Elektrisk felt – Wikipedia

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id="toc-Karakteristiske_feltstyrker-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Elektrostatiske_felt" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Elektrostatiske_felt"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Elektrostatiske felt</span> </div> </a> <ul id="toc-Elektrostatiske_felt-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Elektrisk_fluks" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Elektrisk_fluks"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Elektrisk fluks</span> </div> </a> <button aria-controls="toc-Elektrisk_fluks-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>Vis/skjul underseksjonen Elektrisk fluks</span> </button> <ul id="toc-Elektrisk_fluks-sublist" class="vector-toc-list"> <li id="toc-Eksempel:_Ladet_kule" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Eksempel:_Ladet_kule"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Eksempel: Ladet kule</span> </div> </a> <ul id="toc-Eksempel:_Ladet_kule-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Differensiell_form_av_Gauss'_lov" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Differensiell_form_av_Gauss'_lov"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Differensiell form av Gauss' lov</span> </div> </a> <button aria-controls="toc-Differensiell_form_av_Gauss'_lov-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>Vis/skjul underseksjonen Differensiell form av Gauss' lov</span> </button> <ul id="toc-Differensiell_form_av_Gauss'_lov-sublist" class="vector-toc-list"> <li id="toc-Punktladning" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Punktladning"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Punktladning</span> </div> </a> <ul id="toc-Punktladning-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Ladninger_på_linjer_og_plan" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Ladninger_på_linjer_og_plan"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Ladninger på linjer og plan</span> </div> </a> <button aria-controls="toc-Ladninger_på_linjer_og_plan-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>Vis/skjul underseksjonen Ladninger på linjer og plan</span> </button> <ul id="toc-Ladninger_på_linjer_og_plan-sublist" class="vector-toc-list"> <li id="toc-Linjeladning" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Linjeladning"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Linjeladning</span> </div> </a> <ul id="toc-Linjeladning-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ladet_plan" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ladet_plan"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Ladet plan</span> </div> </a> <ul id="toc-Ladet_plan-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ladet_ring" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ladet_ring"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Ladet ring</span> </div> </a> <ul id="toc-Ladet_ring-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ladet_disk" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ladet_disk"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Ladet disk</span> </div> </a> <ul id="toc-Ladet_disk-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ledere_og_dielektrika" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ledere_og_dielektrika"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Ledere og dielektrika</span> </div> </a> <ul id="toc-Ledere_og_dielektrika-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Elektrisk_potensial" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Elektrisk_potensial"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Elektrisk potensial</span> </div> </a> <ul id="toc-Elektrisk_potensial-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kondensator" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Kondensator"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Kondensator</span> </div> </a> <button aria-controls="toc-Kondensator-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>Vis/skjul underseksjonen Kondensator</span> </button> <ul id="toc-Kondensator-sublist" class="vector-toc-list"> <li id="toc-Elektrisk_feltenergi" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Elektrisk_feltenergi"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Elektrisk feltenergi</span> </div> </a> <ul id="toc-Elektrisk_feltenergi-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Dipolfeltet" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Dipolfeltet"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Dipolfeltet</span> </div> </a> <button aria-controls="toc-Dipolfeltet-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>Vis/skjul underseksjonen Dipolfeltet</span> </button> <ul id="toc-Dipolfeltet-sublist" class="vector-toc-list"> <li id="toc-Dipolpotensialet" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dipolpotensialet"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Dipolpotensialet</span> </div> </a> <ul id="toc-Dipolpotensialet-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multipoler" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multipoler"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.2</span> <span>Multipoler</span> </div> </a> <ul id="toc-Multipoler-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Dipol_i_ytre_felt" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Dipol_i_ytre_felt"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Dipol i ytre felt</span> </div> </a> <button aria-controls="toc-Dipol_i_ytre_felt-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>Vis/skjul underseksjonen Dipol i ytre felt</span> </button> <ul id="toc-Dipol_i_ytre_felt-sublist" class="vector-toc-list"> <li id="toc-Potensiell_energi" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Potensiell_energi"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Potensiell energi</span> </div> </a> <ul id="toc-Potensiell_energi-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inhomogent_felt" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inhomogent_felt"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2</span> <span>Inhomogent felt</span> </div> </a> <ul id="toc-Inhomogent_felt-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Elektrisk_polarisasjon" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Elektrisk_polarisasjon"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Elektrisk polarisasjon</span> </div> </a> <button aria-controls="toc-Elektrisk_polarisasjon-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>Vis/skjul underseksjonen Elektrisk polarisasjon</span> </button> <ul id="toc-Elektrisk_polarisasjon-sublist" class="vector-toc-list"> <li id="toc-Lineært_materiale" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lineært_materiale"> <div class="vector-toc-text"> <span class="vector-toc-numb">10.1</span> <span>Lineært materiale</span> </div> </a> <ul id="toc-Lineært_materiale-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Elektrodynamikk" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Elektrodynamikk"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Elektrodynamikk</span> </div> </a> <ul id="toc-Elektrodynamikk-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kvanteelektrodynamikk" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Kvanteelektrodynamikk"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Kvanteelektrodynamikk</span> </div> </a> <ul