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class="vector-toc-numb">2</span> <span>Electric field</span> </div> </a> <button aria-controls="toc-Electric_field-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 Electric field subsection</span> </button> <ul id="toc-Electric_field-sublist" class="vector-toc-list"> <li id="toc-Gauss's_law" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gauss's_law"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Gauss's law</span> </div> </a> <ul id="toc-Gauss's_law-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Poisson_and_Laplace_equations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Poisson_and_Laplace_equations"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Poisson and Laplace equations</span> </div> </a> <ul id="toc-Poisson_and_Laplace_equations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Electrostatic_approximation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Electrostatic_approximation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Electrostatic approximation</span> </div> </a> <button aria-controls="toc-Electrostatic_approximation-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 Electrostatic approximation subsection</span> </button> <ul id="toc-Electrostatic_approximation-sublist" class="vector-toc-list"> <li id="toc-Electrostatic_potential" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Electrostatic_potential"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Electrostatic potential</span> </div> </a> <ul id="toc-Electrostatic_potential-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Electrostatic_energy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Electrostatic_energy"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Electrostatic energy</span> </div> </a> <ul id="toc-Electrostatic_energy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Electrostatic_pressure" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Electrostatic_pressure"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Electrostatic pressure</span> </div> </a> <ul id="toc-Electrostatic_pressure-sublist" class="vector-toc-list"> </ul> </li> </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">4</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">5</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">6</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Electrostatics</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 77 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-77" 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">77 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Elektrostatik" title="Elektrostatik – Alemannic" lang="gsw" hreflang="gsw" data-title="Elektrostatik" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-anp mw-list-item"><a href="https://anp.wikipedia.org/wiki/%E0%A4%B8%E0%A5%8D%E0%A4%A5%E0%A4%BF%E0%A4%B0%E0%A4%B5%E0%A5%88%E0%A4%A6%E0%A5%8D%E0%A4%AF%E0%A5%81%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A5%80" title="स्थिरवैद्युतिकी – Angika" lang="anp" hreflang="anp" data-title="स्थिरवैद्युतिकी" data-language-autonym="अंगिका" data-language-local-name="Angika" class="interlanguage-link-target"><span>अंगिका</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%83%D9%87%D8%B1%D8%B3%D9%83%D9%88%D9%86%D9%8A%D8%A7%D8%AA" title="كهرسكونيات – Arabic" lang="ar" hreflang="ar" data-title="كهرسكونيات" 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/Electrost%C3%A1tica" title="Electrostática – Asturian" lang="ast" hreflang="ast" data-title="Electrostática" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Elektrostatika" title="Elektrostatika – Azerbaijani" lang="az" hreflang="az" data-title="Elektrostatika" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</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%D0%B0%D1%81%D1%82%D0%B0%D1%82%D1%8B%D0%BA%D0%B0" title="Электрастатыка – Belarusian" lang="be" hreflang="be" data-title="Электрастатыка" 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%D0%B0%D1%81%D1%82%D0%B0%D1%82%D1%8B%D0%BA%D0%B0" title="Электрастатыка – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Электрастатыка" 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%BE%D1%81%D1%82%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Електростатика – Bulgarian" lang="bg" hreflang="bg" data-title="Електростатика" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Elektrostatika" title="Elektrostatika – Bosnian" lang="bs" hreflang="bs" data-title="Elektrostatika" 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/Electroest%C3%A0tica" title="Electroestàtica – Catalan" lang="ca" hreflang="ca" data-title="Electroestàtica" 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%D0%B0" title="Электростатика – Chuvash" lang="cv" hreflang="cv" data-title="Электростатика" 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/Elektrostatika" title="Elektrostatika – Czech" lang="cs" hreflang="cs" data-title="Elektrostatika" 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/Elektrostatik" title="Elektrostatik – Danish" lang="da" hreflang="da" data-title="Elektrostatik" 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/Elektrostatik" title="Elektrostatik – German" lang="de" hreflang="de" data-title="Elektrostatik" 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/Elektrostaatika" title="Elektrostaatika – Estonian" lang="et" hreflang="et" data-title="Elektrostaatika" 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%BF%CF%83%CF%84%CE%B1%CF%84%CE%B9%CE%BA%CE%AE" title="Ηλεκτροστατική – Greek" lang="el" hreflang="el" data-title="Ηλεκτροστατική" 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/Electrost%C3%A1tica" title="Electrostática – Spanish" lang="es" hreflang="es" data-title="Electrostática" 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/Elektrostatiko" title="Elektrostatiko – Esperanto" lang="eo" hreflang="eo" data-title="Elektrostatiko" 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/Elektrostatika" title="Elektrostatika – Basque" lang="eu" hreflang="eu" data-title="Elektrostatika" 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/%D8%A7%D9%84%DA%A9%D8%AA%D8%B1%D9%88%D8%A7%D8%B3%D8%AA%D8%A7%D8%AA%DB%8C%DA%A9" title="الکترواستاتیک – Persian" lang="fa" hreflang="fa" data-title="الکترواستاتیک" 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/%C3%89lectrostatique" title="Électrostatique – French" lang="fr" hreflang="fr" data-title="Électrostatique" 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/Leictreastataic" title="Leictreastataic – Irish" lang="ga" hreflang="ga" data-title="Leictreastataic" 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/Electrost%C3%A1tica" title="Electrostática – Galician" lang="gl" hreflang="gl" data-title="Electrostática" 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%95%EC%A0%84%EA%B8%B0%ED%95%99" title="정전기학 – Korean" lang="ko" hreflang="ko" data-title="정전기학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B7%D5%AC%D5%A5%D5%AF%D5%BF%D6%80%D5%A1%D5%BD%D5%BF%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1" title="Էլեկտրաստատիկա – Armenian" lang="hy" hreflang="hy" data-title="Էլեկտրաստատիկա" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A5%8D%E0%A4%A5%E0%A4%BF%E0%A4%B0%E0%A4%B5%E0%A5%88%E0%A4%A6%E0%A5%8D%E0%A4%AF%E0%A5%81%E0%A4%A4%E0%A4%BF%E0%A4%95%E0%A5%80" title="स्थिरवैद्युतिकी – Hindi" lang="hi" hreflang="hi" data-title="स्थिरवैद्युतिकी" 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/Elektrostatika" title="Elektrostatika – Croatian" lang="hr" hreflang="hr" data-title="Elektrostatika" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Elektrostatiko" title="Elektrostatiko – Ido" lang="io" hreflang="io" data-title="Elektrostatiko" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Elektrostatika" title="Elektrostatika – Indonesian" lang="id" hreflang="id" data-title="Elektrostatika" 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/Elettrostatica" title="Elettrostatica – Italian" lang="it" hreflang="it" data-title="Elettrostatica" 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%90%D7%9C%D7%A7%D7%98%D7%A8%D7%95%D7%A1%D7%98%D7%98%D7%99%D7%A7%D7%94" title="אלקטרוסטטיקה – Hebrew" lang="he" hreflang="he" data-title="אלקטרוסטטיקה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B8%E0%B3%8D%E0%B2%A5%E0%B2%BE%E0%B2%AF%E0%B3%80_%E0%B2%B5%E0%B2%BF%E0%B2%A6%E0%B3%8D%E0%B2%AF%E0%B3%81%E0%B2%A6%E0%B3%8D%E0%B2%B5%E0%B2%BF%E0%B2%9C%E0%B3%8D%E0%B2%9E%E0%B2%BE%E0%B2%A8" title="ಸ್ಥಾಯೀ ವಿದ್ಯುದ್ವಿಜ್ಞಾನ – Kannada" lang="kn" hreflang="kn" data-title="ಸ್ಥಾಯೀ ವಿದ್ಯುದ್ವಿಜ್ಞಾನ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</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%9D%E1%83%A1%E1%83%A2%E1%83%90%E1%83%A2%E1%83%98%E1%83%99%E1%83%90" title="ელექტროსტატიკა – Georgian" lang="ka" hreflang="ka" data-title="ელექტროსტატიკა" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%AD%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D1%81%D1%82%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Электрстатика – Kazakh" lang="kk" hreflang="kk" data-title="Электрстатика" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Elektwostatik" title="Elektwostatik – Haitian Creole" lang="ht" hreflang="ht" data-title="Elektwostatik" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Elektrostatika" title="Elektrostatika – Latvian" lang="lv" hreflang="lv" data-title="Elektrostatika" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Elektrostatika" title="Elektrostatika – Lithuanian" lang="lt" hreflang="lt" data-title="Elektrostatika" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Elektrosztatika" title="Elektrosztatika – Hungarian" lang="hu" hreflang="hu" data-title="Elektrosztatika" 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%BE%D1%81%D1%82%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Електростатика – Macedonian" lang="mk" hreflang="mk" data-title="Електростатика" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%87%E0%B4%B2%E0%B4%95%E0%B5%8D%E0%B4%9F%E0%B5%8D%E0%B4%B0%E0%B5%8B%E0%B4%B8%E0%B5%8D%E0%B4%B1%E0%B5%8D%E0%B4%B1%E0%B4%BE%E0%B4%B1%E0%B5%8D%E0%B4%B1%E0%B4%BF%E0%B4%95%E0%B5%8D%E0%B4%B8%E0%B5%8D" title="ഇലക്ട്രോസ്റ്റാറ്റിക്സ് – Malayalam" lang="ml" hreflang="ml" data-title="ഇലക്ട്രോസ്റ്റാറ്റിക്സ്" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Elektrostatik" title="Elektrostatik – Malay" lang="ms" hreflang="ms" data-title="Elektrostatik" 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-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%9C%E1%80%BB%E1%80%BE%E1%80%95%E1%80%BA%E1%80%84%E1%80%BC%E1%80%AD%E1%80%99%E1%80%BA%E1%80%95%E1%80%8A%E1%80%AC" title="လျှပ်ငြိမ်ပညာ – Burmese" lang="my" hreflang="my" data-title="လျှပ်ငြိမ်ပညာ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Elektrostatica" title="Elektrostatica – Dutch" lang="nl" hreflang="nl" data-title="Elektrostatica" 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%9D%99%E9%9B%BB%E6%B0%97%E5%AD%A6" title="静電気学 – Japanese" lang="ja" hreflang="ja" data-title="静電気学" 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-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Elektrostaatik" title="Elektrostaatik – Northern Frisian" lang="frr" hreflang="frr" data-title="Elektrostaatik" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Elektrostatikk" title="Elektrostatikk – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Elektrostatikk" 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/Elektrostatikk" title="Elektrostatikk – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Elektrostatikk" 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-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Electrostatica" title="Electrostatica – Occitan" lang="oc" hreflang="oc" data-title="Electrostatica" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Elektrostatika" title="Elektrostatika – Uzbek" lang="uz" hreflang="uz" data-title="Elektrostatika" 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-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%87%E0%A8%B2%E0%A9%88%E0%A8%95%E0%A8%9F%E0%A9%8D%E0%A8%B0%E0%A9%8B%E0%A8%B8%E0%A8%9F%E0%A9%88%E0%A8%9F%E0%A8%BF%E0%A8%95%E0%A8%B8" title="ਇਲੈਕਟ੍ਰੋਸਟੈਟਿਕਸ – Punjabi" lang="pa" hreflang="pa" data-title="ਇਲੈਕਟ੍ਰੋਸਟੈਟਿਕਸ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%A7%D9%84%DB%8C%DA%A9%D9%B9%D8%B1%D9%88%D8%B3%D9%B9%DB%8C%D9%B9%DA%A9%D8%B3" title="الیکٹروسٹیٹکس – Western Punjabi" lang="pnb" hreflang="pnb" data-title="الیکٹروسٹیٹکس" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Eletrost%C3%A0tica" title="Eletrostàtica – Piedmontese" lang="pms" hreflang="pms" data-title="Eletrostàtica" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Elektrostatyka" title="Elektrostatyka – Polish" lang="pl" hreflang="pl" data-title="Elektrostatyka" 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/Eletrost%C3%A1tica" title="Eletrostática – Portuguese" lang="pt" hreflang="pt" data-title="Eletrostática" 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/Electrostatic%C4%83" title="Electrostatică – Romanian" lang="ro" hreflang="ro" data-title="Electrostatică" 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%D0%BA%D0%B0" title="Электростатика – Russian" lang="ru" hreflang="ru" data-title="Электростатика" 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/Elektrostatika" title="Elektrostatika – Albanian" lang="sq" hreflang="sq" data-title="Elektrostatika" 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/Electrostatics" title="Electrostatics – Simple English" lang="en-simple" hreflang="en-simple" data-title="Electrostatics" 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/Elektrostatika" title="Elektrostatika – Slovak" lang="sk" hreflang="sk" data-title="Elektrostatika" 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/Elektrostatika" title="Elektrostatika – Slovenian" lang="sl" hreflang="sl" data-title="Elektrostatika" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</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%BE%D1%81%D1%82%D0%B0%D1%82%D0%B8%D0%BA%D0%B0" title="Електростатика – Serbian" lang="sr" hreflang="sr" data-title="Електростатика" 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/Elektrostatika" title="Elektrostatika – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Elektrostatika" 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-su mw-list-item"><a href="https://su.wikipedia.org/wiki/%C3%89l%C3%A9ktrostatika" title="Éléktrostatika – Sundanese" lang="su" hreflang="su" data-title="Éléktrostatika" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target"><span>Sunda</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/S%C3%A4hk%C3%B6statiikka" title="Sähköstatiikka – Finnish" lang="fi" hreflang="fi" data-title="Sähköstatiikka" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Elektrostatik" title="Elektrostatik – Swedish" lang="sv" hreflang="sv" data-title="Elektrostatik" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A8%E0%AE%BF%E0%AE%B2%E0%AF%88%E0%AE%AE%E0%AE%BF%E0%AE%A9%E0%AF%8D%E0%AE%A9%E0%AE%BF%E0%AE%AF%E0%AE%B2%E0%AF%8D" title="நிலைமின்னியல் – Tamil" lang="ta" hreflang="ta" data-title="நிலைமின்னியல்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</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%B0" title="Электростатика – Tatar" lang="tt" hreflang="tt" data-title="Электростатика" 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-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%8E%E0%B0%B2%E0%B1%86%E0%B0%95%E0%B1%8D%E0%B0%9F%E0%B1%8D%E0%B0%B0%E0%B1%8B%E0%B0%B8%E0%B1%8D%E0%B0%9F%E0%B0%BE%E0%B0%9F%E0%B0%BF%E0%B0%95%E0%B1%8D%E0%B0%B8%E0%B1%8D" title="ఎలెక్ట్రోస్టాటిక్స్ – Telugu" lang="te" hreflang="te" data-title="ఎలెక్ట్రోస్టాటిక్స్" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AA%E0%B8%96%E0%B8%B4%E0%B8%95%E0%B8%A2%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C%E0%B9%84%E0%B8%9F%E0%B8%9F%E0%B9%89%E0%B8%B2" title="สถิตยศาสตร์ไฟฟ้า – Thai" lang="th" hreflang="th" data-title="สถิตยศาสตร์ไฟฟ้า" 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/Elektrostatik" title="Elektrostatik – Turkish" lang="tr" hreflang="tr" data-title="Elektrostatik" 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%D0%BA%D0%B0" title="Електростатика – Ukrainian" lang="uk" hreflang="uk" data-title="Електростатика" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%A8%D8%B1%D9%82%DB%8C_%D8%B3%DA%A9%D9%88%D9%86%DB%8C%D8%A7%D8%AA" title="برقی سکونیات – Urdu" lang="ur" hreflang="ur" data-title="برقی سکونیات" 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/T%C4%A9nh_%C4%91i%E1%BB%87n_h%E1%BB%8Dc" title="Tĩnh điện học – Vietnamese" lang="vi" hreflang="vi" data-title="Tĩnh điện học" 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-wo mw-list-item"><a href="https://wo.wikipedia.org/wiki/Mb%C3%ABjtekkaaral" title="Mbëjtekkaaral – Wolof" lang="wo" hreflang="wo" data-title="Mbëjtekkaaral" data-language-autonym="Wolof" data-language-local-name="Wolof" class="interlanguage-link-target"><span>Wolof</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E9%9D%99%E7%94%B5%E5%AD%A6" title="静电学 – Wu" lang="wuu" hreflang="wuu" data-title="静电学" 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%9D%9C%E9%9B%BB%E5%AD%B8" title="靜電學 – Cantonese" lang="yue" hreflang="yue" data-title="靜電學" 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%9D%9C%E9%9B%BB%E5%AD%B8" title="靜電學 – Chinese" lang="zh" hreflang="zh" data-title="靜電學" 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 href="https://www.wikidata.org/wiki/Special:EntityPage/Q26336#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div 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.sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle">Articles about</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Electromagnetism" title="Electromagnetism">Electromagnetism</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File/Frameless"><a href="/wiki/File:VFPt_Solenoid_correct2.svg" class="mw-file-description" title="Solenoid"><img alt="Solenoid" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/VFPt_Solenoid_correct2.svg/190px-VFPt_Solenoid_correct2.svg.png" decoding="async" width="190" height="78" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/VFPt_Solenoid_correct2.svg/285px-VFPt_Solenoid_correct2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0d/VFPt_Solenoid_correct2.svg/380px-VFPt_Solenoid_correct2.svg.png 2x" data-file-width="490" data-file-height="200" /></a></span></td></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Electricity" title="Electricity">Electricity</a></li> <li><a href="/wiki/Magnetism" title="Magnetism">Magnetism</a></li> <li><a href="/wiki/Optics" title="Optics">Optics</a></li> <li><a href="/wiki/History_of_electromagnetic_theory" title="History of electromagnetic theory">History</a></li> <li><a href="/wiki/Computational_electromagnetics" title="Computational electromagnetics">Computational</a></li> <li><a href="/wiki/List_of_textbooks_in_electromagnetism" title="List of textbooks in electromagnetism">Textbooks</a></li> <li><a href="/wiki/List_of_electrical_phenomena" title="List of electrical phenomena">Phenomena</a></li></ul></td> </tr><tr><td class="sidebar-content hlist"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;;color: var(--color-base)"><a class="mw-selflink selflink">Electrostatics</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Charge_density" title="Charge density">Charge density</a></li> <li><a href="/wiki/Electrical_conductor" title="Electrical conductor">Conductor</a></li> <li><a href="/wiki/Coulomb%27s_law" title="Coulomb's law">Coulomb law</a></li> <li><a href="/wiki/Electret" title="Electret">Electret</a></li> <li><a href="/wiki/Electric_charge" title="Electric charge">Electric charge</a></li> <li><a href="/wiki/Electric_dipole_moment" title="Electric dipole moment">Electric dipole</a></li> <li><a href="/wiki/Electric_field" title="Electric field">Electric field</a></li> <li><a href="/wiki/Electric_flux" title="Electric flux">Electric flux</a></li> <li><a href="/wiki/Electric_potential" title="Electric potential">Electric potential</a></li> <li><a href="/wiki/Electrostatic_discharge" title="Electrostatic discharge">Electrostatic discharge</a></li> <li><a href="/wiki/Electrostatic_induction" title="Electrostatic induction">Electrostatic induction</a></li> <li><a href="/wiki/Gauss%27s_law" title="Gauss's law">Gauss law</a></li> <li><a href="/wiki/Insulator_(electricity)" title="Insulator (electricity)">Insulator</a></li> <li><a href="/wiki/Permittivity" title="Permittivity">Permittivity</a></li> <li><a href="/wiki/Polarization_density" title="Polarization density">Polarization</a></li> <li><a href="/wiki/Electric_potential_energy" title="Electric potential energy">Potential energy</a></li> <li><a href="/wiki/Static_electricity" title="Static electricity">Static electricity</a></li> <li><a href="/wiki/Triboelectric_effect" title="Triboelectric effect">Triboelectricity</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/Magnetostatics" title="Magnetostatics">Magnetostatics</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Amp%C3%A8re%27s_circuital_law" title="Ampère's circuital law">Ampère law</a></li> <li><a href="/wiki/Biot%E2%80%93Savart_law" title="Biot–Savart law">Biot–Savart