of the objects. </p><p>An object may possess a property known as <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a>. Since an <a href="/wiki/Electric_field" title="Electric field">electric field</a> exerts force on a charged object, if the object has a positive charge, the force will be in the direction of the <a href="/wiki/Electric_field_vector" class="mw-redirect" title="Electric field vector">electric field vector</a> at the location of the charge; if the charge is negative, the force will be in the opposite direction. </p><p>The magnitude of force is given by the quantity of the charge multiplied by the magnitude of the electric field vector, </p><p><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 {F} |=q|\mathbf {E} |.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathbf {F} |=q|\mathbf {E} |.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff2202138fd0cffd4a8a454e8352ff1766e30743" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.842ex; height:2.843ex;" alt="{\displaystyle |\mathbf {F} |=q|\mathbf {E} |.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Electrostatics">Electrostatics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electric_potential&action=edit&section=2" title="Edit section: Electrostatics"><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/Electrostatics" title="Electrostatics">Electrostatics</a></div><style data-mw-deduplicate="TemplateStyles:r1273380762/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:292px;max-width:292px"><div class="trow"><div class="tsingle" style="width:144px;max-width:144px"><div class="thumbimage" style="height:142px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:VFPt_plus_thumb_potential%2Bcontour.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/VFPt_plus_thumb_potential%2Bcontour.svg/142px-VFPt_plus_thumb_potential%2Bcontour.svg.png" decoding="async" width="142" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/VFPt_plus_thumb_potential%2Bcontour.svg/213px-VFPt_plus_thumb_potential%2Bcontour.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/VFPt_plus_thumb_potential%2Bcontour.svg/284px-VFPt_plus_thumb_potential%2Bcontour.svg.png 2x" data-file-width="180" data-file-height="180" /></a></span></div></div><div class="tsingle" style="width:144px;max-width:144px"><div class="thumbimage" style="height:142px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:VFPt_minus_thumb_potential%2Bcontour.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/VFPt_minus_thumb_potential%2Bcontour.svg/142px-VFPt_minus_thumb_potential%2Bcontour.svg.png" decoding="async" width="142" height="142" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/VFPt_minus_thumb_potential%2Bcontour.svg/213px-VFPt_minus_thumb_potential%2Bcontour.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7a/VFPt_minus_thumb_potential%2Bcontour.svg/284px-VFPt_minus_thumb_potential%2Bcontour.svg.png 2x" data-file-width="180" data-file-height="180" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Electric potential of separate positive and negative point charges shown as color range from magenta (+), through yellow (0), to cyan (−). Circular contours are equipotential lines. Electric field lines leave the positive charge and enter the negative charge.</div></div></div></div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1273380762/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:292px;max-width:292px"><div class="trow"><div class="tsingle" style="width:290px;max-width:290px"><div class="thumbimage" style="height:216px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:VFPt_charges_plus_minus_potential%2Bcontour.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/VFPt_charges_plus_minus_potential%2Bcontour.svg/288px-VFPt_charges_plus_minus_potential%2Bcontour.svg.png" decoding="async" width="288" height="216" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/VFPt_charges_plus_minus_potential%2Bcontour.svg/432px-VFPt_charges_plus_minus_potential%2Bcontour.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e0/VFPt_charges_plus_minus_potential%2Bcontour.svg/576px-VFPt_charges_plus_minus_potential%2Bcontour.svg.png 2x" data-file-width="800" data-file-height="600" /></a></span></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Electric potential in the vicinity of two opposite point charges.</div></div></div></div> <p>An electric potential at a point <span class="texhtml"><b>r</b></span> in a static <a href="/wiki/Electric_field" title="Electric field">electric field</a> <span class="texhtml"><b>E</b></span> is given by the <a href="/wiki/Line_integral" title="Line integral">line integral</a> </p> <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 5px; border-width:2px; border-style: solid; border-color: #50C478; color: inherit;text-align: center; display: table"> <p><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 V_{\mathbf {E} }=-\int _{\mathcal {C}}\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">C</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ℓ</mi> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathbf {E} }=-\int _{\mathcal {C}}\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d41f889ccbb8b7cfb5be343a52f9f7da09fc61fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:17.129ex; height:5.676ex;" alt="{\displaystyle V_{\mathbf {E} }=-\int _{\mathcal {C}}\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}\,}"></span> </p> </div> <p>where <span class="texhtml mvar" style="font-style:italic;">C</span> is an arbitrary path from some fixed reference point to <span class="texhtml"><b>r</b></span>; it is uniquely determined up to a constant that is added or subtracted from the integral. In electrostatics, the <a href="/wiki/Maxwell-Faraday_equation" class="mw-redirect" title="Maxwell-Faraday equation">Maxwell-Faraday equation</a> reveals that the <a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">curl</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="{\textstyle \nabla \times \mathbf {E} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \nabla \times \mathbf {E} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f75af4853f5b7d2c35e4ab4178f4b3f79ec90aff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.533ex; height:2.176ex;" alt="{\textstyle \nabla \times \mathbf {E} }"></span> is zero, making the electric field <a href="/wiki/Conservative_vector_field" title="Conservative vector field">conservative</a>. Thus, the line integral above does not depend on the specific path <span class="texhtml mvar" style="font-style:italic;">C</span> chosen but only on its endpoints, making <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="{\textstyle V_{\mathbf {E} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle V_{\mathbf {E} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3037f2735f72b76fb133deff890a85f6466928c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.83ex; height:2.509ex;" alt="{\textstyle V_{\mathbf {E} }}"></span> well-defined everywhere. The <a href="/wiki/Gradient_theorem" title="Gradient theorem">gradient theorem</a> then allows us to write: </p> <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 5px; border-width:2px; border-style: solid; border-color: #50C478; color: inherit;text-align: center; display: table"> <p><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} =-\mathbf {\nabla } V_{\mathbf {E} }\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} =-\mathbf {\nabla } V_{\mathbf {E} }\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a43c3ef9e446e11f228af803a19aa4242668d921" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.817ex; height:2.509ex;" alt="{\displaystyle \mathbf {E} =-\mathbf {\nabla } V_{\mathbf {E} }\,}"></span> </p> </div> <p>This states that the electric field points "downhill" towards lower voltages. By <a href="/wiki/Gauss%27s_law" title="Gauss's law">Gauss's law</a>, the potential can also be found to satisfy <a href="/wiki/Poisson%27s_equation" title="Poisson's equation">Poisson's equation</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 {\nabla } \cdot \mathbf {E} =\mathbf {\nabla } \cdot \left(-\mathbf {\nabla } V_{\mathbf {E} }\right)=-\nabla ^{2}V_{\mathbf {E} }=\rho /\varepsilon _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\nabla } \cdot \mathbf {E} =\mathbf {\nabla } \cdot \left(-\mathbf {\nabla } V_{\mathbf {E} }\right)=-\nabla ^{2}V_{\mathbf {E} }=\rho /\varepsilon _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1d70be3bf255ebc1b6d84cb032688e55a3540c7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.796ex; height:3.176ex;" alt="{\displaystyle \mathbf {\nabla } \cdot \mathbf {E} =\mathbf {\nabla } \cdot \left(-\mathbf {\nabla } V_{\mathbf {E} }\right)=-\nabla ^{2}V_{\mathbf {E} }=\rho /\varepsilon _{0}}"></span> </p><p>where <span class="texhtml mvar" style="font-style:italic;">ρ</span> is the total <a href="/wiki/Charge_density" title="Charge density">charge density</a> and <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="{\textstyle \mathbf {\nabla } \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mathbf {\nabla } \cdot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa3674f6b9254dfa32d14b43c7f25e33fc2dc26d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.583ex; height:2.176ex;" alt="{\textstyle \mathbf {\nabla } \cdot }"></span> denotes the <a href="/wiki/Divergence" title="Divergence">divergence</a>. </p><p>The concept of electric potential is closely linked with <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a>. A <a href="/wiki/Test_charge" class="mw-redirect" title="Test charge">test charge</a>, <span class="texhtml"><i>q</i></span>, has an <a href="/wiki/Electric_potential_energy" title="Electric potential energy">electric potential energy</a>, <span class="texhtml"><i>U</i><sub><b>E</b></sub></span>, given by </p><p><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 U_{\mathbf {E} }=q\,V.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>q</mi> <mspace width="thinmathspace" /> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\mathbf {E} }=q\,V.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/772aa23e7fbc9632ab276763b187d603803e3966" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.051ex; height:2.509ex;" alt="{\displaystyle U_{\mathbf {E} }=q\,V.}"></span> </p><p>The potential energy and hence, also the electric potential, is only defined up to an additive constant: one must arbitrarily choose a position where the potential energy and the electric potential are zero. </p><p>These equations cannot be used if <span class="nowrap"><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="{\textstyle \nabla \times \mathbf {E} \neq \mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \nabla \times \mathbf {E} \neq \mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76810a2ec3b3f8adc90c4acbee55166e96dca8f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.968ex; height:2.676ex;" alt="{\textstyle \nabla \times \mathbf {E} \neq \mathbf {0} }"></span>,</span> i.e., in the case of a <i>non-conservative electric field</i> (caused by a changing <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a>; see <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a>). The generalization of electric potential to this case is described in the section <a href="#Generalization_to_electrodynamics">§ Generalization to electrodynamics</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Electric_potential_due_to_a_point_charge">Electric potential due to a point charge</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electric_potential&action=edit&section=3" title="Edit section: Electric potential due to a point charge"><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">See also: <a href="/wiki/Coulomb%27s_law" title="Coulomb's law">Coulomb's law</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Electric_potential_varying_charge.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Electric_potential_varying_charge.gif/150px-Electric_potential_varying_charge.gif" decoding="async" width="150" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Electric_potential_varying_charge.gif/225px-Electric_potential_varying_charge.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/f/f8/Electric_potential_varying_charge.gif 2x" data-file-width="300" data-file-height="300" /></a><figcaption>The electric potential created by a charge, <i>Q</i>, is <i>V</i> = <i>Q</i>/(4πε<sub>0</sub><i>r</i>). Different values of <i>Q</i> yield different values of electric potential, <i>V</i>, (shown in the image).