id="toc-Kvanteelektrodynamikk-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Se_også" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Se_også"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Se også</span> </div> </a> <ul id="toc-Se_også-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Referanser" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Referanser"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>Referanser</span> </div> </a> <ul id="toc-Referanser-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Eksterne_lenker" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Eksterne_lenker"> <div class="vector-toc-text"> <span class="vector-toc-numb">15</span> <span>Eksterne lenker</span> </div> </a> <ul id="toc-Eksterne_lenker-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="Innhold" 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="Vis/skjul innholdsfortegnelsen" > <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">Vis/skjul innholdsfortegnelsen</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">Elektrisk felt</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="Gå til en artikkel på et annet språk. Tilgjengelig på 98 språk" > <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-98" 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">98 språk</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Elektrisk_felt" title="Elektrisk felt – norsk nynorsk" lang="nn" hreflang="nn" data-title="Elektrisk felt" data-language-autonym="Norsk nynorsk" data-language-local-name="norsk nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Elektrisk_felt" title="Elektrisk felt – dansk" lang="da" hreflang="da" data-title="Elektrisk felt" data-language-autonym="Dansk" data-language-local-name="dansk" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Elektriskt_f%C3%A4lt" title="Elektriskt fält – svensk" lang="sv" hreflang="sv" data-title="Elektriskt fält" data-language-autonym="Svenska" data-language-local-name="svensk" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Rafsvi%C3%B0" title="Rafsvið – islandsk" lang="is" hreflang="is" data-title="Rafsvið" data-language-autonym="Íslenska" data-language-local-name="islandsk" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Elektrisches_Feld" title="Elektrisches Feld – sveitsertysk" lang="gsw" hreflang="gsw" data-title="Elektrisches Feld" data-language-autonym="Alemannisch" data-language-local-name="sveitsertysk" class="interlanguage-link-target"><span>Alemannisch</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%8A%A4%E1%88%8C%E1%8A%AD%E1%89%B5%E1%88%AA%E1%8A%AD_%E1%88%98%E1%88%B5%E1%8A%AD" title="ኤሌክትሪክ መስክ – amharisk" lang="am" hreflang="am" data-title="ኤሌክትሪክ መስክ" data-language-autonym="አማርኛ" data-language-local-name="amharisk" class="interlanguage-link-target"><span>አማርኛ</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AD%D9%82%D9%84_%D9%83%D9%87%D8%B1%D8%A8%D8%A7%D8%A6%D9%8A" title="حقل كهربائي – arabisk" lang="ar" hreflang="ar" data-title="حقل كهربائي" data-language-autonym="العربية" data-language-local-name="arabisk" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%AC%E0%A7%88%E0%A6%A6%E0%A7%8D%E0%A6%AF%E0%A7%81%E0%A6%A4%E0%A6%BF%E0%A6%95_%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A7%87%E0%A6%A4%E0%A7%8D%E0%A7%B0" title="বৈদ্যুতিক ক্ষেত্ৰ – assamesisk" lang="as" hreflang="as" data-title="বৈদ্যুতিক ক্ষেত্ৰ" data-language-autonym="অসমীয়া" data-language-local-name="assamesisk" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Campu_ll%C3%A9tricu" title="Campu llétricu – asturisk" lang="ast" hreflang="ast" data-title="Campu llétricu" data-language-autonym="Asturianu" data-language-local-name="asturisk" 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/Elektrik_sah%C9%99si" title="Elektrik sahəsi – aserbajdsjansk" lang="az" hreflang="az" data-title="Elektrik sahəsi" data-language-autonym="Azərbaycanca" data-language-local-name="aserbajdsjansk" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D8%A7%D9%84%DA%A9%D8%AA%D8%B1%DB%8C%DA%A9_%D9%85%D8%A6%DB%8C%D8%AF%D8%A7%D9%86%DB%8C" title="الکتریک مئیدانی – søraserbajdsjansk" lang="azb" hreflang="azb" data-title="الکتریک مئیدانی" data-language-autonym="تۆرکجه" data-language-local-name="søraserbajdsjansk" class="interlanguage-link-target"><span>تۆرکجه</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%A4%E0%A6%A1%E0%A6%BC%E0%A6%BF%E0%A7%8E_%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A7%87%E0%A6%A4%E0%A7%8D%E0%A6%B0" title="তড়িৎ ক্ষেত্র – bengali" lang="bn" hreflang="bn" data-title="তড়িৎ ক্ষেত্র" data-language-autonym="বাংলা" data-language-local-name="bengali" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Ti%C4%81n-ti%C3%BB%E2%81%BF" title="Tiān-tiûⁿ – minnan" lang="nan" hreflang="nan" data-title="Tiān-tiûⁿ" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="minnan" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%AD%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D1%8B%D1%87%D0%BD%D0%B0%D0%B5_%D0%BF%D0%BE%D0%BB%D0%B5" title="Электрычнае поле – belarusisk" lang="be" hreflang="be" data-title="Электрычнае поле" data-language-autonym="Беларуская" data-language-local-name="belarusisk" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%AD%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D1%8B%D1%87%D0%BD%D0%B0%D0%B5_%D0%BF%D0%BE%D0%BB%D0%B5" title="Электрычнае поле – belarusisk (klassisk ortografi)" lang="be-tarask" hreflang="be-tarask" data-title="Электрычнае поле" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="belarusisk (klassisk ortografi)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bh mw-list-item"><a href="https://bh.wikipedia.org/wiki/%E0%A4%AC%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%AF%E0%A5%81%E0%A4%A4_%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0" title="बिद्युत क्षेत्र – bihari" lang="bh" hreflang="bh" data-title="बिद्युत क्षेत्र" data-language-autonym="भोजपुरी" data-language-local-name="bihari" class="interlanguage-link-target"><span>भोजपुरी</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%95%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE_%D0%BF%D0%BE%D0%BB%D0%B5" title="Електрическо поле – bulgarsk" lang="bg" hreflang="bg" data-title="Електрическо поле" data-language-autonym="Български" data-language-local-name="bulgarsk" 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/Elektri%C4%8Dno_polje" title="Električno polje – bosnisk" lang="bs" hreflang="bs" data-title="Električno polje" data-language-autonym="Bosanski" data-language-local-name="bosnisk" 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/Camp_el%C3%A8ctric" title="Camp elèctric – katalansk" lang="ca" hreflang="ca" data-title="Camp elèctric" data-language-autonym="Català" data-language-local-name="katalansk" 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%83%D0%B9" title="Электро уй – tsjuvasjisk" lang="cv" hreflang="cv" data-title="Электро уй" data-language-autonym="Чӑвашла" data-language-local-name="tsjuvasjisk" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Elektrick%C3%A9_pole" title="Elektrické pole – tsjekkisk" lang="cs" hreflang="cs" data-title="Elektrické pole" data-language-autonym="Čeština" data-language-local-name="tsjekkisk" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-sn mw-list-item"><a href="https://sn.wikipedia.org/wiki/Dandare_remagetsi" title="Dandare remagetsi – shona" lang="sn" hreflang="sn" data-title="Dandare remagetsi" data-language-autonym="ChiShona" data-language-local-name="shona" class="interlanguage-link-target"><span>ChiShona</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Elektrisches_Feld" title="Elektrisches Feld – tysk" lang="de" hreflang="de" data-title="Elektrisches Feld" data-language-autonym="Deutsch" data-language-local-name="tysk" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Elektriv%C3%A4li" title="Elektriväli – estisk" lang="et" hreflang="et" data-title="Elektriväli" data-language-autonym="Eesti" data-language-local-name="estisk" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%97%CE%BB%CE%B5%CE%BA%CF%84%CF%81%CE%B9%CE%BA%CF%8C_%CF%80%CE%B5%CE%B4%CE%AF%CE%BF" title="Ηλεκτρικό πεδίο – gresk" lang="el" hreflang="el" data-title="Ηλεκτρικό πεδίο" data-language-autonym="Ελληνικά" data-language-local-name="gresk" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Electric_field" title="Electric field – engelsk" lang="en" hreflang="en" data-title="Electric field" data-language-autonym="English" data-language-local-name="engelsk" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Campo_el%C3%A9ctrico" title="Campo eléctrico – spansk" lang="es" hreflang="es" data-title="Campo eléctrico" data-language-autonym="Español" data-language-local-name="spansk" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Elektra_kampo" title="Elektra kampo – esperanto" lang="eo" hreflang="eo" data-title="Elektra kampo" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-ext mw-list-item"><a href="https://ext.wikipedia.org/wiki/Campu_el%C3%A9tricu" title="Campu elétricu – ekstremaduransk" lang="ext" hreflang="ext" data-title="Campu elétricu" data-language-autonym="Estremeñu" data-language-local-name="ekstremaduransk" class="interlanguage-link-target"><span>Estremeñu</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Eremu_elektriko" title="Eremu elektriko – baskisk" lang="eu" hreflang="eu" data-title="Eremu elektriko" data-language-autonym="Euskara" data-language-local-name="baskisk" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%DB%8C%D8%AF%D8%A7%D9%86_%D8%A7%D9%84%DA%A9%D8%AA%D8%B1%DB%8C%DA%A9%DB%8C" title="میدان الکتریکی – persisk" lang="fa" hreflang="fa" data-title="میدان الکتریکی" data-language-autonym="فارسی" data-language-local-name="persisk" 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/Champ_%C3%A9lectrique" title="Champ électrique – fransk" lang="fr" hreflang="fr" data-title="Champ électrique" data-language-autonym="Français" data-language-local-name="fransk" 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/R%C3%A9imse_leictreach" title="Réimse leictreach – irsk" lang="ga" hreflang="ga" data-title="Réimse leictreach" data-language-autonym="Gaeilge" data-language-local-name="irsk" 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/Campo_el%C3%A9ctrico" title="Campo eléctrico – galisisk" lang="gl" hreflang="gl" data-title="Campo eléctrico" data-language-autonym="Galego" data-language-local-name="galisisk" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%84%EA%B8%B0%EC%9E%A5" title="전기장 – koreansk" lang="ko" hreflang="ko" data-title="전기장" data-language-autonym="한국어" data-language-local-name="koreansk" 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%AF%D5%A1%D5%B6_%D5%A4%D5%A1%D5%B7%D5%BF" title="Էլեկտրական դաշտ – armensk" lang="hy" hreflang="hy" data-title="Էլեկտրական դաշտ" data-language-autonym="Հայերեն" data-language-local-name="armensk" 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%B5%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%AF%E0%A5%81%E0%A4%A4%E0%A5%8D-%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0" title="विद्युत्-क्षेत्र – hindi" lang="hi" hreflang="hi" data-title="विद्युत्-क्षेत्र" data-language-autonym="हिन्दी" data-language-local-name="hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Elektri%C4%8Dno_polje" title="Električno polje – kroatisk" lang="hr" hreflang="hr" data-title="Električno polje" data-language-autonym="Hrvatski" data-language-local-name="kroatisk" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-gor mw-list-item"><a href="https://gor.wikipedia.org/wiki/Medan_listrik" title="Medan listrik – gorontalo" lang="gor" hreflang="gor" data-title="Medan listrik" data-language-autonym="Bahasa Hulontalo" data-language-local-name="gorontalo" class="interlanguage-link-target"><span>Bahasa Hulontalo</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Medan_listrik" title="Medan listrik – indonesisk" lang="id" hreflang="id" data-title="Medan listrik" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesisk" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-zu mw-list-item"><a href="https://zu.wikipedia.org/wiki/Umzuba" title="Umzuba – zulu" lang="zu" hreflang="zu" data-title="Umzuba" data-language-autonym="IsiZulu" data-language-local-name="zulu" class="interlanguage-link-target"><span>IsiZulu</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Campo_elettrico" title="Campo elettrico – italiensk" lang="it" hreflang="it" data-title="Campo elettrico" data-language-autonym="Italiano" data-language-local-name="italiensk" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A9%D7%93%D7%94_%D7%97%D7%A9%D7%9E%D7%9C%D7%99" title="שדה חשמלי – hebraisk" lang="he" hreflang="he" data-title="שדה חשמלי" data-language-autonym="עברית" data-language-local-name="hebraisk" 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%B5%E0%B2%BF%E0%B2%A6%E0%B3%8D%E0%B2%AF%E0%B3%81%E0%B2%A4%E0%B3%8D_%E0%B2%95%E0%B3%8D%E0%B2%B7%E0%B3%87%E0%B2%A4%E0%B3%8D%E0%B2%B0" title="ವಿದ್ಯುತ್ ಕ್ಷೇತ್ರ – kannada" lang="kn" hreflang="kn" data-title="ವಿದ್ಯುತ್ ಕ್ಷೇತ್ರ" 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%A3%E1%83%9A%E1%83%98_%E1%83%95%E1%83%94%E1%83%9A%E1%83%98" title="ელექტრული ველი – georgisk" lang="ka" hreflang="ka" data-title="ელექტრული ველი" data-language-autonym="ქართული" data-language-local-name="georgisk" 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_%D3%A9%D1%80%D1%96%D1%81%D1%96" title="Электр өрісі – kasakhisk" lang="kk" hreflang="kk" data-title="Электр өрісі" data-language-autonym="Қазақша" data-language-local-name="kasakhisk" 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/Chan_elektrik" title="Chan elektrik – haitisk" lang="ht" hreflang="ht" data-title="Chan elektrik" data-language-autonym="Kreyòl ayisyen" data-language-local-name="haitisk" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Campus_electricus" title="Campus electricus – latin" lang="la" hreflang="la" data-title="Campus electricus" data-language-autonym="Latina" data-language-local-name="latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Elektriskais_lauks" title="Elektriskais lauks – latvisk" lang="lv" hreflang="lv" data-title="Elektriskais lauks" data-language-autonym="Latviešu" data-language-local-name="latvisk" 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/Elektrinis_laukas" title="Elektrinis laukas – litauisk" lang="lt" hreflang="lt" data-title="Elektrinis laukas" data-language-autonym="Lietuvių" data-language-local-name="litauisk" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Lektrikveldj" title="Lektrikveldj – limburgsk" lang="li" hreflang="li" data-title="Lektrikveldj" data-language-autonym="Limburgs" data-language-local-name="limburgsk" class="interlanguage-link-target"><span>Limburgs</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Elektromos_mez%C5%91" title="Elektromos