law</a></li> <li><a href="/wiki/Gauss%27s_law_for_magnetism" title="Gauss's law for magnetism">Gauss magnetic law</a></li> <li><a href="/wiki/Magnetic_moment" title="Magnetic moment">Magnetic dipole</a></li> <li><a href="/wiki/Magnetic_field" title="Magnetic field">Magnetic field</a></li> <li><a href="/wiki/Magnetic_flux" title="Magnetic flux">Magnetic flux</a></li> <li><a href="/wiki/Magnetic_scalar_potential" title="Magnetic scalar potential">Magnetic scalar potential</a></li> <li><a href="/wiki/Magnetic_vector_potential" title="Magnetic vector potential">Magnetic vector potential</a></li> <li><a href="/wiki/Magnetization" title="Magnetization">Magnetization</a></li> <li><a href="/wiki/Permeability_(electromagnetism)" title="Permeability (electromagnetism)">Permeability</a></li> <li><a href="/wiki/Right-hand_rule#Electromagnetism" title="Right-hand rule">Right-hand rule</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/Classical_electromagnetism" title="Classical electromagnetism">Electrodynamics</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Bremsstrahlung" title="Bremsstrahlung">Bremsstrahlung</a></li> <li><a href="/wiki/Cyclotron_radiation" title="Cyclotron radiation">Cyclotron radiation</a></li> <li><a href="/wiki/Displacement_current" title="Displacement current">Displacement current</a></li> <li><a href="/wiki/Eddy_current" title="Eddy current">Eddy current</a></li> <li><a href="/wiki/Electromagnetic_field" title="Electromagnetic field">Electromagnetic field</a></li> <li><a href="/wiki/Electromagnetic_induction" title="Electromagnetic induction">Electromagnetic induction</a></li> <li><a href="/wiki/Electromagnetic_pulse" title="Electromagnetic pulse">Electromagnetic pulse</a></li> <li><a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">Electromagnetic radiation</a></li> <li><a href="/wiki/Faraday%27s_law_of_induction" title="Faraday's law of induction">Faraday law</a></li> <li><a href="/wiki/Jefimenko%27s_equations" title="Jefimenko's equations">Jefimenko equations</a></li> <li><a href="/wiki/Larmor_formula" title="Larmor formula">Larmor formula</a></li> <li><a href="/wiki/Lenz%27s_law" title="Lenz's law">Lenz law</a></li> <li><a href="/wiki/Li%C3%A9nard%E2%80%93Wiechert_potential" title="Liénard–Wiechert potential">Liénard–Wiechert potential</a></li> <li><a href="/wiki/London_equations" title="London equations">London equations</a></li> <li><a href="/wiki/Lorentz_force" title="Lorentz force">Lorentz force</a></li> <li><a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell equations</a></li> <li><a href="/wiki/Maxwell_stress_tensor" title="Maxwell stress tensor">Maxwell tensor</a></li> <li><a href="/wiki/Poynting_vector" title="Poynting vector">Poynting vector</a></li> <li><a href="/wiki/Synchrotron_radiation" title="Synchrotron radiation">Synchrotron radiation</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/Electrical_network" title="Electrical network">Electrical network</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Alternating_current" title="Alternating current">Alternating current</a></li> <li><a href="/wiki/Capacitance" title="Capacitance">Capacitance</a></li> <li><a href="/wiki/Current_density" title="Current density">Current density</a></li> <li><a href="/wiki/Direct_current" title="Direct current">Direct current</a></li> <li><a href="/wiki/Electric_current" title="Electric current">Electric current</a></li> <li><a href="/wiki/Electric_power" title="Electric power">Electric power</a></li> <li><a href="/wiki/Electrolysis" title="Electrolysis">Electrolysis</a></li> <li><a href="/wiki/Electromotive_force" title="Electromotive force">Electromotive force</a></li> <li><a href="/wiki/Electrical_impedance" title="Electrical impedance">Impedance</a></li> <li><a href="/wiki/Inductance" title="Inductance">Inductance</a></li> <li><a href="/wiki/Joule_heating" title="Joule heating">Joule heating</a></li> <li><a href="/wiki/Kirchhoff%27s_circuit_laws" title="Kirchhoff's circuit laws">Kirchhoff laws</a></li> <li><a href="/wiki/Network_analysis_(electrical_circuits)" title="Network analysis (electrical circuits)">Network analysis</a></li> <li><a href="/wiki/Ohm%27s_law" title="Ohm's law">Ohm law</a></li> <li><a href="/wiki/Series_and_parallel_circuits#Parallel_circuits" title="Series and parallel circuits">Parallel circuit</a></li> <li><a href="/wiki/Electrical_resistance_and_conductance" title="Electrical resistance and conductance">Resistance</a></li> <li><a href="/wiki/Resonator#Electromagnetics" title="Resonator">Resonant cavities</a></li> <li><a href="/wiki/Series_and_parallel_circuits#Series_circuits" title="Series and parallel circuits">Series circuit</a></li> <li><a href="/wiki/Voltage" title="Voltage">Voltage</a></li> <li><a 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> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Cat_demonstrating_static_cling_with_styrofoam_peanuts.jpg" class="mw-file-description"><img alt="A tabby cat covered in packing peanuts." src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Cat_demonstrating_static_cling_with_styrofoam_peanuts.jpg/310px-Cat_demonstrating_static_cling_with_styrofoam_peanuts.jpg" decoding="async" width="310" height="205" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/e/e0/Cat_demonstrating_static_cling_with_styrofoam_peanuts.jpg 1.5x" data-file-width="436" data-file-height="289" /></a><figcaption><a href="/wiki/Foam_peanut" title="Foam peanut">Foam peanuts</a> clinging to a cat's fur due to <a href="/wiki/Static_electricity" title="Static electricity">static electricity</a>. The electric field of the charged fur causes polarization of the molecules of the foam due to <a href="/wiki/Electrostatic_induction" title="Electrostatic induction">electrostatic induction</a>, resulting in a slight attraction of the light plastic pieces to the fur.<sup id="cite_ref-Ling_1-0" class="reference"><a href="#cite_note-Ling-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Bloomfield_2-0" class="reference"><a href="#cite_note-Bloomfield-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-PhysicsClassroom_3-0" class="reference"><a href="#cite_note-PhysicsClassroom-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Thompson_4-0" class="reference"><a href="#cite_note-Thompson-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> This effect is also the cause of <a href="/wiki/Static_cling" title="Static cling">static cling</a> in clothes.</figcaption></figure> <p><b>Electrostatics</b> is a branch of <a href="/wiki/Physics" title="Physics">physics</a> that studies slow-moving or stationary <a href="/wiki/Electric_charge" title="Electric charge">electric charges</a>. </p><p>Since <a href="/wiki/Classical_antiquity" title="Classical antiquity">classical times</a>, it has been known that some materials, such as <a href="/wiki/Amber" title="Amber">amber</a>, attract lightweight particles after <a href="/wiki/Triboelectric_effect" title="Triboelectric effect">rubbing</a>. The <a href="/wiki/Greek_language" title="Greek language">Greek</a> word for amber, <span title="Greek-language text"><span lang="el">ἤλεκτρον</span></span> (<span title="Greek-language romanization"><i lang="el-Latn">ḗlektron</i></span>), was thus the source of the word <i><a href="/wiki/Electricity" title="Electricity">electricity</a></i>. Electrostatic phenomena arise from the <a href="/wiki/Force" title="Force">forces</a> that electric charges exert on each other. Such <a href="/wiki/Forces" class="mw-redirect" title="Forces">forces</a> are described by <a href="/wiki/Coulomb%27s_law" title="Coulomb's law">Coulomb's law</a>. </p><p>There are many examples of electrostatic phenomena, from those as simple as the attraction of plastic wrap to one's hand after it is removed from a package, to the apparently spontaneous explosion of grain silos, the damage of electronic components during manufacturing, and <a href="/wiki/Photocopier" title="Photocopier">photocopier</a> and <a href="/wiki/Laser_printing" title="Laser printing">laser printer</a> operation. </p><p>The electrostatic model accurately predicts electrical phenomena in "classical" cases where the velocities are low and the system is macroscopic so no quantum effects are involved. It also plays a role in quantum mechanics, where additional terms also need to be included. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Coulomb's_law"><span id="Coulomb.27s_law"></span>Coulomb's law</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electrostatics&action=edit&section=1" title="Edit section: Coulomb's law"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">Main article: <a href="/wiki/Coulomb%27s_law" title="Coulomb's law">Coulomb's law</a></div> <p><a href="/wiki/Coulomb%27s_law" title="Coulomb's law">Coulomb's law</a> states that:<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> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>The magnitude of the electrostatic force of attraction or repulsion between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them.</p></blockquote> <p>The force is along the straight line joining them. If the two charges have the same sign, the electrostatic force between them is repulsive; if they have different signs, the force between them is attractive. </p><p>If <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 r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"></span> is the distance (in <a href="/wiki/Meters" class="mw-redirect" title="Meters">meters</a>) between two charges, then the force between two point charges <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> 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 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/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> is: </p> <dl><dd><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 F={1 \over 4\pi \varepsilon _{0}}{|Qq| \over r^{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <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> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>Q</mi> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F={1 \over 4\pi \varepsilon _{0}}{|Qq| \over r^{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72093695989eae7555a600dcb3df971d4d66a535" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.992ex; height:6.176ex;" alt="{\displaystyle F={1 \over 4\pi \varepsilon _{0}}{|Qq| \over r^{2}},}"></span></dd></dl> <p>where <i>ε</i><sub>0</sub> = <span class="nowrap"><span data-sort-value="6988885418781880000♠"></span>8.854<span style="margin-left:.25em;">187</span><span style="margin-left:.25em;">8188</span>(14)<span style="margin-left:0.25em;margin-right:0.15em;">×</span>10<sup>−12</sup> F⋅m<sup>−1</sup></span><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span><sup id="cite_ref-physconst-eps0_6-0" class="reference"><a href="#cite_note-physconst-eps0-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> is the <a href="/wiki/Vacuum_permittivity" title="Vacuum permittivity">vacuum permittivity</a>.