</figcaption></figure> <p>The electric potential arising from a point charge, <span class="texhtml"><i>Q</i></span>, at a distance, <span class="texhtml"><i>r</i></span>, from the location of <span class="texhtml"><i>Q</i></span> is observed to be <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 V_{\mathbf {E} }={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Q}{r}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <mi>r</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathbf {E} }={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Q}{r}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bf69e52ce87fda9227841372479d16de9f83aa9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.718ex; height:5.843ex;" alt="{\displaystyle V_{\mathbf {E} }={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Q}{r}},}"></span> where <span class="texhtml"><i>ε</i><sub>0</sub></span> is the <a href="/wiki/Permittivity_of_vacuum" class="mw-redirect" title="Permittivity of vacuum">permittivity of vacuum</a><sup id="cite_ref-physconst-eps0_4-0" class="reference"><a href="#cite_note-physconst-eps0-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>, <span class="texhtml"><i>V</i><sub><b>E</b></sub></span> is known as the <b>Coulomb potential</b>. Note that, in contrast to the magnitude of an <a href="/wiki/Electric_field" title="Electric field">electric field</a> due to a point charge, the electric potential scales respective to the reciprocal of the radius, rather than the radius squared. </p><p>The electric potential at any location, <span class="texhtml"><b>r</b></span>, in a system of point charges is equal to the sum of the individual electric potentials due to every point charge in the system. This fact simplifies calculations significantly, because addition of potential (scalar) fields is much easier than addition of the electric (vector) fields. Specifically, the potential of a set of discrete point charges <span class="texhtml mvar" style="font-style:italic;">q<sub>i</sub></span> at points <span class="texhtml"><b>r</b><sub><i>i</i></sub></span> becomes </p><p><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 V_{\mathbf {E} }(\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\sum _{i=1}^{n}{\frac {q_{i}}{|\mathbf {r} -\mathbf {r} _{i}|}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathbf {E} }(\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\sum _{i=1}^{n}{\frac {q_{i}}{|\mathbf {r} -\mathbf {r} _{i}|}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13e69bcfb528f912837f7ede76d562853cafed59" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.798ex; height:6.843ex;" alt="{\displaystyle V_{\mathbf {E} }(\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\sum _{i=1}^{n}{\frac {q_{i}}{|\mathbf {r} -\mathbf {r} _{i}|}}\,}"></span> </p><p>where </p> <ul><li><span class="texhtml"><b>r</b></span> is a point at which the potential is evaluated;</li> <li><span class="texhtml"><b>r</b><sub><i>i</i></sub></span> is a point at which there is a nonzero charge; and</li> <li><span class="texhtml mvar" style="font-style:italic;">q<sub>i</sub></span> is the charge at the point <span class="texhtml"><b>r</b><sub><i>i</i></sub></span>.</li></ul> <p>And the potential of a continuous charge distribution <span class="texhtml"><i>ρ</i>(<b>r</b>)</span> becomes </p><p><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 V_{\mathbf {E} }(\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{R}{\frac {\rho (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\mathrm {d} ^{3}r'\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>r</mi> <mo>′</mo> </msup> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{\mathbf {E} }(\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{R}{\frac {\rho (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\mathrm {d} ^{3}r'\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11cc61689bbb2367df8688d764c69cdffe1f0aae" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.828ex; height:6.509ex;" alt="{\displaystyle V_{\mathbf {E} }(\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{R}{\frac {\rho (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\mathrm {d} ^{3}r'\,,}"></span> </p><p>where </p> <ul><li><span class="texhtml"><b>r</b></span> is a point at which the potential is evaluated;</li> <li><span class="texhtml mvar" style="font-style:italic;">R</span> is a region containing all the points at which the charge density is nonzero;</li> <li><span class="texhtml"><b>r</b><span class="nowrap" style="padding-left:0.1em;">'</span></span> is a point inside <span class="texhtml mvar" style="font-style:italic;">R</span>; and</li> <li><span class="texhtml"><i>ρ</i>(<b>r</b><span class="nowrap" style="padding-left:0.1em;">'</span>)</span> is the charge density at the point <span class="texhtml"><b>r</b><span class="nowrap" style="padding-left:0.1em;">'</span></span>.</li></ul> <p>The equations given above for the electric potential (and all the equations used here) are in the forms required by <a href="/wiki/SI_units" class="mw-redirect" title="SI units">SI units</a>. In some other (less common) systems of units, such as <a href="/wiki/Gaussian_units" title="Gaussian units">CGS-Gaussian</a>, many of these equations would be altered. </p> <div class="mw-heading mw-heading2"><h2 id="Generalization_to_electrodynamics">Generalization to electrodynamics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electric_potential&action=edit&section=4" title="Edit section: Generalization to electrodynamics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When time-varying magnetic fields are present (which is true whenever there are time-varying electric fields and vice versa), it is not possible to describe the electric field simply as a scalar potential <span class="texhtml"><i>V</i></span> because the electric field is no longer <a href="/wiki/Conservative_force" title="Conservative force">conservative</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 \textstyle \int _{C}\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ℓ</mi> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \int _{C}\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30f2fd30b7e3bc3c4f8132a36d59ae994ad59d51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.796ex; height:3.176ex;" alt="{\displaystyle \textstyle \int _{C}\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}}"></span> is path-dependent because <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 {\nabla } \times \mathbf {E} \neq \mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\nabla } \times \mathbf {E} \neq \mathbf {0} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5301ad79e42848ce3be67570a5a2399dfec61e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.968ex; height:2.676ex;" alt="{\displaystyle \mathbf {\nabla } \times \mathbf {E} \neq \mathbf {0} }"></span> (due to the <a href="/wiki/Maxwell-Faraday_equation" class="mw-redirect" title="Maxwell-Faraday equation">Maxwell-Faraday equation</a>). </p><p>Instead, one can still define a scalar potential by also including the <a href="/wiki/Magnetic_vector_potential" title="Magnetic vector potential">magnetic vector potential</a> <span class="texhtml"><b>A</b></span>. In