mező – ungarsk" lang="hu" hreflang="hu" data-title="Elektromos mező" data-language-autonym="Magyar" data-language-local-name="ungarsk" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%95%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%BE_%D0%BF%D0%BE%D0%BB%D0%B5" title="Електрично поле – makedonsk" lang="mk" hreflang="mk" data-title="Електрично поле" data-language-autonym="Македонски" data-language-local-name="makedonsk" 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%B5%E0%B5%88%E0%B4%A6%E0%B5%8D%E0%B4%AF%E0%B5%81%E0%B4%A4%E0%B4%95%E0%B5%8D%E0%B4%B7%E0%B5%87%E0%B4%A4%E0%B5%8D%E0%B4%B0%E0%B4%82" title="വൈദ്യുതക്ഷേത്രം – malayalam" lang="ml" hreflang="ml" data-title="വൈദ്യുതക്ഷേത്രം" data-language-autonym="മലയാളം" data-language-local-name="malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%A6%E0%A5%8D%E0%A4%AF%E0%A5%81%E0%A4%A4_%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0" title="विद्युत क्षेत्र – marathi" lang="mr" hreflang="mr" data-title="विद्युत क्षेत्र" data-language-autonym="मराठी" data-language-local-name="marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-mzn mw-list-item"><a href="https://mzn.wikipedia.org/wiki/%D9%85%DB%8C%D8%AF%D8%A7%D9%86_%D8%A7%D9%84%DA%A9%D8%AA%D8%B1%DB%8C%DA%A9%DB%8C" title="میدان الکتریکی – mazandarani" lang="mzn" hreflang="mzn" data-title="میدان الکتریکی" data-language-autonym="مازِرونی" data-language-local-name="mazandarani" class="interlanguage-link-target"><span>مازِرونی</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Medan_elektrik" title="Medan elektrik – malayisk" lang="ms" hreflang="ms" data-title="Medan elektrik" data-language-autonym="Bahasa Melayu" data-language-local-name="malayisk" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A6%D0%B0%D1%85%D0%B8%D0%BB%D0%B3%D0%B0%D0%B0%D0%BD_%D0%BE%D1%80%D0%BE%D0%BD" title="Цахилгаан орон – mongolsk" lang="mn" hreflang="mn" data-title="Цахилгаан орон" data-language-autonym="Монгол" data-language-local-name="mongolsk" class="interlanguage-link-target"><span>Монгол</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%85%E1%80%85%E1%80%BA%E1%80%85%E1%80%80%E1%80%BA%E1%80%80%E1%80%BD%E1%80%84%E1%80%BA%E1%80%B8" title="လျှပ်စစ်စက်ကွင်း – burmesisk" lang="my" hreflang="my" data-title="လျှပ်စစ်စက်ကွင်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="burmesisk" 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/Elektrisch_veld" title="Elektrisch veld – nederlandsk" lang="nl" hreflang="nl" data-title="Elektrisch veld" data-language-autonym="Nederlands" data-language-local-name="nederlandsk" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E9%9B%BB%E5%A0%B4" title="電場 – japansk" lang="ja" hreflang="ja" data-title="電場" data-language-autonym="日本語" data-language-local-name="japansk" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-frr badge-Q70894304 mw-list-item" title=""><a href="https://frr.wikipedia.org/wiki/Elektrisk_fial" title="Elektrisk fial – nordfrisisk" lang="frr" hreflang="frr" data-title="Elektrisk fial" data-language-autonym="Nordfriisk" data-language-local-name="nordfrisisk" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Camp_electric" title="Camp electric – oksitansk" lang="oc" hreflang="oc" data-title="Camp electric" data-language-autonym="Occitan" data-language-local-name="oksitansk" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%AC%E0%AD%88%E0%AC%A6%E0%AD%81%E0%AC%A4%E0%AC%BF%E0%AC%95_%E0%AC%95%E0%AD%8D%E0%AC%B7%E0%AD%87%E0%AC%A4%E0%AD%8D%E0%AC%B0" title="ବୈଦୁତିକ କ୍ଷେତ୍ର – odia" lang="or" hreflang="or" data-title="ବୈଦୁତିକ କ୍ଷେତ୍ର" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="odia" class="interlanguage-link-target"><span>ଓଡ଼ିଆ</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Elektr_maydon" title="Elektr maydon – usbekisk" lang="uz" hreflang="uz" data-title="Elektr maydon" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="usbekisk" 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%A8%BF%E0%A8%95_%E0%A8%AB%E0%A9%80%E0%A8%B2%E0%A8%A1" title="ਇਲੈਕਟ੍ਰਿਕ ਫੀਲਡ – panjabi" lang="pa" hreflang="pa" data-title="ਇਲੈਕਟ੍ਰਿਕ ਫੀਲਡ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="panjabi" 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%DA%A9_%D9%81%DB%8C%D9%84%DA%88" title="الیکٹرک فیلڈ – vestpunjabi" lang="pnb" hreflang="pnb" data-title="الیکٹرک فیلڈ" data-language-autonym="پنجابی" data-language-local-name="vestpunjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%A8%D8%B1%DB%90%DA%9A%D9%86%D8%A7%D9%8A%D9%8A_%D8%B3%D8%A7%D8%AD%D9%87" title="برېښنايي ساحه – pashto" lang="ps" hreflang="ps" data-title="برېښنايي ساحه" data-language-autonym="پښتو" data-language-local-name="pashto" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%8A%E1%9F%82%E1%9E%93%E1%9E%A2%E1%9E%82%E1%9F%92%E1%9E%82%E1%9E%B8%E1%9E%9F%E1%9E%93%E1%9E%B8" title="ដែនអគ្គីសនី – khmer" lang="km" hreflang="km" data-title="ដែនអគ្គីសនី" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="khmer" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Camp_el%C3%A9trich" title="Camp elétrich – piemontesisk" lang="pms" hreflang="pms" data-title="Camp elétrich" data-language-autonym="Piemontèis" data-language-local-name="piemontesisk" 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/Pole_elektryczne" title="Pole elektryczne – polsk" lang="pl" hreflang="pl" data-title="Pole elektryczne" data-language-autonym="Polski" data-language-local-name="polsk" 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/Campo_el%C3%A9trico" title="Campo elétrico – portugisisk" lang="pt" hreflang="pt" data-title="Campo elétrico" data-language-autonym="Português" data-language-local-name="portugisisk" 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/C%C3%A2mp_electric" title="Câmp electric – rumensk" lang="ro" hreflang="ro" data-title="Câmp electric" data-language-autonym="Română" data-language-local-name="rumensk" 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%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D0%BE%D0%BB%D0%B5" title="Электрическое поле – russisk" lang="ru" hreflang="ru" data-title="Электрическое поле" data-language-autonym="Русский" data-language-local-name="russisk" 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/Fusha_elektrike" title="Fusha elektrike – albansk" lang="sq" hreflang="sq" data-title="Fusha elektrike" data-language-autonym="Shqip" data-language-local-name="albansk" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-scn badge-Q17437796 badge-featuredarticle mw-list-item" title="utmerket artikkel-merke"><a href="https://scn.wikipedia.org/wiki/Campu_el%C3%A8ttricu" title="Campu elèttricu – siciliansk" lang="scn" hreflang="scn" data-title="Campu elèttricu" data-language-autonym="Sicilianu" data-language-local-name="siciliansk" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Electric_field" title="Electric field – enkel engelsk" lang="en-simple" hreflang="en-simple" data-title="Electric field" data-language-autonym="Simple English" data-language-local-name="enkel engelsk" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Elektrick%C3%A9_pole" title="Elektrické pole – slovakisk" lang="sk" hreflang="sk" data-title="Elektrické pole" data-language-autonym="Slovenčina" data-language-local-name="slovakisk" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Elektri%C4%8Dno_polje" title="Električno polje – slovensk" lang="sl" hreflang="sl" data-title="Električno