<sup id="cite_ref-Sadiku_7-0" class="reference"><a href="#cite_note-Sadiku-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI unit</a> of <i>ε</i><sub>0</sub> is equivalently <a href="/wiki/Ampere" title="Ampere">A</a><sup>2</sup>⋅<a href="/wiki/Second" title="Second">s</a><sup>4</sup> ⋅kg<sup>−1</sup>⋅m<sup>−3</sup> or <a href="/wiki/Coulomb" title="Coulomb">C</a><sup>2</sup>⋅<a href="/wiki/Newton_(unit)" title="Newton (unit)">N</a><sup>−1</sup>⋅m<sup>−2</sup> or <a href="/wiki/Farad" title="Farad">F</a>⋅m<sup>−1</sup>. </p> <div class="mw-heading mw-heading2"><h2 id="Electric_field">Electric field</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electrostatics&action=edit&section=2" title="Edit section: Electric field"><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/Electric_field" title="Electric field">Electric field</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Electrostatic_induction.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Electrostatic_induction.svg/440px-Electrostatic_induction.svg.png" decoding="async" width="440" height="302" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Electrostatic_induction.svg/660px-Electrostatic_induction.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/Electrostatic_induction.svg/880px-Electrostatic_induction.svg.png 2x" data-file-width="951" data-file-height="652" /></a><figcaption>The <a href="/wiki/Electrostatic_field" class="mw-redirect" title="Electrostatic field">electrostatic field</a> <i>(lines with arrows)</i> of a nearby positive charge <i><span style="color:red;">(+)</span></i> causes the mobile charges in conductive objects to separate due to <a href="/wiki/Electrostatic_induction" title="Electrostatic induction">electrostatic induction</a>. Negative charges <i><span style="color:blue;">(blue)</span></i> are attracted and move to the surface of the object facing the external charge. Positive charges <i><span style="color:red;">(red)</span></i> are repelled and move to the surface facing away. These induced surface charges are exactly the right size and shape so their opposing electric field cancels the electric field of the external charge throughout the interior of the metal. Therefore, the electrostatic field everywhere inside a conductive object is zero, and the <a href="/wiki/Electrostatic_potential" class="mw-redirect" title="Electrostatic potential">electrostatic potential</a> is constant.</figcaption></figure> <p>The electric field, <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>, in units of <a href="/wiki/Newton_(unit)" title="Newton (unit)">Newtons</a> per <a href="/wiki/Coulomb" title="Coulomb">Coulomb</a> or <a href="/wiki/Volt" title="Volt">volts</a> per meter, is a <a href="/wiki/Vector_field" title="Vector field">vector field</a> that can be defined everywhere, except at the location of point charges (where it diverges to infinity).<sup id="cite_ref-Purcell_8-0" class="reference"><a href="#cite_note-Purcell-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> It is defined as the electrostatic force <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 {,} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold">,</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {,} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c6bdeebe3d5ba936181a4e500261ee769fc0dfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:0.742ex; height:1.343ex;" alt="{\displaystyle \mathbf {,} }"></span> on a hypothetical small <a href="/wiki/Test_charge" class="mw-redirect" title="Test charge">test charge</a> at the point due to Coulomb's law, divided by the charge <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/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span> </p> <dl><dd><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 {F} \over q}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mi>q</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} ={\mathbf {F} \over q}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c8c50f9dcdfd2efd422a0d573039d8c24839f3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:7.374ex; height:5.676ex;" alt="{\displaystyle \mathbf {E} ={\mathbf {F} \over q}}"></span></dd></dl> <p><a href="/wiki/Field_line" title="Field line">Electric field lines</a> are useful for visualizing the electric field. Field lines begin on positive charge and terminate on negative charge. They are parallel to the direction of the electric field at each point, and the density of these field lines is a measure of the magnitude of the electric field at any given point. </p><p>A collection 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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> particles of charge <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2752dcbff884354069fe332b8e51eb0a70a531b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.837ex; height:2.009ex;" alt="{\displaystyle q_{i}}"></span>, located at points <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 {r} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed603561819ebd007acd75a0931d3ba401ad677a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.902ex; height:2.009ex;" alt="{\displaystyle \mathbf {r} _{i}}"></span> (called <i>source points</i>) generates the electric field at <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 {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"></span> (called the <i>field point</i>) of:<sup id="cite_ref-Purcell_8-1" class="reference"><a href="#cite_note-Purcell-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><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 {r} )={1 \over 4\pi \varepsilon _{0}}\sum _{i=1}^{n}q_{i}{{\hat {\mathbf {r} }}_{i} \over {|\mathbf {r} _{i}|}^{2}}={1 \over 4\pi \varepsilon _{0}}\sum _{i=1}^{n}q_{i}{\mathbf {r} _{i} \over {|\mathbf {r} _{i}|}^{3}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <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> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <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> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} (\mathbf {r} )={1 \over 4\pi \varepsilon _{0}}\sum _{i=1}^{n}q_{i}{{\hat {\mathbf {r} }}_{i} \over {|\mathbf {r} _{i}|}^{2}}={1 \over 4\pi \varepsilon _{0}}\sum _{i=1}^{n}q_{i}{\mathbf {r} _{i} \over {|\mathbf {r} _{i}|}^{3}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a4f93dd426699c6191ccaf02850f0c4fcda95d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.552ex; height:6.843ex;" alt="{\displaystyle \mathbf {E} (\mathbf {r} )={1 \over 4\pi \varepsilon _{0}}\sum _{i=1}^{n}q_{i}{{\hat {\mathbf {r} }}_{i} \over {|\mathbf {r} _{i}|}^{2}}={1 \over 4\pi \varepsilon _{0}}\sum _{i=1}^{n}q_{i}{\mathbf {r} _{i} \over {|\mathbf {r} _{i}|}^{3}},}"></span></dd></dl> <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="{\textstyle \mathbf {r} _{i}=\mathbf {r} -\mathbf {r} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mathbf {r} _{i}=\mathbf {r} -\mathbf {r} _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14eca4a376ddf1c710a7a0d784af73f7502b3d43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.844ex; height:2.343ex;" alt="{\textstyle \mathbf {r} _{i}=\mathbf {r} -\mathbf {r} _{i}}"></span> is the <a href="/wiki/Displacement_vector" class="mw-redirect" title="Displacement vector">displacement vector</a> from a <i>source point</i> <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 {r} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed603561819ebd007acd75a0931d3ba401ad677a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.902ex; height:2.009ex;" alt="{\displaystyle \mathbf {r} _{i}}"></span> to the <i>field point</i> <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 {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"></span>, 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 {\hat {\mathbf {r} }}_{i}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\mathbf {r} _{i}}{|\mathbf {r} _{i}|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">f</mi> </mrow> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\hat {\mathbf {r} }}_{i}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\mathbf {r} _{i}}{|\mathbf {r} _{i}|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d045a394d0d948d37b47b636a0e59822600515f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.534ex; height:4.843ex;" alt="{\textstyle {\hat {\mathbf {r} }}_{i}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\mathbf {r} _{i}}{|\mathbf {r} _{i}|}}}"></span> is a <a href="/wiki/Unit_vector" title="Unit vector">unit vector</a> that indicates the direction of the field. For a single point charge, <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/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>, at the origin, the magnitude of this electric field is <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 E=q/4\pi \varepsilon _{0}r^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</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>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=q/4\pi \varepsilon _{0}r^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f602ba4cbb8857ee012f1dc5d02f83d37995a56a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.841ex; height:3.176ex;" alt="{\displaystyle E=q/4\pi \varepsilon _{0}r^{2}}"></span> and points away from that charge if it is positive. The fact that the force (and hence the field) can be calculated by summing over all the contributions due to individual source particles is an example of the <a href="/wiki/Superposition_principle" title="Superposition principle">superposition principle</a>. The electric field produced by a distribution of charges is given by the <a href="/wiki/Volume_charge_density" class="mw-redirect" title="Volume charge density">volume charge density</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 \rho (\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (\mathbf {r} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77f477411625125978c0a18946bdfae2c1f13bcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.113ex; height:2.843ex;" alt="{\displaystyle \rho (\mathbf {r} )}"></span> and can be obtained by converting this sum into a <a href="/wiki/Triple_integral" class="mw-redirect" title="Triple integral">triple