particular, <span class="texhtml"><b>A</b></span> is defined to satisfy: </p><p><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 {B} =\mathbf {\nabla } \times \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} =\mathbf {\nabla } \times \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06e479269ae003ed92c057eecdcf35f2b060cf70" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.795ex; height:2.176ex;" alt="{\displaystyle \mathbf {B} =\mathbf {\nabla } \times \mathbf {A} }"></span> </p><p>where <span class="texhtml"><b>B</b></span> is the <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a>. By the <a href="/wiki/Fundamental_theorem_of_vector_calculus" class="mw-redirect" title="Fundamental theorem of vector calculus">fundamental theorem of vector calculus</a>, such an <span class="texhtml"><b>A</b></span> can always be found, since <a href="/wiki/Gauss%27s_law_for_magnetism" title="Gauss's law for magnetism">the divergence of the magnetic field is always zero</a> due to the absence of <a href="/wiki/Magnetic_monopole" title="Magnetic monopole">magnetic monopoles</a>. Now, the quantity <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 {F} =\mathbf {E} +{\frac {\partial \mathbf {A} }{\partial t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =\mathbf {E} +{\frac {\partial \mathbf {A} }{\partial t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32cae04e8a9b89bf1f0fba780a5f0c5d99e32098" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.552ex; height:5.509ex;" alt="{\displaystyle \mathbf {F} =\mathbf {E} +{\frac {\partial \mathbf {A} }{\partial t}}}"></span> <i>is</i> a conservative field, since the curl of <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} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7f22b39d51f780fc02859059c1757c606b9de2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.757ex; height:2.176ex;" alt="{\displaystyle \mathbf {E} }"></span> is canceled by the curl of <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 {\frac {\partial \mathbf {A} }{\partial t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \mathbf {A} }{\partial t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84f6620056a339f68903a65333c2555a1e9e9a33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:4.174ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \mathbf {A} }{\partial t}}}"></span> according to the <a href="/wiki/Maxwell%E2%80%93Faraday_equation" class="mw-redirect" title="Maxwell–Faraday equation">Maxwell–Faraday equation</a>. One can therefore write <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 {E} =-\mathbf {\nabla } V-{\frac {\partial \mathbf {A} }{\partial t}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mi>V</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} =-\mathbf {\nabla } V-{\frac {\partial \mathbf {A} }{\partial t}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd2a8e7fa25cae3de47e7543b143a1c34bfabf29" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.048ex; height:5.509ex;" alt="{\displaystyle \mathbf {E} =-\mathbf {\nabla } V-{\frac {\partial \mathbf {A} }{\partial t}},}"></span> </p><p>where <span class="texhtml"><i>V</i></span> is the scalar potential defined by the conservative field <span class="texhtml"><b>F</b></span>. </p><p>The electrostatic potential is simply the special case of this definition where <span class="texhtml"><b>A</b></span> is time-invariant. On the other hand, for time-varying fields, <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 -\int _{a}^{b}\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}\neq V_{(b)}-V_{(a)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ℓ</mi> </mrow> <mo>≠</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>−</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\int _{a}^{b}\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}\neq V_{(b)}-V_{(a)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5640bbccbd0429bc04011383c7d3e20fdac3c7fd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.062ex; height:6.343ex;" alt="{\displaystyle -\int _{a}^{b}\mathbf {E} \cdot \mathrm {d} {\boldsymbol {\ell }}\neq V_{(b)}-V_{(a)}}"></span> unlike electrostatics. </p> <div class="mw-heading mw-heading3"><h3 id="Gauge_freedom">Gauge freedom</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electric_potential&action=edit&section=5" title="Edit section: Gauge freedom"><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/Gauge_fixing" title="Gauge fixing">Gauge fixing</a></div> <p>The electrostatic potential could have any constant added to it without affecting the electric field. In electrodynamics, the electric potential has infinitely many degrees of freedom. For any (possibly time-varying or space-varying) scalar field, <span class="texhtml">𝜓</span>, we can perform the following <a href="/wiki/Gauge_Transformation" class="mw-redirect" title="Gauge Transformation">gauge transformation</a> to find a new set of potentials that produce exactly the same electric and magnetic fields:<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p><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 {\begin{aligned}V^{\prime }&=V-{\frac {\partial \psi }{\partial t}}\\\mathbf {A} ^{\prime }&=\mathbf {A} +\nabla \psi \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>V</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ψ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mi mathvariant="normal">∇</mi> <mi>ψ</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}V^{\prime }&=V-{\frac {\partial \psi }{\partial t}}\\\mathbf {A} ^{\prime }&=\mathbf {A} +\nabla \psi \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d7c37c69f9b06820bb24f83351f87b2d14c1fce" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:14.863ex; height:8.843ex;" alt="{\displaystyle {\begin{aligned}V^{\prime }&=V-{\frac {\partial \psi }{\partial t}}\\\mathbf {A} ^{\prime }&=\mathbf {A} +\nabla \psi \end{aligned}}}"></span> </p><p>Given different choices of gauge, the electric potential could have quite different properties. In the <a href="/wiki/Coulomb_Gauge" class="mw-redirect" title="Coulomb Gauge">Coulomb gauge</a>, the electric potential is given by <a href="/wiki/Poisson%27s_equation" title="Poisson's equation">Poisson's equation</a> </p><p><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 \nabla ^{2}V=-{\frac {\rho }{\varepsilon _{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>V</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ρ</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}V=-{\frac {\rho }{\varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13d9fab741f179f53b0fe5b3fe4c4c8862e7e9c2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.658ex; height:5.176ex;" alt="{\displaystyle \nabla ^{2}V=-{\frac {\rho }{\varepsilon _{0}}}}"></span> </p><p>just like in electrostatics. However, in the <a href="/wiki/Lorenz_gauge_condition" title="Lorenz gauge condition">Lorenz gauge</a>, the electric potential is a <a href="/wiki/Retarded_potential" title="Retarded potential">retarded potential</a> that propagates at the speed of light and is the solution to an <a href="/wiki/Inhomogeneous_electromagnetic_wave_equation#A_and_φ_potential_fields" title="Inhomogeneous electromagnetic wave equation">inhomogeneous wave equation</a>: </p><p><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 \nabla ^{2}V-{\frac {1}{c^{2}}}{\frac {\partial ^{2}V}{\partial t^{2}}}=-{\frac {\rho }{\varepsilon _{0}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>V</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ρ</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}V-{\frac {1}{c^{2}}}{\frac {\partial ^{2}V}{\partial t^{2}}}=-{\frac {\rho }{\varepsilon _{0}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c61c218bfac75330085108bbf7bfdce751c264b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:23.416ex; height:6.009ex;" alt="{\displaystyle \nabla ^{2}V-{\frac {1}{c^{2}}}{\frac {\partial ^{2}V}{\partial t^{2}}}=-{\frac {\rho }{\varepsilon _{0}}}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Units">Units</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electric_potential&action=edit&section=6" title="Edit section: Units"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/SI_derived_unit" title="SI derived unit">SI derived unit</a> of electric potential is the <a href="/wiki/Volt" title="Volt">volt</a> (in honor of <a href="/wiki/Alessandro_Volta" title="Alessandro Volta">Alessandro Volta</a>), denoted as V, which is why the electric potential difference between two points in space is known as a <a href="/wiki/Voltage" title="Voltage">voltage</a>. Older units are rarely used today. Variants of the <a href="/wiki/Centimetre%E2%80%93gram%E2%80%93second_system_of_units" title="Centimetre–gram–second system of units">centimetre–gram–second system of units</a> included a number of different units for electric potential, including the <a href="/wiki/Conversion_of_units#Electromotive_force,_electric_potential_difference" title="Conversion of units">abvolt</a> and the <a href="/wiki/Statvolt" title="Statvolt">statvolt</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Galvani_potential_versus_electrochemical_potential">Galvani potential versus electrochemical potential</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electric_potential&action=edit&section=7" title="Edit section: Galvani potential versus electrochemical potential"><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 articles: <a href="/wiki/Galvani_potential" title="Galvani potential">Galvani potential</a>, <a href="/wiki/Electrochemical_potential" title="Electrochemical potential">Electrochemical potential</a>, and <a href="/wiki/Fermi_level" title="Fermi level">Fermi level</a></div> <p>Inside metals (and other solids and liquids), the energy of an electron is affected not only by the electric potential, but also by the specific atomic environment that it is in. When a <a href="/wiki/Voltmeter" title="Voltmeter">voltmeter</a> is connected between two different types of metal, it measures the <b>potential difference</b> corrected for the different atomic environments.<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 quantity measured by a voltmeter is called <a href="/wiki/Electrochemical_potential" title="Electrochemical potential">electrochemical potential</a> or <a href="/wiki/Fermi_level" title="Fermi level">fermi level</a>, while the pure unadjusted electric potential, <span class="texhtml"><i>V</i></span>, is sometimes called the <a href="/wiki/Galvani_potential" title="Galvani potential">Galvani potential</a>, <span class="texhtml mvar" style="font-style:italic;">ϕ</span>. The terms "voltage" and "electric potential" are a bit ambiguous but one may refer to <em>either</em> of these in different contexts. </p> <div class="mw-heading mw-heading2"><h2 id="Common_formulas">Common formulas</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electric_potential&action=edit&section=8" title="Edit section: Common formulas"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable"> <caption> </caption> <tbody><tr> <th>Charge configuration </th> <th>Figure </th> <th colspan="2">Electric potential </th></tr> <tr> <td>Infinite wire </td> <td><span typeof="mw:File/Frameless"><a href="/wiki/File:Charged_infinite_wire_problem.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/Charged_infinite_wire_problem.svg/235px-Charged_infinite_wire_problem.svg.png" decoding="async" width="235" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/01/Charged_infinite_wire_problem.svg/353px-Charged_infinite_wire_problem.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/01/Charged_infinite_wire_problem.svg/470px-Charged_infinite_wire_problem.svg.png 2x" data-file-width="531" data-file-height="531" /></a></span> </td> <td colspan="2"><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 V=-{\frac {\lambda }{2\pi \varepsilon _{0}}}\ln x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mrow> <mn>2</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=-{\frac {\lambda }{2\pi \varepsilon _{0}}}\ln x,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e12250743dc2ef70d2f64eabb46b9e987001088" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.852ex; height:5.676ex;" alt="{\displaystyle V=-{\frac {\lambda }{2\pi \varepsilon _{0}}}\ln x,}"></span> <p>where <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 \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> is uniform linear charge density. </p> </td></tr> <tr> <td>Infinitely large surface </td> <td><span typeof="mw:File/Frameless"><a href="/wiki/File:Charged_infinite_plane_problem.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Charged_infinite_plane_problem.svg/235px-Charged_infinite_plane_problem.svg.png" decoding="async" width="235" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/17/Charged_infinite_plane_problem.svg/353px-Charged_infinite_plane_problem.