polje" data-language-autonym="Slovenščina" data-language-local-name="slovensk" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A8%D9%88%D8%A7%D8%B1%DB%8C_%DA%A9%D8%A7%D8%B1%DB%95%D8%A8%D8%A7%DB%8C%DB%8C" title="بواری کارەبایی – sentralkurdisk" lang="ckb" hreflang="ckb" data-title="بواری کارەبایی" data-language-autonym="کوردی" data-language-local-name="sentralkurdisk" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%95%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%BE_%D0%BF%D0%BE%D1%99%D0%B5" title="Електрично поље – serbisk" lang="sr" hreflang="sr" data-title="Електрично поље" data-language-autonym="Српски / srpski" data-language-local-name="serbisk" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Elektri%C4%8Dno_polje" title="Električno polje – serbokroatisk" lang="sh" hreflang="sh" data-title="Električno polje" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbokroatisk" 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/M%C3%A9dan_Listrik" title="Médan Listrik – sundanesisk" lang="su" hreflang="su" data-title="Médan Listrik" data-language-autonym="Sunda" data-language-local-name="sundanesisk" 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%B6kentt%C3%A4" title="Sähkökenttä – finsk" lang="fi" hreflang="fi" data-title="Sähkökenttä" data-language-autonym="Suomi" data-language-local-name="finsk" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-tl badge-Q70893996 mw-list-item" title=""><a href="https://tl.wikipedia.org/wiki/Elektrikong_field" title="Elektrikong field – tagalog" lang="tl" hreflang="tl" data-title="Elektrikong field" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AE%BF%E0%AE%A9%E0%AF%8D%E0%AE%AA%E0%AF%81%E0%AE%B2%E0%AE%AE%E0%AF%8D" title="மின்புலம் – tamil" lang="ta" hreflang="ta" data-title="மின்புலம்" 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%BA%D1%8B%D1%80%D1%8B" title="Электр кыры – tatarisk" lang="tt" hreflang="tt" data-title="Электр кыры" data-language-autonym="Татарча / tatarça" data-language-local-name="tatarisk" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AA%E0%B8%99%E0%B8%B2%E0%B8%A1%E0%B9%84%E0%B8%9F%E0%B8%9F%E0%B9%89%E0%B8%B2" title="สนามไฟฟ้า – thai" lang="th" hreflang="th" data-title="สนามไฟฟ้า" data-language-autonym="ไทย" data-language-local-name="thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Elektrik_alan%C4%B1" title="Elektrik alanı – tyrkisk" lang="tr" hreflang="tr" data-title="Elektrik alanı" data-language-autonym="Türkçe" data-language-local-name="tyrkisk" 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%B8%D1%87%D0%BD%D0%B5_%D0%BF%D0%BE%D0%BB%D0%B5" title="Електричне поле – ukrainsk" lang="uk" hreflang="uk" data-title="Електричне поле" data-language-autonym="Українська" data-language-local-name="ukrainsk" 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_%D9%85%DB%8C%D8%AF%D8%A7%D9%86" title="برقی میدان – urdu" lang="ur" hreflang="ur" data-title="برقی میدان" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90i%E1%BB%87n_tr%C6%B0%E1%BB%9Dng" title="Điện trường – vietnamesisk" lang="vi" hreflang="vi" data-title="Điện trường" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamesisk" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E9%9B%BB%E5%A0%B4" title="電場 – klassisk kinesisk" lang="lzh" hreflang="lzh" data-title="電場" data-language-autonym="文言" data-language-local-name="klassisk kinesisk" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Palibot_han_elektriko" title="Palibot han elektriko – waray-waray" lang="war" hreflang="war" data-title="Palibot han elektriko" data-language-autonym="Winaray" data-language-local-name="waray-waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wo mw-list-item"><a href="https://wo.wikipedia.org/wiki/Toolu_mb%C3%ABj" title="Toolu mbëj – wolof" lang="wo" hreflang="wo" data-title="Toolu mbëj" 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/%E7%94%B5%E5%9C%BA" title="电场 – wu" lang="wuu" hreflang="wuu" data-title="电场" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%9B%BB%E5%A0%B4" title="電場 – kantonesisk" lang="yue" hreflang="yue" data-title="電場" data-language-autonym="粵語" data-language-local-name="kantonesisk" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%9B%BB%E5%A0%B4" title="電場 – kinesisk" lang="zh" hreflang="zh" data-title="電場" data-language-autonym="中文" data-language-local-name="kinesisk" 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/Q46221#sitelinks-wikipedia" title="Rediger lenker til artikkelen på andre språk" class="wbc-editpage">Rediger lenker</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Navnerom"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li 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src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/VFPt_charges_plus_minus_thumb.svg/250px-VFPt_charges_plus_minus_thumb.svg.png" decoding="async" width="250" height="188" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/VFPt_charges_plus_minus_thumb.svg/375px-VFPt_charges_plus_minus_thumb.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/VFPt_charges_plus_minus_thumb.svg/500px-VFPt_charges_plus_minus_thumb.svg.png 2x" data-file-width="220" data-file-height="165" /></a><figcaption>Elektriske feltlinjer fra en positiv (rød) og en like stor, men negativ (blå) ladning.</figcaption></figure> <p><b>Elektrisk felt</b> (også kalt <b>elektrisk feltstyrke</b>) gir <a href="/wiki/Kraft" title="Kraft">kraften</a> som virker i hvert punkt i rommet på en <a href="/wiki/Elektrisk_ladning" title="Elektrisk ladning">elektrisk ladet</a> partikkel som der befinner seg i ro. Det er et <a href="/wiki/Vektorfelt" title="Vektorfelt">vektorfelt</a> som skapes av andre elektriske ladninger ifølge <a href="/wiki/Coulombs_lov" title="Coulombs lov">Coulombs lov</a> eller fra et magnetfelt som varierer med tiden og beskrevet ved <a href="/wiki/Faradays_induksjonslov" title="Faradays induksjonslov">Faradays induksjonslov</a>. </p><p>Det elektriske feltet er vanligvis betegnet ved vektorsymbolet <b>E</b>. En ladning <i>q</i>  som er i ro i dette <a href="/wiki/Felt_(fysikk)" title="Felt (fysikk)">feltet</a>, vil bli påvirket av kraften <span class="nowrap"><b>F</b> = <i>q</i> <b>E</b></span>. I <a href="/wiki/SI-systemet" title="SI-systemet">SI-systemet</a> er derfor enheten for elektrisk feltstyrke <a href="/wiki/Newton_(enhet)" title="Newton (enhet)">Newton</a> per <a href="/wiki/Coulomb" title="Coulomb">Coulomb</a> N/C som er lik <a href="/wiki/Volt" title="Volt">Volt</a> per <a href="/wiki/Meter" title="Meter">meter</a>, V/m. Begrepet elektrisk felt ble først introdusert av <a href="/wiki/Michael_Faraday" title="Michael Faraday">Michael Faraday</a> på midten av 1800-tallet. </p><p>Varierer det elektriske feltet med tiden, vil det generere <a href="/wiki/Maxwells_forskyvningsstr%C3%B8m" title="Maxwells forskyvningsstrøm">Maxwells forskyvningsstrøm</a> som igjen skaper et magnetisk felt. Dette samspillet omtales som et <a href="/wiki/Elektromagnetisk_felt" title="Elektromagnetisk felt">elektromagnetisk felt</a> og er beskrevet av <a href="/wiki/Maxwells_ligninger" class="mw-redirect" title="Maxwells ligninger">Maxwells ligninger</a> som danner grunnlaget for <a href="/wiki/Elektromagnetisme" title="Elektromagnetisme">elektromagnetisk teori</a>. <a href="/wiki/Spesiell_relativitetsteori" class="mw-redirect" title="Spesiell relativitetsteori">Spesiell relativitetsteori</a> forener det elektriske og magnetiske vektorfeltet i <a href="/wiki/Kovariant_relativitetsteori#Kovariant_bevegelsesligning" title="Kovariant