integral</a>: </p> <dl><dd><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 {r} )={\frac {1}{4\pi \varepsilon _{0}}}\iiint \,\rho (\mathbf {r} '){\mathbf {r} ' \over {|\mathbf {r} '|}^{3}}\mathrm {d} ^{3}|\mathbf {r} '|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <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> <mo>∭</mo> <mspace width="thinmathspace" /> <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 class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} (\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\iiint \,\rho (\mathbf {r} '){\mathbf {r} ' \over {|\mathbf {r} '|}^{3}}\mathrm {d} ^{3}|\mathbf {r} '|}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04a227fbe7333a8c3ec8506cb369c1f94c1cae8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.192ex; height:6.676ex;" alt="{\displaystyle \mathbf {E} (\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\iiint \,\rho (\mathbf {r} '){\mathbf {r} ' \over {|\mathbf {r} '|}^{3}}\mathrm {d} ^{3}|\mathbf {r} '|}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Gauss's_law"><span id="Gauss.27s_law"></span>Gauss's law</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electrostatics&action=edit&section=3" title="Edit section: Gauss's law"><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/Gauss%27s_law" title="Gauss's law">Gauss's law</a> and <a href="/wiki/Gaussian_surface" title="Gaussian surface">Gaussian surface</a></div> <p><a href="/wiki/Gauss%27s_law" title="Gauss's law">Gauss's law</a><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> states that "the total <a href="/wiki/Electric_flux" title="Electric flux">electric flux</a> through any closed surface in free space of any shape drawn in an electric field is proportional to the total <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a> enclosed by the surface." Many numerical problems can be solved by considering a <a href="/wiki/Gaussian_surface" title="Gaussian surface">Gaussian surface</a> around a body. Mathematically, Gauss's law takes the form of an integral equation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{E}=\oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={Q_{\text{enclosed}} \over \varepsilon _{0}}=\int _{V}{\rho \over \varepsilon _{0}}\mathrm {d} ^{3}r,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>=</mo> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</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">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>enclosed</mtext> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{E}=\oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={Q_{\text{enclosed}} \over \varepsilon _{0}}=\int _{V}{\rho \over \varepsilon _{0}}\mathrm {d} ^{3}r,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bae631c3339965ce7ca2a6287773c9985b3712d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:41.225ex; height:5.843ex;" alt="{\displaystyle \Phi _{E}=\oint _{S}\mathbf {E} \cdot \mathrm {d} \mathbf {A} ={Q_{\text{enclosed}} \over \varepsilon _{0}}=\int _{V}{\rho \over \varepsilon _{0}}\mathrm {d} ^{3}r,}"></span></dd></dl> <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 \mathrm {d} ^{3}r=\mathrm {d} x\ \mathrm {d} y\ \mathrm {d} z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>x</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>y</mi> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} ^{3}r=\mathrm {d} x\ \mathrm {d} y\ \mathrm {d} z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17eecba4079ffd6f536b7572ea78563db92e7455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.106ex; height:3.009ex;" alt="{\displaystyle \mathrm {d} ^{3}r=\mathrm {d} x\ \mathrm {d} y\ \mathrm {d} z}"></span> is a volume element. If the charge is distributed over a surface or along a line, replace <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 \rho \,\mathrm {d} ^{3}r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho \,\mathrm {d} ^{3}r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/580640c6416ccc3899c680df2a9240d73312a095" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.984ex; height:3.176ex;" alt="{\displaystyle \rho \,\mathrm {d} ^{3}r}"></span> by <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 \,\mathrm {d} A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma \,\mathrm {d} A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d973cf8ed2d0d0dba98b97b08bb6993c1982b031" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.752ex; height:2.176ex;" alt="{\displaystyle \sigma \,\mathrm {d} A}"></span> or <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 \,\mathrm {d} \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda \,\mathrm {d} \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05703830118ff432e9cad77cda9825823ca57127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.005ex; height:2.176ex;" alt="{\displaystyle \lambda \,\mathrm {d} \ell }"></span>. The <a href="/wiki/Divergence_theorem" title="Divergence theorem">divergence theorem</a> allows Gauss's Law to be written in differential form: </p> <dl><dd><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 \nabla \cdot \mathbf {E} ={\rho \over \varepsilon _{0}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" 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"> <mfrac> <mi>ρ</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot \mathbf {E} ={\rho \over \varepsilon _{0}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e51ec7e9ee1d40626ad7c3b5360d705794d39db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.091ex; height:5.176ex;" alt="{\displaystyle \nabla \cdot \mathbf {E} ={\rho \over \varepsilon _{0}}.}"></span></dd></dl> <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 \nabla \cdot }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6df6024211b717870f07844116e116b2eb314d12" 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="{\displaystyle \nabla \cdot }"></span> is the <a href="/wiki/Divergence" title="Divergence">divergence operator</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Poisson_and_Laplace_equations">Poisson and Laplace equations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electrostatics&action=edit&section=4" title="Edit section: Poisson and Laplace equations"><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/Poisson%27s_equation" title="Poisson's equation">Poisson's equation</a> and <a href="/wiki/Laplace%27s_equation" title="Laplace's equation">Laplace's equation</a></div> <p>The definition of electrostatic potential, combined with the differential form of Gauss's law (above), provides a relationship between the potential Φ and the charge density <i>ρ</i>: </p> <dl><dd><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 {\nabla }^{2}\phi =-{\rho \over \varepsilon _{0}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ϕ</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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\nabla }^{2}\phi =-{\rho \over \varepsilon _{0}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60cabacab3cdd1b843f790807e6680031cf93355" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.903ex; height:5.176ex;" alt="{\displaystyle {\nabla }^{2}\phi =-{\rho \over \varepsilon _{0}}.}"></span></dd></dl> <p>This relationship is a form of <a href="/wiki/Poisson%27s_equation" title="Poisson's equation">Poisson's equation</a>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> In the absence of unpaired electric charge, the equation becomes <a href="/wiki/Laplace%27s_equation" title="Laplace's equation">Laplace's equation</a>: </p> <dl><dd><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 {\nabla }^{2}\phi =0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>ϕ</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\nabla }^{2}\phi =0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b452c3d82756a36cbbeef05929c644b2fb633c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.283ex; height:3.009ex;" alt="{\displaystyle {\nabla }^{2}\phi =0,}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Electrostatic_approximation">Electrostatic approximation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electrostatics&action=edit&section=5" title="Edit section: Electrostatic approximation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Electrostatics_relation_triangle.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Electrostatics_relation_triangle.svg/220px-Electrostatics_relation_triangle.svg.png" decoding="async" width="220" height="154" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Electrostatics_relation_triangle.svg/330px-Electrostatics_relation_triangle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Electrostatics_relation_triangle.svg/440px-Electrostatics_relation_triangle.svg.png 2x" data-file-width="1134" data-file-height="794" /></a><figcaption>Summary of electrostatic relations between electric potential, electric field and charge density. Here, <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 {r} =\mathbf {x} -\mathbf {x'} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">x</mi> <mo>′</mo> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} =\mathbf {x} -\mathbf {x'} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d61d046aa7c9d77c053fe520f77b78f374ec286a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.548ex; height:2.676ex;" alt="{\displaystyle \mathbf {r} =\mathbf {x} -\mathbf {x'} }"></span>.</figcaption></figure> <p>The validity of the electrostatic approximation rests on the assumption that the electric field is <a href="/wiki/Irrotational" class="mw-redirect" title="Irrotational">irrotational</a>: </p> <dl><dd><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 \nabla \times \mathbf {E} =0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \times \mathbf {E} =0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/777b3ce74fff575f90387d5a5f45e8758c825eb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.441ex; height:2.176ex;" alt="{\displaystyle \nabla \times \mathbf {E} =0.}"></span></dd></dl> <p>From <a href="/wiki/Faraday%27s_law_of_induction" title="Faraday's law of induction">Faraday's law</a>, this assumption implies the absence or near-absence of time-varying magnetic fields: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\partial \mathbf {B} \over \partial t}=0.}"> <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">B</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\partial \mathbf {B} \over \partial t}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17cc5fa8e5d5939aa5be407dc13b237a3b20b993" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.963ex; height:5.509ex;" alt="{\displaystyle {\partial \mathbf {B} \over \partial t}=0.