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/17/Charged_infinite_plane_problem.svg/470px-Charged_infinite_plane_problem.svg.png 2x" data-file-width="531" data-file-height="531" /></a></span> </td> <td colspan="2"><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 V=-{\frac {\sigma x}{2\varepsilon _{0}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>σ</mi> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=-{\frac {\sigma x}{2\varepsilon _{0}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c8f1166fad8b44c6223a37760851b649216846e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.477ex; height:5.009ex;" alt="{\displaystyle V=-{\frac {\sigma x}{2\varepsilon _{0}}},}"></span> <p>where <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> is uniform surface charge density. </p> </td></tr> <tr> <td>Infinitely long cylindrical volume </td> <td><span typeof="mw:File/Frameless"><a href="/wiki/File:Charged_infinite_cylinder_problem.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Charged_infinite_cylinder_problem.svg/235px-Charged_infinite_cylinder_problem.svg.png" decoding="async" width="235" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Charged_infinite_cylinder_problem.svg/353px-Charged_infinite_cylinder_problem.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Charged_infinite_cylinder_problem.svg/470px-Charged_infinite_cylinder_problem.svg.png 2x" data-file-width="531" data-file-height="531" /></a></span> </td> <td colspan="2"><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 V=-{\frac {\lambda }{2\pi \varepsilon _{0}}}\ln x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mrow> <mn>2</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=-{\frac {\lambda }{2\pi \varepsilon _{0}}}\ln x,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e12250743dc2ef70d2f64eabb46b9e987001088" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.852ex; height:5.676ex;" alt="{\displaystyle V=-{\frac {\lambda }{2\pi \varepsilon _{0}}}\ln x,}"></span> <p>where <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 \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> is uniform linear charge density. </p> </td></tr> <tr> <td>Spherical volume </td> <td><span typeof="mw:File/Frameless"><a href="/wiki/File:Charged_solid_sphere_problem.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Charged_solid_sphere_problem.svg/235px-Charged_solid_sphere_problem.svg.png" decoding="async" width="235" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Charged_solid_sphere_problem.svg/353px-Charged_solid_sphere_problem.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ef/Charged_solid_sphere_problem.svg/470px-Charged_solid_sphere_problem.svg.png 2x" data-file-width="531" data-file-height="531" /></a></span> </td> <td><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 V={\frac {Q}{4\pi \varepsilon _{0}x}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {Q}{4\pi \varepsilon _{0}x}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e1cf4b8509e6b7a4ae2895656461315551c222c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.331ex; height:5.843ex;" alt="{\displaystyle V={\frac {Q}{4\pi \varepsilon _{0}x}},}"></span> <p>outside the sphere, where <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 Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> is the total charge uniformly distributed in the volume. </p> </td> <td><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 V={\frac {Q(3R^{2}-r^{2})}{8\pi \varepsilon _{0}R^{3}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Q</mi> <mo stretchy="false">(</mo> <mn>3</mn> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>8</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {Q(3R^{2}-r^{2})}{8\pi \varepsilon _{0}R^{3}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab2bd8990a5b4480c32b9e752ed81ac556fc7e1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.94ex; height:6.509ex;" alt="{\displaystyle V={\frac {Q(3R^{2}-r^{2})}{8\pi \varepsilon _{0}R^{3}}},}"></span> <p>inside the sphere, where <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 Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> is the total charge uniformly distributed in the volume. </p> </td></tr> <tr> <td>Spherical surface </td> <td><span typeof="mw:File/Frameless"><a href="/wiki/File:Charged_spherical_surface_problem.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Charged_spherical_surface_problem.svg/235px-Charged_spherical_surface_problem.svg.png" decoding="async" width="235" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Charged_spherical_surface_problem.svg/353px-Charged_spherical_surface_problem.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Charged_spherical_surface_problem.svg/470px-Charged_spherical_surface_problem.svg.png 2x" data-file-width="531" data-file-height="531" /></a></span> </td> <td><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 V={\frac {Q}{4\pi \varepsilon _{0}x}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {Q}{4\pi \varepsilon _{0}x}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e1cf4b8509e6b7a4ae2895656461315551c222c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.331ex; height:5.843ex;" alt="{\displaystyle V={\frac {Q}{4\pi \varepsilon _{0}x}},}"></span> <p>outside the sphere, where <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 Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> is the total charge uniformly distributed on the surface. </p> </td> <td><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 V={\frac {Q}{4\pi \varepsilon _{0}R}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>R</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {Q}{4\pi \varepsilon _{0}R}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f236fb38e3657315106d5fd2c54aec956c3f6c73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.765ex; height:5.843ex;" alt="{\displaystyle V={\frac {Q}{4\pi \varepsilon _{0}R}},}"></span> <p>inside the sphere for uniform charge distribution. </p> </td></tr> <tr> <td>Charged Ring </td> <td><span typeof="mw:File/Frameless"><a href="/wiki/File:Charged_ring_problem.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Charged_ring_problem.svg/235px-Charged_ring_problem.svg.png" decoding="async" width="235" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Charged_ring_problem.svg/353px-Charged_ring_problem.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/54/Charged_ring_problem.svg/470px-Charged_ring_problem.svg.png 2x" data-file-width="531" data-file-height="531" /></a></span> </td> <td colspan="2"><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 V={\frac {Q}{4\pi \varepsilon _{0}{\sqrt {R^{2}+x^{2}}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {Q}{4\pi \varepsilon _{0}{\sqrt {R^{2}+x^{2}}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08dff70709d6424ae7b34ff01785a1ccb6f0785a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:21.367ex; height:6.676ex;" alt="{\displaystyle V={\frac {Q}{4\pi \varepsilon _{0}{\sqrt {R^{2}+x^{2}}}}},}"></span> <p>on the axis, where <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 Q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span> is the total charge uniformly distributed on the ring. </p> </td></tr> <tr> <td>Charged Disc </td> <td><span typeof="mw:File/Frameless"><a href="/wiki/File:Charged_disc_problem.