relativitetsteori">Faradays felttensor</a>. <a href="/wiki/Lys" title="Lys">Lys</a> og annen <a href="/wiki/Elektromagnetisk_str%C3%A5ling" title="Elektromagnetisk stråling">elektromagnetisk stråling</a> er <a href="/wiki/B%C3%B8lge" title="Bølge">bølger</a> av disse feltene. I <a href="/wiki/Kvantemekanikk" title="Kvantemekanikk">kvantemekanikken</a> erstattes denne klassiske beskrivelsen med <a href="/wiki/Kvanteelektrodynamikk" title="Kvanteelektrodynamikk">kvanteelektrodynamikk</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Karakteristiske_feltstyrker">Karakteristiske feltstyrker</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektrisk_felt&veaction=edit&section=1" title="Rediger avsnitt: Karakteristiske feltstyrker" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektrisk_felt&action=edit&section=1" title="Rediger kildekoden til seksjonen Karakteristiske feltstyrker"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>I dagliglivet er vi omgitt av spenninger med størrelser som typisk er mellom et par volt opp til noen hundre volt over avstander på noen meter. Derfor vil vi fra elektriske installasjoner omkring oss være utsatt for elektriske feltstyrker <i>E</i> = (0.1 - 100) V/m. Nær høyspentlinjer vil man kunne ha mer enn 1000 V/m = 1kV/m. Feltene foran gammeldagse fjernsynskjermer kunne være mye sterkere. Likedan er vi hele tiden bombardert med elektromagnetisk stråling fra <a href="/wiki/Radio" title="Radio">radioer</a> og <a href="/wiki/Mobiltelefon" title="Mobiltelefon">mobiltelefoner</a>. For at signalene skal være hørbare, må disse bølgene inneholde felltstyrker større enn 10<sup>-5</sup> V/m. </p><p>Fra Naturens side er vi i tillegg utsatt for et mer konstant elektrisk felt med omtrentlig størrelse 130 V/m nær <a href="/wiki/Jorden" title="Jorden">Jorden</a> på grunn av <a href="/wiki/Atmosf%C3%A6risk_elektrisitet" title="Atmosfærisk elektrisitet">atmosfærisk elektrisitet</a>. Men det er likevel lite i forhold til hva som kan oppstå som <a href="/wiki/Statisk_elektrisitet" title="Statisk elektrisitet">statisk elektrisitet</a> ved <a href="/wiki/Triboelektrisk_effekt" title="Triboelektrisk effekt">triboelektriske effekter</a>. Ved så høye feltstyrker som 10<sup>6</sup> V/m kan man få <a href="/wiki/Elektrisk_gjennomslag" title="Elektrisk gjennomslag">elektrisk gjennomslag</a> i luften når elektroner blir revet løs fra atomene. Det kan arte seg som gnister eller en lysbue.<sup id="cite_ref-YF_1-0" class="reference"><a href="#cite_note-YF-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>Enda sterkere felt finnes inne i <a href="/wiki/Atom" title="Atom">atomene</a>. Elektronet i et <a href="/wiki/Hydrogen" title="Hydrogen">hydrogen</a>-atom beveger seg i det elektriske feltet fra et <a href="/wiki/Proton" title="Proton">proton</a> i en avstand som er gitt ved en <a href="/wiki/Bohrs_atommodell" title="Bohrs atommodell">Bohr-radius</a>. Det gir typiske feltstyrker av størrelsesorden 10<sup>12</sup> V/m. Det er dette elektriske feltet som delvis er grunnen til <a href="/wiki/Finstruktur" title="Finstruktur">finstrukturen</a> til forskjellige <a href="/wiki/Spektrallinje" title="Spektrallinje">spektrallinjer</a> i atomenes <a href="/wiki/Emisjonsspekter" title="Emisjonsspekter">emisjonsspekter</a>. </p><p>Ved enda høyere feltstyrker vil partikler og deres <a href="/wiki/Antipartikkel" title="Antipartikkel">antipartikler</a> oppstå spontant fra <a href="/wiki/Vakuum" title="Vakuum">vakuum</a>.<sup id="cite_ref-IZ_2-0" class="reference"><a href="#cite_note-IZ-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Julian_Schwinger" class="mw-redirect" title="Julian Schwinger">Julian Schwinger</a> har vist ved bruk av <a href="/wiki/Kvanteelektrodynamikk" title="Kvanteelektrodynamikk">kvanteelektrodynamikk</a> at slik <a href="/wiki/Pardannelse" title="Pardannelse">pardannelse</a> av <a href="/wiki/Elektron" title="Elektron">elektron</a>-<a href="/wiki/Positron" title="Positron">positronpar</a> vil opptre for elektriske felt </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 E>E_{c}={\frac {m_{e}^{2}c^{3}}{e\hbar }}=1.3\times 10^{18}{\rm {V/m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>></mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msubsup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <mi>e</mi> <mi class="MJX-variant">ℏ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1.3</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>18</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">V</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">m</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E>E_{c}={\frac {m_{e}^{2}c^{3}}{e\hbar }}=1.3\times 10^{18}{\rm {V/m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e419f763d0de703a1c99bbb3d613773ccee9783" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:34.577ex; height:5.843ex;" alt="{\displaystyle E>E_{c}={\frac {m_{e}^{2}c^{3}}{e\hbar }}=1.3\times 10^{18}{\rm {V/m}}}"></span></dd></dl> <p>Her er <i>e </i> <a href="/wiki/Element%C3%A6rladning" title="Elementærladning">elektronets ladning</a>, <i>m<sub>e</sub></i>  dets <a href="/wiki/Masse" title="Masse">masse</a>, <i>c </i> er <a href="/wiki/Lyshastigheten" class="mw-redirect" title="Lyshastigheten">lyshastigheten</a> og <i>ħ = h</i>/2<i>π </i> er den reduserte <a href="/wiki/Plancks_konstant" title="Plancks konstant">Planck-konstanten</a>. Dette fenomenet gir løsningen på <a href="/w/index.php?title=Kleins_paradoks&action=edit&redlink=1" class="new" title="Kleins paradoks (ikke skrevet ennå)">Kleins paradoks</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Men det er ikke kjent hvor så sterke felt vil kunne opptre. Tilnærmet så høye felt kan finnes på overflaten av de aller tyngste <a href="/wiki/Atomkjerne" title="Atomkjerne">atomkjerner</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Elektrostatiske_felt">Elektrostatiske felt</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektrisk_felt&veaction=edit&section=2" title="Rediger avsnitt: Elektrostatiske felt" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektrisk_felt&action=edit&section=2" title="Rediger kildekoden til seksjonen Elektrostatiske felt"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:E_FieldOnePointCharge.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/E_FieldOnePointCharge.svg/250px-E_FieldOnePointCharge.svg.png" decoding="async" width="250" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f0/E_FieldOnePointCharge.svg/375px-E_FieldOnePointCharge.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f0/E_FieldOnePointCharge.svg/500px-E_FieldOnePointCharge.svg.png 2x" data-file-width="295" data-file-height="295" /></a><figcaption>Elektriske feltvektorer for en positiv punktladning peker utover og avtar i størrelse med kvadratet til avstanden fra ladningen.