}"></span></dd></dl> <p>In other words, electrostatics does not require the absence of magnetic fields or electric currents. Rather, if magnetic fields or electric currents <i>do</i> exist, they must not change with time, or in the worst-case, they must change with time only <i>very slowly</i>. In some problems, both electrostatics and <a href="/wiki/Magnetostatics" title="Magnetostatics">magnetostatics</a> may be required for accurate predictions, but the coupling between the two can still be ignored. Electrostatics and magnetostatics can both be seen as non-relativistic <a href="/wiki/Galilean_electromagnetism" title="Galilean electromagnetism">Galilean limits</a> for electromagnetism.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> In addition, conventional electrostatics ignore quantum effects which have to be added for a complete description.<sup id="cite_ref-Purcell_8-2" class="reference"><a href="#cite_note-Purcell-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 2">: 2 </span></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Electrostatic_potential">Electrostatic potential</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electrostatics&action=edit&section=6" title="Edit section: Electrostatic 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 article: <a href="/wiki/Electrostatic_potential" class="mw-redirect" title="Electrostatic potential">Electrostatic potential</a></div> <p>As the electric field is <a href="/wiki/Irrotational" class="mw-redirect" title="Irrotational">irrotational</a>, it is possible to express the electric field as the <a href="/wiki/Gradient" title="Gradient">gradient</a> of a scalar function, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></span>, called the <a href="/wiki/Electrostatic_potential" class="mw-redirect" title="Electrostatic potential">electrostatic potential</a> (also known as the <a href="/wiki/Voltage" title="Voltage">voltage</a>). An electric field, <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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></span>, points from regions of high electric potential to regions of low electric potential, expressed mathematically as </p> <dl><dd><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} =-\nabla \phi .}"> <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> <mi mathvariant="normal">∇</mi> <mi>ϕ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} =-\nabla \phi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d218ceab4c18f343f212a8fa2be83df61c8caaa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.632ex; height:2.509ex;" alt="{\displaystyle \mathbf {E} =-\nabla \phi .}"></span></dd></dl> <p>The <a href="/wiki/Gradient_theorem" title="Gradient theorem">gradient theorem</a> can be used to establish that the electrostatic potential is the amount of <a href="/wiki/Work_(physics)" title="Work (physics)">work</a> per unit charge required to move a charge from point <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 a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> to point <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 b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"></span> with the following <a href="/wiki/Line_integral" title="Line integral">line integral</a>: </p> <dl><dd><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 -\int _{a}^{b}{\mathbf {E} \cdot \mathrm {d} \mathbf {\ell } }=\phi (\mathbf {b} )-\phi (\mathbf {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"> <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>ℓ</mi> </mrow> </mrow> <mo>=</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\int _{a}^{b}{\mathbf {E} \cdot \mathrm {d} \mathbf {\ell } }=\phi (\mathbf {b} )-\phi (\mathbf {a} ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d3d0ad0e0f54df799222451ac5cd5c70f6fb946" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.442ex; height:6.343ex;" alt="{\displaystyle -\int _{a}^{b}{\mathbf {E} \cdot \mathrm {d} \mathbf {\ell } }=\phi (\mathbf {b} )-\phi (\mathbf {a} ).}"></span></dd></dl> <p>From these equations, we see that the electric potential is constant in any region for which the electric field vanishes (such as occurs inside a conducting object). </p> <div class="mw-heading mw-heading3"><h3 id="Electrostatic_energy">Electrostatic energy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electrostatics&action=edit&section=7" title="Edit section: Electrostatic energy"><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/Electric_potential_energy" title="Electric potential energy">Electric potential energy</a> and <a href="/wiki/Energy_density" title="Energy density">Energy density</a></div> <p>A <a href="/wiki/Test_particle" title="Test particle">test particle</a>'s potential energy, <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 U_{\mathrm {E} }^{\text{single}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>single</mtext> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\mathrm {E} }^{\text{single}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29233da60c12a0f901a594fc1ef9d12306ca7048" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.102ex; height:3.509ex;" alt="{\displaystyle U_{\mathrm {E} }^{\text{single}}}"></span>, can be calculated from a <a href="/wiki/Line_integral" title="Line integral">line integral</a> of the work, <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_{n}\mathbf {E} \cdot \mathrm {d} \mathbf {\ell } }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</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>ℓ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{n}\mathbf {E} \cdot \mathrm {d} \mathbf {\ell } }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6bba24441dafb93281fad3cb41146f1cb5727f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.954ex; height:2.509ex;" alt="{\displaystyle q_{n}\mathbf {E} \cdot \mathrm {d} \mathbf {\ell } }"></span>. We integrate from a point at infinity, and assume a collection 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 N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span> particles of charge <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_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/503d0af3998f76cd4eaf8b3cc5e8834e254cb71b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.057ex; height:2.509ex;" alt="{\displaystyle Q_{n}}"></span>, are already situated at the points <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 {r} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed603561819ebd007acd75a0931d3ba401ad677a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.902ex; height:2.009ex;" alt="{\displaystyle \mathbf {r} _{i}}"></span>. This potential energy (in <a href="/wiki/Joule" title="Joule">Joules</a>) is: </p> <dl><dd><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 U_{\mathrm {E} }^{\text{single}}=q\phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon _{0}}}\sum _{i=1}^{N}{\frac {Q_{i}}{\left\|{\mathcal {\mathbf {R} _{i}}}\right\|}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>single</mtext> </mrow> </msubsup> <mo>=</mo> <mi>q</mi> <mi>ϕ</mi> <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> <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> </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> <mo symmetric="true">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">i</mi> </mrow> </msub> </mrow> </mrow> <mo symmetric="true">‖</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\mathrm {E} }^{\text{single}}=q\phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon _{0}}}\sum _{i=1}^{N}{\frac {Q_{i}}{\left\|{\mathcal {\mathbf {R} _{i}}}\right\|}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9678bb7651f8519cd5820af67fb300dd401a7e2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.227ex; height:7.343ex;" alt="{\displaystyle U_{\mathrm {E} }^{\text{single}}=q\phi (\mathbf {r} )={\frac {q}{4\pi \varepsilon _{0}}}\sum _{i=1}^{N}{\frac {Q_{i}}{\left\|{\mathcal {\mathbf {R} _{i}}}\right\|}}}"></span></dd></dl> <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 \mathbf {\mathcal {R_{i}}} =\mathbf {r} -\mathbf {r} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi class="MJX-tex-caligraphic" mathvariant="script">R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">i</mi> </mrow> </msub> </mrow> </mrow> <mo>=</mo> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {\mathcal {R_{i}}} =\mathbf {r} -\mathbf {r} _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25af15693c811c7a9ca91df3ec05d93a6a9d2bfc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.713ex; height:2.509ex;" alt="{\displaystyle \mathbf {\mathcal {R_{i}}} =\mathbf {r} -\mathbf {r} _{i}}"></span> is the distance of each charge <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9f7193081d440425e522698e80817b5d558df03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.638ex; height:2.509ex;" alt="{\displaystyle Q_{i}}"></span> from the <a href="/wiki/Test_charge" class="mw-redirect" title="Test charge">test charge</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 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/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"></span>, which situated at the point <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 {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"></span>, 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 \phi (\mathbf {r} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi (\mathbf {r} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b691651163c74d532577e18847c7cdd92c7c1b26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.297ex; height:2.843ex;" alt="{\displaystyle \phi (\mathbf {r} )}"></span> is the electric potential that would be at <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 {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"></span> if the <a href="/wiki/Test_charge" class="mw-redirect" title="Test charge">test charge</a> were not present. If only two charges are present, the potential energy is <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_{1}Q_{2}/(4\pi \varepsilon _{0}r)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>r</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{1}Q_{2}/(4\pi \varepsilon _{0}r)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92a42f2376627b469f3aa0e2456ac5deb9045112" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.438ex; height:2.843ex;" alt="{\displaystyle Q_{1}Q_{2}/(4\pi \varepsilon _{0}r)}"></span>. The total <a href="/wiki/Electric_potential_energy" title="Electric