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Charged_disc_problem.svg/235px-Charged_disc_problem.svg.png" decoding="async" width="235" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Charged_disc_problem.svg/353px-Charged_disc_problem.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/12/Charged_disc_problem.svg/470px-Charged_disc_problem.svg.png 2x" data-file-width="531" data-file-height="531" /></a></span> </td> <td colspan="2"><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 V={\frac {\sigma }{2\varepsilon _{0}}}\left[{\sqrt {x^{2}+R^{2}}}-x\right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>σ</mi> <mrow> <mn>2</mn> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>−</mo> <mi>x</mi> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {\sigma }{2\varepsilon _{0}}}\left[{\sqrt {x^{2}+R^{2}}}-x\right],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/695453b082cfb396f9a933038b6e8028e832ba9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:27.174ex; height:5.176ex;" alt="{\displaystyle V={\frac {\sigma }{2\varepsilon _{0}}}\left[{\sqrt {x^{2}+R^{2}}}-x\right],}"></span> <p>on the axis, where <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> is the uniform surface charge density. </p> </td></tr> <tr> <td>Electric Dipole </td> <td><span typeof="mw:File/Frameless"><a href="/wiki/File:Electric_dipole_-_axial_and_equatorial_problem.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Electric_dipole_-_axial_and_equatorial_problem.svg/235px-Electric_dipole_-_axial_and_equatorial_problem.svg.png" decoding="async" width="235" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Electric_dipole_-_axial_and_equatorial_problem.svg/353px-Electric_dipole_-_axial_and_equatorial_problem.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Electric_dipole_-_axial_and_equatorial_problem.svg/470px-Electric_dipole_-_axial_and_equatorial_problem.svg.png 2x" data-file-width="531" data-file-height="531" /></a></span> </td> <td><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 V=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cec2530198028fb5a11a3afa59462e6641230d65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.695ex; height:2.509ex;" alt="{\displaystyle V=0,}"></span> <p>on the equatorial plane. </p> </td> <td><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 V={\frac {p}{4\pi \varepsilon _{0}x^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {p}{4\pi \varepsilon _{0}x^{2}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1d1f318389ac5e9d8ae7776d1e99ded37928c12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.385ex; height:5.509ex;" alt="{\displaystyle V={\frac {p}{4\pi \varepsilon _{0}x^{2}}},}"></span> <p>on the axis (given that <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 x\gg d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≫</mo> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\gg d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea2f5f0b8d182d112400df210455b75e3fe6a37c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.16ex; height:2.176ex;" alt="{\displaystyle x\gg d}"></span>), where <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" 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 x}"></span> can also be negative to indicate position at the opposite direction on the axis, and <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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> is the magnitude of <a href="/wiki/Electric_dipole_moment" title="Electric dipole moment">electric dipole moment</a>. </p> </td></tr></tbody></table> <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=Electric_potential&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Absolute_electrode_potential" title="Absolute electrode potential">Absolute electrode potential</a></li> <li><a href="/wiki/Electrochemical_potential" title="Electrochemical potential">Electrochemical potential</a></li> <li><a href="/wiki/Electrode_potential" title="Electrode potential">Electrode potential</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electric_potential&action=edit&section=10" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</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="CITEREFGoldstein1959" class="citation book cs1"><a href="/wiki/Herbert_Goldstein" title="Herbert Goldstein">Goldstein, Herbert</a> (June 1959). <i>Classical Mechanics</i>. United States: Addison-Wesley. p. 383. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0201025108" title="Special:BookSources/0201025108"><bdi>0201025108</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Mechanics&rft.place=United+States&rft.pages=383&rft.pub=Addison-Wesley&rft.date=1959-06&rft.isbn=0201025108&rft.aulast=Goldstein&rft.aufirst=Herbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectric+potential" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGriffiths1999" class="citation book cs1">Griffiths, David J. (1999). <i>Introduction to Electrodynamics</i>. Pearson Prentice Hall. pp. <span class="nowrap">416–</span>417. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-203-1601-0" title="Special:BookSources/978-81-203-1601-0"><bdi>978-81-203-1601-0</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.pages=%3Cspan+class%3D%22nowrap%22%3E416-%3C%2Fspan%3E417&rft.pub=Pearson+Prentice+Hall&rft.date=1999&rft.isbn=978-81-203-1601-0&rft.aulast=Griffiths&rft.aufirst=David+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectric+potential" 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="CITEREFYoungFreedman2012" class="citation book cs1">Young, Hugh A.; Freedman, Roger D. (2012). <i>Sears and Zemansky's University Physics with Modern Physics</i> (13th ed.). Boston: Addison-Wesley. p. 754.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sears+and+Zemansky%27s+University+Physics+with+Modern+Physics&rft.place=Boston&rft.pages=754&rft.edition=13th&rft.pub=Addison-Wesley&rft.date=2012&rft.aulast=Young&rft.aufirst=Hugh+A.