</figcaption></figure> <p>Når alle elektriske ladninger og strømmer i et system ikke forandrer seg med tiden, vil de resulterende feltene også være konstante eller «statiske». Alle elektriske felt kan da i utgangspunktet beregnes fra <a href="/wiki/Coulombs_lov" title="Coulombs lov">Coulombs lov</a> som danner grunnlaget for <a href="/wiki/Elektrostatikk" title="Elektrostatikk">elektrostatikken</a>. Den sier at en ladning <i>q </i> i vakuum vil skape et elektrisk felt i et punkt <b>r</b> = (<i>x,y,z</i>) som er gitt ved </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} )={q \over 4\pi \varepsilon _{0}r^{2}}{\hat {\mathbf {r} }}={q\,\mathbf {r} \over 4\pi \varepsilon _{0}|\mathbf {r} |^{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> <mi>q</mi> <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> </mrow> </mfrac> </mrow> <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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} (\mathbf {r} )={q \over 4\pi \varepsilon _{0}r^{2}}{\hat {\mathbf {r} }}={q\,\mathbf {r} \over 4\pi \varepsilon _{0}|\mathbf {r} |^{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e668676f01a79c2f1b759ad6fcc48abb413d12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.517ex; height:6.009ex;" alt="{\displaystyle \mathbf {E} (\mathbf {r} )={q \over 4\pi \varepsilon _{0}r^{2}}{\hat {\mathbf {r} }}={q\,\mathbf {r} \over 4\pi \varepsilon _{0}|\mathbf {r} |^{3}}}"></span></dd></dl> <p>når den befinner seg i origo og <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 {\hat {\mathbf {r} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {r} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9740464b71653e12932278ee944540be8caa5b96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.343ex;" alt="{\displaystyle {\hat {\mathbf {r} }}}"></span> = <b>r</b>/<i>r</i>  er en enhetsvektor i radiell retning med <i>r</i> = |<b>r</b>|. Her er 1/4<i>π ε</i><sub>0</sub>  <a href="/wiki/Coulombs_konstant" title="Coulombs konstant">Coulombs konstant</a> i <a href="/wiki/SI-systemet" title="SI-systemet">SI-systemet</a> som oftest brukes i dag. Feltvektorene peker derfor ut fra origo og med lengder som avtar med kvadratet av avstanden til ladningen. I stedet for å beskrive feltet med slike feltvektorer, er det mer vanlig å bruke <a href="/wiki/Feltlinje" title="Feltlinje">feltlinjer</a>. Det er kontinuerlige <a href="/wiki/Kurve" title="Kurve">kurver</a> som i hvert punkt har vektorfeltet som <a href="/wiki/Kurve#Buelengde_og_tangentvektor" title="Kurve">tangent</a>.<sup id="cite_ref-HR_4-0" class="reference"><a href="#cite_note-HR-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Når man har mange ladninger <i>q<sub>i</sub> </i> plassert i faste posisjoner <b>r</b><sub><i>i</i></sub>, blir det resulterende feltet i posisjon <b>r</b>  gitt med <a href="/wiki/Vektor_(matematikk)" title="Vektor (matematikk)">vektorsummen</a> av feltene fra hver av dem, </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} )=\sum _{i}{q_{i}(\mathbf {r} -\mathbf {r} _{i}) \over 4\pi \varepsilon _{0}|\mathbf {r} -\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> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</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> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} (\mathbf {r} )=\sum _{i}{q_{i}(\mathbf {r} -\mathbf {r} _{i}) \over 4\pi \varepsilon _{0}|\mathbf {r} -\mathbf {r} _{i}|^{3}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7780588962479a480a600ae1c1212d793baa77ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.816ex; height:6.843ex;" alt="{\displaystyle \mathbf {E} (\mathbf {r} )=\sum _{i}{q_{i}(\mathbf {r} -\mathbf {r} _{i}) \over 4\pi \varepsilon _{0}|\mathbf {r} -\mathbf {r} _{i}|^{3}}.}"></span></dd></dl> <p>Det elektriske feltet oppfyller derfor <a href="/wiki/Superposisjonsprinsippet" title="Superposisjonsprinsippet">superposisjonsprinsippet</a>. Ofte kan det benyttes til å forenkle beregning av feltet.<sup id="cite_ref-Griffiths_5-0" class="reference"><a href="#cite_note-Griffiths-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>I mange tilfeller kan ladningene som skaper elektriske felt, betraktes å være kontinuerlig fordelt. En slik fordeling i rommet vil ha en <a href="/wiki/Elektrisk_ladning" title="Elektrisk ladning">ladningstetthet</a> <i>ρ</i>  slik at ladningen i et lite volumelement rundt kildepunktet <b>r</b> ' kan skrives på den differensialle formen <i>dq</i> ' = <i>ρ</i>(<b>r</b> ')<i>d</i><sup> 3</sup><i>x</i> '. Ved integrasjon over hele ladningsfordelingen finnes det totale feltet, </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} )=\int d^{3}x'{\rho (\mathbf {r} ')(\mathbf {r} -\mathbf {r} ') \over 4\pi \varepsilon _{0}|\mathbf {r} -\mathbf {r} '|^{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> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>x</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} (\mathbf {r} )=\int d^{3}x'{\rho (\mathbf {r} ')(\mathbf {r} -\mathbf {r} ') \over 4\pi \varepsilon _{0}|\mathbf {r} -\mathbf {r} '|^{3}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29cc6b539ce1eba84d257f546e0f6f2d3baffb65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.827ex; height:6.843ex;" alt="{\displaystyle \mathbf {E} (\mathbf {r} )=\int d^{3}x'{\rho (\mathbf {r} ')(\mathbf {r} -\mathbf {r} ') \over 4\pi \varepsilon _{0}|\mathbf {r} -\mathbf {r} '|^{3}}.}"></span></dd></dl> <p>På same måte kan man beregne feltet fra kontinuerlige fordelinger av elektrisk ladning på <a href="/wiki/Flate" title="Flate">flater</a>, langs <a href="/wiki/Kurve" title="Kurve">kurver</a> eller på <a href="/wiki/Linje" title="Linje">linjer</a>. Todimensjonal ladningsfordeling på en flate betegnes vanligvis med symbolet <i>σ</i>, mens langs en linje bruker man oftest <i>λ</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Elektrisk_fluks">Elektrisk fluks</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektrisk_felt&veaction=edit&section=3" title="Rediger avsnitt: Elektrisk fluks" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektrisk_felt&action=edit&section=3" title="Rediger kildekoden til seksjonen Elektrisk fluks"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Desto tettere de elektriske feltlinjene er i et område av rommet, desto sterkere er feltet. Et kvalitativt mål for dette kan man få ved å beregne hvor mange feltlinjer som går gjennom en fiktiv <a href="/wiki/Flate" title="Flate">flate</a> <i>S</i>  i denne delen av rommet. Denne størrelsen er gitt ved integralet </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}(S)=\int _{S}\mathbf {E} \cdot d\mathbf {S} }"> <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 stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <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> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{E}(S)=\int _{S}\mathbf {E} \cdot d\mathbf {S} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d915967a0dfd4e4f724897a775db595789bc9201" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.682ex; height:5.676ex;" alt="{\displaystyle \Phi _{E}(S)=\int _{S}\mathbf {E} \cdot d\mathbf {S} }"></span></dd></dl> <p>hvor <i>d</i> <b>S</b> = <i>dS</i> <b>n</b>  er et lite flateelement som har enhetsvektoren <b>n</b>  som normal. Verdien av integralet kalles den <b>elektriske fluksen</b> gjennom flaten og tilsvarer definisjonen av <a href="/wiki/Magnetisk_fluks" class="mw-redirect" title="Magnetisk fluks">magnetisk fluks</a>. Navnet kommer fra det analoge integralet i <a href="/wiki/Hydrodynamikk" title="Hydrodynamikk">hydrodynamikk</a> hvor det gir mengden av vann som strømmer gjennom en slik flate. Feltstyrken <b>E</b> kan derfor også kalles for «elektrisk flukstetthet». </p><p>I utgangspunktet sier <a href="/wiki/Coulombs_lov" title="Coulombs lov">Coulombs lov</a> hva det elektriske feltet er utenfor en punktladning. En mer generell formulering er <a href="/wiki/Gauss%27_lov" class="mw-redirect" title="Gauss' lov">Gauss' lov</a> som