potential energy">electric potential energy</a> due a collection of <i>N</i> charges is calculating by assembling these particles <a href="/wiki/Electric_Potential_Energy" class="mw-redirect" title="Electric Potential Energy">one at a time</a>: </p> <dl><dd><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 U_{\mathrm {E} }^{\text{total}}={\frac {1}{4\pi \varepsilon _{0}}}\sum _{j=1}^{N}Q_{j}\sum _{i=1}^{j-1}{\frac {Q_{i}}{r_{ij}}}={\frac {1}{2}}\sum _{i=1}^{N}Q_{i}\phi _{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>total</mtext> </mrow> </msubsup> <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>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </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> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\mathrm {E} }^{\text{total}}={\frac {1}{4\pi \varepsilon _{0}}}\sum _{j=1}^{N}Q_{j}\sum _{i=1}^{j-1}{\frac {Q_{i}}{r_{ij}}}={\frac {1}{2}}\sum _{i=1}^{N}Q_{i}\phi _{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee98a53179441fd6265f0d1c1855570498390ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:43.198ex; height:7.676ex;" alt="{\displaystyle U_{\mathrm {E} }^{\text{total}}={\frac {1}{4\pi \varepsilon _{0}}}\sum _{j=1}^{N}Q_{j}\sum _{i=1}^{j-1}{\frac {Q_{i}}{r_{ij}}}={\frac {1}{2}}\sum _{i=1}^{N}Q_{i}\phi _{i},}"></span></dd></dl> <p>where the following sum from, <i>j</i> = 1 to <i>N</i>, excludes <i>i</i> = <i>j</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{i}={\frac {1}{4\pi \varepsilon _{0}}}\sum _{\stackrel {j=1}{j\neq i}}^{N}{\frac {Q_{j}}{r_{ij}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </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> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mi>j</mi> <mo>≠</mo> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> </mover> </mrow> </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>j</mi> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{i}={\frac {1}{4\pi \varepsilon _{0}}}\sum _{\stackrel {j=1}{j\neq i}}^{N}{\frac {Q_{j}}{r_{ij}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d3d1e0102d0d21b03398accb1164c1ad8da0e91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:19.112ex; height:9.343ex;" alt="{\displaystyle \phi _{i}={\frac {1}{4\pi \varepsilon _{0}}}\sum _{\stackrel {j=1}{j\neq i}}^{N}{\frac {Q_{j}}{r_{ij}}}.}"></span></dd></dl> <p>This electric potential, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0182dbf29b54844c92fd9b0311778a02a38398ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.185ex; height:2.509ex;" alt="{\displaystyle \phi _{i}}"></span> is what would be measured at <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 {r} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed603561819ebd007acd75a0931d3ba401ad677a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.902ex; height:2.009ex;" alt="{\displaystyle \mathbf {r} _{i}}"></span> if the charge <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9f7193081d440425e522698e80817b5d558df03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.638ex; height:2.509ex;" alt="{\displaystyle Q_{i}}"></span> were missing. This formula obviously excludes the (infinite) energy that would be required to assemble each point charge from a disperse cloud of charge. The sum over charges can be converted into an integral over charge density using the prescription <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 \sum (\cdots )\rightarrow \int (\cdots )\rho \,\mathrm {d} ^{3}r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∑</mo> <mo stretchy="false">(</mo> <mo>⋯</mo> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo>∫</mo> <mo stretchy="false">(</mo> <mo>⋯</mo> <mo stretchy="false">)</mo> <mi>ρ</mi> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum (\cdots )\rightarrow \int (\cdots )\rho \,\mathrm {d} ^{3}r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/810974406685d25164ed1da0ff1d7e167f17862f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.535ex; height:3.343ex;" alt="{\textstyle \sum (\cdots )\rightarrow \int (\cdots )\rho \,\mathrm {d} ^{3}r}"></span>: </p> <dl><dd><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 U_{\mathrm {E} }^{\text{total}}={\frac {1}{2}}\int \rho (\mathbf {r} )\phi (\mathbf {r} )\,\mathrm {d} ^{3}r={\frac {\varepsilon _{0}}{2}}\int \left|{\mathbf {E} }\right|^{2}\,\mathrm {d} ^{3}r,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">E</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>total</mtext> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>∫</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mi>ϕ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mn>2</mn> </mfrac> </mrow> <mo>∫</mo> <msup> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{\mathrm {E} }^{\text{total}}={\frac {1}{2}}\int \rho (\mathbf {r} )\phi (\mathbf {r} )\,\mathrm {d} ^{3}r={\frac {\varepsilon _{0}}{2}}\int \left|{\mathbf {E} }\right|^{2}\,\mathrm {d} ^{3}r,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e28d6674b2bbf0a99433ce9880966e224a0c58e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:43.286ex; height:5.676ex;" alt="{\displaystyle U_{\mathrm {E} }^{\text{total}}={\frac {1}{2}}\int \rho (\mathbf {r} )\phi (\mathbf {r} )\,\mathrm {d} ^{3}r={\frac {\varepsilon _{0}}{2}}\int \left|{\mathbf {E} }\right|^{2}\,\mathrm {d} ^{3}r,}"></span></dd></dl> <p>This second expression for <a href="/wiki/Electrostatic_energy" class="mw-redirect" title="Electrostatic energy">electrostatic energy</a> uses the fact that the electric field is the negative <a href="/wiki/Gradient" title="Gradient">gradient</a> of the electric potential, as well as <a href="/wiki/Vector_calculus_identities" title="Vector calculus identities">vector calculus identities</a> in a way that resembles <a href="/wiki/Integration_by_parts" title="Integration by parts">integration by parts</a>. These two integrals for electric field energy seem to indicate two mutually exclusive formulas for electrostatic energy density, namely <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 {\frac {1}{2}}\rho \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>ρ</mi> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {1}{2}}\rho \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c03b75973ee21773c14bb6da5e36fd379a450914" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.245ex; height:3.509ex;" alt="{\textstyle {\frac {1}{2}}\rho \phi }"></span> 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 {\frac {1}{2}}\varepsilon _{0}E^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {1}{2}}\varepsilon _{0}E^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f5775c1a93417829ec3ad3c63743e54029f6a6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.644ex; height:3.509ex;" alt="{\textstyle {\frac {1}{2}}\varepsilon _{0}E^{2}}"></span>; they yield equal values for the total electrostatic energy only if both are integrated over all space. </p> <div class="mw-heading mw-heading3"><h3 id="Electrostatic_pressure">Electrostatic pressure</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electrostatics&action=edit&section=8" title="Edit section: Electrostatic pressure"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>On a <a href="/wiki/Electrical_conductor" title="Electrical conductor">conductor</a>, a surface charge will experience a force in the presence of an <a href="/wiki/Electric_field" title="Electric field">electric field</a>. This force is the average of the discontinuous electric field at the surface charge. This average in terms of the field just outside the surface amounts to: </p> <dl><dd><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={\frac {\varepsilon _{0}}{2}}E^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mn>2</mn> </mfrac> </mrow> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P={\frac {\varepsilon _{0}}{2}}E^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8dc738ef523127990e4e6599ccb85d77b39c641" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.313ex; height:4.676ex;" alt="{\displaystyle P={\frac {\varepsilon _{0}}{2}}E^{2},}"></span></dd></dl> <p>This pressure tends to draw the conductor into the field, regardless of the sign of the surface charge. </p> <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=Electrostatics&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/Electromagnetism" title="Electromagnetism">Electromagnetism</a> – Fundamental interaction between charged particles</li> <li><a href="/wiki/Electrostatic_generator" title="Electrostatic generator">Electrostatic generator</a>, machines that create static electricity.</li> <li><a href="/wiki/Electrostatic_induction" title="Electrostatic induction">Electrostatic induction</a>, separation of charges due to electric fields.</li> <li><a href="/wiki/Permittivity" title="Permittivity">Permittivity</a> and <a href="/wiki/Relative_permittivity" title="Relative permittivity">relative permittivity</a>, the electric polarizability of materials.</li> <li><a href="/wiki/Elementary_charge" title="Elementary charge">Quantisation of charge</a>, the charge units carried by electrons or protons.</li> <li><a href="/wiki/Static_electricity" title="Static electricity">Static electricity</a>, stationary charge accumulated on a material.</li> <li><a href="/wiki/Triboelectric_effect" title="Triboelectric effect">Triboelectric effect</a>, separation of charges due to sliding or contact.</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=Electrostatics&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 mw-references-columns"><ol class="references"> <li id="cite_note-Ling-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Ling_1-0">^</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="CITEREFLingMoebsSanny2019" class="citation book cs1">Ling, Samuel J.; Moebs, William; Sanny, Jeff (2019). <a rel="nofollow" class="external text" href="https://opentextbc.ca/universityphysicsv2openstax/chapter/conductors-insulators-and-charging-by-induction/"><i>University Physics, Vol. 2</i></a>. OpenStax. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781947172210" title="Special:BookSources/9781947172210"><bdi>9781947172210</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=University+Physics%2C+Vol.+2&rft.pub=OpenStax&rft.date=2019&rft.isbn=9781947172210&rft.aulast=Ling&rft.aufirst=Samuel+J.&rft.au=Moebs%2C+William&rft.au=Sanny%2C+Jeff&rft_id=https%3A%2F%2Fopentextbc.ca%2Funiversityphysicsv2openstax%2Fchapter%2Fconductors-insulators-and-charging-by-induction%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrostatics" class="Z3988"></span> Ch.30: Conductors, Insulators, and Charging by Induction</span> </li> <li id="cite_note-Bloomfield-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bloomfield_2-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBloomfield2015" class="citation book cs1">Bloomfield, Louis A. 