&rft.au=Freedman%2C+Roger+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectric+potential" class="Z3988"></span></span> </li> <li id="cite_note-physconst-eps0-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-physconst-eps0_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://physics.nist.gov/cgi-bin/cuu/Value?ep0">"2022 CODATA Value: vacuum electric permittivity"</a>. <i>The NIST Reference on Constants, Units, and Uncertainty</i>. <a href="/wiki/National_Institute_of_Standards_and_Technology" title="National Institute of Standards and Technology">NIST</a>. May 2024<span class="reference-accessdate">. Retrieved <span class="nowrap">2024-05-18</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+NIST+Reference+on+Constants%2C+Units%2C+and+Uncertainty&rft.atitle=2022+CODATA+Value%3A+vacuum+electric+permittivity&rft.date=2024-05&rft_id=https%3A%2F%2Fphysics.nist.gov%2Fcgi-bin%2Fcuu%2FValue%3Fep0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectric+potential" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGriffiths1999" class="citation book cs1">Griffiths, David J. (1999). <i>Introduction to Electrodynamics</i> (3rd ed.). Prentice Hall. p. 420. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/013805326X" title="Special:BookSources/013805326X"><bdi>013805326X</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.pages=420&rft.edition=3rd&rft.pub=Prentice+Hall&rft.date=1999&rft.isbn=013805326X&rft.aulast=Griffiths&rft.aufirst=David+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectric+potential" 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="CITEREFBagotskii2006" class="citation book cs1">Bagotskii VS (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=09QI-assq1cC&pg=PA22"><i>Fundamentals of electrochemistry</i></a>. John Wiley & Sons. p. 22. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-70058-6" title="Special:BookSources/978-0-471-70058-6"><bdi>978-0-471-70058-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+electrochemistry&rft.pages=22&rft.pub=John+Wiley+%26+Sons&rft.date=2006&rft.isbn=978-0-471-70058-6&rft.aulast=Bagotskii&rft.aufirst=Vladimir+Sergeevich&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D09QI-assq1cC%26pg%3DPA22&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectric+potential" class="Z3988"></span></span> </li> </ol></div></div> <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=Electric_potential&action=edit&section=11" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Electric_potential" class="extiw" title="commons:Category:Electric potential">Electric potential</a></span>.</div></div> </div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPolitzerTruhlar1981" class="citation book cs1">Politzer P, Truhlar DG (1981). <i>Chemical Applications of Atomic and Molecular Electrostatic Potentials: Reactivity, Structure, Scattering, and Energetics of Organic, Inorganic, and Biological Systems</i>. Boston, MA: Springer US. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4757-9634-6" title="Special:BookSources/978-1-4757-9634-6"><bdi>978-1-4757-9634-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Chemical+Applications+of+Atomic+and+Molecular+Electrostatic+Potentials%3A+Reactivity%2C+Structure%2C+Scattering%2C+and+Energetics+of+Organic%2C+Inorganic%2C+and+Biological+Systems&rft.place=Boston%2C+MA&rft.pub=Springer+US&rft.date=1981&rft.isbn=978-1-4757-9634-6&rft.aulast=Politzer&rft.aufirst=Peter&rft.au=Truhlar%2C+Donald+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectric+potential" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSenMurray1996" class="citation book cs1">Sen K, Murray JS (1996). <i>Molecular Electrostatic Potentials: Concepts and Applications</i>. Amsterdam: Elsevier. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-444-82353-3" title="Special:BookSources/978-0-444-82353-3"><bdi>978-0-444-82353-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Molecular+Electrostatic+Potentials%3A+Concepts+and+Applications&rft.place=Amsterdam&rft.pub=Elsevier&rft.date=1996&rft.isbn=978-0-444-82353-3&rft.aulast=Sen&rft.aufirst=Kalidas&rft.au=Murray%2C+Jane+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectric+potential" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGriffiths1999" class="citation book cs1">Griffiths DJ (1999). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoel00grif_0"><i>Introduction to Electrodynamics</i></a></span> (3rd. ed.). Prentice Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-805326-X" title="Special:BookSources/0-13-805326-X"><bdi>0-13-805326-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=Introduction+to+Electrodynamics&rft.edition=3rd.&rft.pub=Prentice+Hall&rft.date=1999&rft.isbn=0-13-805326-X&rft.aulast=Griffiths&rft.aufirst=David+J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoel00grif_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectric+potential" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJackson1999" class="citation book cs1">Jackson JD (1999). <i>Classical Electrodynamics</i> (3rd. ed.). USA: John Wiley & Sons, Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-30932-1" title="Special:BookSources/978-0-471-30932-1"><bdi>978-0-471-30932-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Electrodynamics&rft.place=USA&rft.edition=3rd.&rft.pub=John+Wiley+%26+Sons%2C+Inc.&rft.date=1999&rft.isbn=978-0-471-30932-1&rft.aulast=Jackson&rft.aufirst=John+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectric+potential" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWangsness1986" class="citation book cs1">Wangsness RK (1986). <i>Electromagnetic Fields</i> (2nd., Revised, illustrated ed.). Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-81186-2" title="Special:BookSources/978-0-471-81186-2"><bdi>978-0-471-81186-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electromagnetic+Fields&rft.edition=2nd.%2C+Revised%2C+illustrated&rft.pub=Wiley&rft.date=1986&rft.isbn=978-0-471-81186-2&rft.aulast=Wangsness&rft.aufirst=Roald+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectric+potential" class="Z3988"></span></li></ul> </div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style></div><div role="navigation" class="navbox authority-control" aria-label="Navbox728" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q55451#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4128753-8">Germany</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Electrostatics"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85042463">United States</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007540857605171">Israel</a></span></li></ul></div></td></tr></tbody></table></div>    </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; 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