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Cambridge University Press. pp. 296–354. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2F9781108333511.008">10.1017/9781108333511.008</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-108-33351-1" title="Special:BookSources/978-1-108-33351-1"><bdi>978-1-108-33351-1</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-08-11</span></span>.</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=296-354&rft.pub=Cambridge+University+Press&rft.date=2017&rft_id=info%3Adoi%2F10.1017%2F9781108333511.008&rft.isbn=978-1-108-33351-1&rft.aulast=J&rft.aufirst=Griffiths&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1017%2F9781108333511.008&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrostatics" class="Z3988"></span></span> </li> <li id="cite_note-physconst-eps0-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-physconst-eps0_6-0">^</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%3AElectrostatics" class="Z3988"></span></span> </li> <li id="cite_note-Sadiku-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Sadiku_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMatthew_Sadiku2009" class="citation book cs1">Matthew Sadiku (2009). <i>Elements of electromagnetics</i>. Oxford University Press. p. 104. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780195387759" title="Special:BookSources/9780195387759"><bdi>9780195387759</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+electromagnetics&rft.pages=104&rft.pub=Oxford+University+Press&rft.date=2009&rft.isbn=9780195387759&rft.au=Matthew+Sadiku&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrostatics" class="Z3988"></span></span> </li> <li id="cite_note-Purcell-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-Purcell_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Purcell_8-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Purcell_8-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPurcell2013" class="citation book cs1">Purcell, Edward M. (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=A2rS5vlSFq0C&q=%22electric+field%22&pg=PA16"><i>Electricity and Magnetism</i></a>. Cambridge University Press. pp. 16–18. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1107014022" title="Special:BookSources/978-1107014022"><bdi>978-1107014022</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electricity+and+Magnetism&rft.pages=16-18&rft.pub=Cambridge+University+Press&rft.date=2013&rft.isbn=978-1107014022&rft.aulast=Purcell&rft.aufirst=Edward+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DA2rS5vlSFq0C%26q%3D%2522electric%2Bfield%2522%26pg%3DPA16&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrostatics" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation journal cs1"><a rel="nofollow" class="external text" href="https://dx.doi.org/10.1163/9789004460409_mor2-b29447057">"Sur l'attraction des sphéroides elliptiques, par M. de La Grange"</a>. <i>Mathematics General Collection</i>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1163%2F9789004460409_mor2-b29447057">10.1163/9789004460409_mor2-b29447057</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2023-08-11</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+General+Collection&rft.atitle=Sur+l%27attraction+des+sph%C3%A9roides+elliptiques%2C+par+M.+de+La+Grange.&rft_id=info%3Adoi%2F10.1163%2F9789004460409_mor2-b29447057&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1163%2F9789004460409_mor2-b29447057&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrostatics" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGauss1877" class="citation cs2">Gauss, Carl Friedrich (1877), <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1007/978-3-642-49319-5_8">"Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum, methodo nova tractata"</a>, <i>Werke</i>, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 279–286, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-49319-5_8">10.1007/978-3-642-49319-5_8</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-49320-1" title="Special:BookSources/978-3-642-49320-1"><bdi>978-3-642-49320-1</bdi></a><span class="reference-accessdate">, retrieved <span class="nowrap">2023-08-11</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Werke&rft.atitle=Theoria+attractionis+corporum+sphaeroidicorum+ellipticorum+homogeneorum%2C+methodo+nova+tractata&rft.pages=279-286&rft.date=1877&rft_id=info%3Adoi%2F10.1007%2F978-3-642-49319-5_8&rft.isbn=978-3-642-49320-1&rft.aulast=Gauss&rft.aufirst=Carl+Friedrich&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1007%2F978-3-642-49319-5_8&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrostatics" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPoissonsciences_(France)1827" class="citation book cs1">Poisson, M; sciences (France), Académie royale des (1827). <a rel="nofollow" class="external text" href="https://www.biodiversitylibrary.org/item/55214"><i>Mémoires de l'Académie (royale) des sciences de l'Institut (imperial) de France</i></a>. Vol. 6. Paris.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=M%C3%A9moires+de+l%27Acad%C3%A9mie+%28royale%29+des+sciences+de+l%27Institut+%28imperial%29+de+France&rft.place=Paris&rft.date=1827&rft.aulast=Poisson&rft.aufirst=M&rft.au=sciences+%28France%29%2C+Acad%C3%A9mie+royale+des&rft_id=https%3A%2F%2Fwww.biodiversitylibrary.org%2Fitem%2F55214&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrostatics" class="Z3988"></span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeras2010" class="citation journal cs1">Heras, J. A. (2010). "The Galilean limits of Maxwell's equations". <i><a href="/wiki/American_Journal_of_Physics" title="American Journal of Physics">American Journal of Physics</a></i>. <b>78</b> (10): 1048–1055. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1012.1068">1012.1068</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2010AmJPh..78.1048H">2010AmJPh..78.1048H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.3442798">10.1119/1.3442798</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118443242">118443242</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Physics&rft.atitle=The+Galilean+limits+of+Maxwell%27s+equations&rft.volume=78&rft.issue=10&rft.pages=1048-1055&rft.date=2010&rft_id=info%3Aarxiv%2F1012.1068&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118443242%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1119%2F1.3442798&rft_id=info%3Abibcode%2F2010AmJPh..78.1048H&rft.aulast=Heras&rft.aufirst=J.+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrostatics" 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=Electrostatics&action=edit&section=11" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHermann_A._HausJames_R._Melcher1989" class="citation book cs1">Hermann A. Haus; James R. Melcher (1989). <i>Electromagnetic Fields and Energy</i>. Englewood Cliffs, NJ: Prentice-Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-249020-X" title="Special:BookSources/0-13-249020-X"><bdi>0-13-249020-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=Electromagnetic+Fields+and+Energy&rft.place=Englewood+Cliffs%2C+NJ&rft.pub=Prentice-Hall&rft.date=1989&rft.isbn=0-13-249020-X&rft.au=Hermann+A.+Haus&rft.au=James+R.+Melcher&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrostatics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHalliday,_DavidRobert_ResnickKenneth_S._Krane1992" class="citation book cs1">Halliday, David; Robert Resnick; Kenneth S. Krane (1992). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/isbn_9780471559184"><i>Physics</i></a></span>. New York: John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-80457-6" title="Special:BookSources/0-471-80457-6"><bdi>0-471-80457-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics&rft.place=New+York&rft.pub=John+Wiley+%26+Sons&rft.date=1992&rft.isbn=0-471-80457-6&rft.au=Halliday%2C+David&rft.au=Robert+Resnick&rft.au=Kenneth+S.+Krane&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fisbn_9780471559184&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrostatics" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGriffiths,_David_J.1999" class="citation book cs1"><a href="/wiki/David_J._Griffiths" title="David J. Griffiths">Griffiths, David J.</a> (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>. Upper Saddle River, NJ: 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.place=Upper+Saddle+River%2C+NJ&rft.pub=Prentice+Hall&rft.date=1999&rft.isbn=0-13-805326-X&rft.au=Griffiths%2C+David+J.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoel00grif_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrostatics" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Electrostatics&action=edit&section=12" title="Edit section: External links"><span>edit</span></a><span 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src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Electrostatics" class="extiw" title="commons:Category:Electrostatics">Electrostatics</a> at Wikimedia Commons</li></ul> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237033735"><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/Special:Search/electrostatics" class="extiw" title="wiktionary:Special:Search/electrostatics">electrostatics</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <ul><li><a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/II_04.html">The Feynman Lectures on Physics Vol. II Ch. 4: Electrostatics</a></li> <li><a rel="nofollow" class="external text" href="http://physics.gmu.edu/~joe/PHYS685/Topic1.pdf">Introduction to Electrostatics</a>: Point charges can be treated as a distribution using the <a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a></li></ul> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><div class="side-box metadata side-box-right"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-abovebelow"> <a href="/wiki/Wikipedia:The_Wikipedia_Library" title="Wikipedia:The Wikipedia Library">Library resources</a> about <br /> <b>Electrostatics</b> <hr /></div> <div class="side-box-flex"> <div class="side-box-text plainlist"><ul><li><a class="external text" href="https://ftl.toolforge.org/cgi-bin/ftl?st=wp&su=Electrostatics">Resources in your library</a></li> </ul></div></div> </div> <p><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wikiversity_logo_2017.svg" 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