kan benyttes i mange andre sammenhenger. Den sier at den totale, elektriske fluksen gjennom enhver lukket flate er gitt ved den totale ladningen innenfor flaten. På integralform skrives den som </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 d\mathbf {S} ={Q \over \varepsilon _{0}}}"> <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> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi _{E}=\oint _{S}\mathbf {E} \cdot d\mathbf {S} ={Q \over \varepsilon _{0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e390c67a4f9bc65efc514e8390be0508208eb48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.446ex; height:5.843ex;" alt="{\displaystyle \Phi _{E}=\oint _{S}\mathbf {E} \cdot d\mathbf {S} ={Q \over \varepsilon _{0}}}"></span></dd></dl> <p>Den tenkte flaten <i>S</i>  kan legges hvor man måtte ønske og kalles en «Gauss-flate». Vanligvis velges den på en slik måte at integrasjonen blir enklest mulig. Det valget er i stor grad bestemt av symmetrien i problemet. Befinner det seg ingen ladninger innenfor flaten, sier loven derfor at like mye fluks må gå inn i flaten som det går ut av den. </p> <div class="mw-heading mw-heading3"><h3 id="Eksempel:_Ladet_kule">Eksempel: Ladet kule</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektrisk_felt&veaction=edit&section=4" title="Rediger avsnitt: Eksempel: Ladet kule" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektrisk_felt&action=edit&section=4" title="Rediger kildekoden til seksjonen Eksempel: Ladet kule"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:GaussSphere.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/GaussSphere.svg/250px-GaussSphere.svg.png" decoding="async" width="250" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e9/GaussSphere.svg/375px-GaussSphere.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e9/GaussSphere.svg/500px-GaussSphere.svg.png 2x" data-file-width="290" data-file-height="290" /></a><figcaption>Sfærisk ladningsfordeling med radius <i>R</i> samt to Gauss-flater med radius <i>r</i> innenfor og <i>r' </i> er utenfor.</figcaption></figure> <p>For å beregne feltet inne i en kule med radius <i>R </i> og konstant ladningstetthet <i>ρ</i>, kan man legge en sfærisk Gauss-flate med radius <i>r </i> og med sentrum i kulens midtpunkt. På grunn av symmetrien må feltet være rettet i radiell retning. Fluksen gjennom Gauss-flaten er dermed <i>E</i>⋅4<i>π r</i><sup>2</sup>, mens den totale ladning innenfor flaten er (4/3)⋅<i>ρπ r</i><sup>3</sup>. Gauss' lov gir da at feltet inni kulen er </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 E={\rho r \over 3\varepsilon _{0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <mi>r</mi> </mrow> <mrow> <mn>3</mn> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={\rho r \over 3\varepsilon _{0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e387e1464a8c44f1bd231388b86025bf406651d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:9.01ex; height:5.176ex;" alt="{\displaystyle E={\rho r \over 3\varepsilon _{0}}}"></span></dd></dl> <p>Det er null i kulens sentrum <i>r</i> = 0  og har verdien <i>ρR</i>/3<i>ε</i><sub>0</sub> på dens overflate <i>r = R</i>. Hvis Gauss-flaten har radius <span class="nowrap"><i>r' > R</i></span>, omslutter den hele ladningen <span class="nowrap"><i>Q</i> = (4/3)⋅<i>ρπ R</i><sup>3</sup> </span> til kulen. I dette området utenfor kulen avtar feltet derfor som </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 E={Q \over 4\pi \varepsilon _{0}r^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>Q</mi> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={Q \over 4\pi \varepsilon _{0}r^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81c15c5b1a64ce8de70dc26fb612349be29c14a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.445ex; height:6.009ex;" alt="{\displaystyle E={Q \over 4\pi \varepsilon _{0}r^{2}}}"></span></dd></dl> <p>i overensstemmelse med Coulombs lov. Setter man her <i>r = R</i>, finner man igjen verdien av feltet som resulterte fra beregningen inni kula. Det elektriske feltet er kontinuerlig på overflaten da det ikke finnes ladninger der. </p><p>At det elektriske feltet utenfor en sfærisk ladningsfordeling med totalladning <i>Q </i> er det samme som feltet fra en punktladning <i>Q </i> i kulens sentrum, er et eksempel på <a href="/wiki/Newtons_skallteorem" title="Newtons skallteorem">Newtons skallteorem</a>. Grunnen er at Coulombs lov har samme matematiske form som <a href="/wiki/Newtons_gravitasjonslov" title="Newtons gravitasjonslov">Newtons gravitasjonslov</a>. En annen konsekvens av dette teoremet er da at det elektriske feltet innenfor et kuleskall er null. Utenfor er feltet det samme som om kuleskallets totalladning var plassert i dets sentrum. Dette er også i overensstemmelse med Gauss' teorem. </p> <div class="mw-heading mw-heading2"><h2 id="Differensiell_form_av_Gauss'_lov"><span id="Differensiell_form_av_Gauss.27_lov"></span>Differensiell form av Gauss' lov</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektrisk_felt&veaction=edit&section=5" title="Rediger avsnitt: Differensiell form av Gauss' lov" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektrisk_felt&action=edit&section=5" title="Rediger kildekoden til seksjonen Differensiell form av Gauss' lov"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Det lukkete flateintegralet som inngår på venstre side av Gauss' lov, kan omskrives ved bruk av <a href="/wiki/Divergensteoremet" class="mw-redirect" title="Divergensteoremet">divergensteoremet</a>. Det sier at </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 \oint _{S}d\mathbf {S} \cdot \mathbf {E} (\mathbf {x} )=\int dV\,{\boldsymbol {\nabla }}\cdot \mathbf {E} (\mathbf {x} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <mi>d</mi> <mi>V</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{S}d\mathbf {S} \cdot \mathbf {E} (\mathbf {x} )=\int dV\,{\boldsymbol {\nabla }}\cdot \mathbf {E} (\mathbf {x} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df11c2fe63aa9fec84289485c0d1c388ed087e10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.282ex; height:5.676ex;" alt="{\displaystyle \oint _{S}d\mathbf {S} \cdot \mathbf {E} (\mathbf {x} )=\int dV\,{\boldsymbol {\nabla }}\cdot \mathbf {E} (\mathbf {x} )}"></span></dd></dl> <p>hvor <i>V </i> er volumet som flaten <i>S </i> omslutter. Men den totale ladningen <i>Q </i> som denne flaten omslutter, kan også skrives som et volumintegral over den elektriske ladningstettheten <i>ρ</i>(<b>x</b>), </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 Q=\int dV\,\rho (\mathbf {x} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mo>∫</mo> <mi>d</mi> <mi>V</mi> <mspace width="thinmathspace" /> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=\int dV\,\rho (\mathbf {x} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/940fa5030eecb49986a842b1c70c198b20cede0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.33ex; height:5.676ex;" alt="{\displaystyle Q=\int dV\,\rho (\mathbf {x} )}"></span></dd></dl> <p>Begge sider av loven er dermed gitt ved integral over det samme volumet. Siden dette kan velges fritt, må de to integrandene være de samme. Det betyr at </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 {\boldsymbol {\nabla }}\cdot \mathbf {E} (\mathbf {x} )={1 \over \varepsilon _{0}}\rho (\mathbf {x} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {E} (\mathbf {x} )={1 \over \varepsilon _{0}}\rho (\mathbf {x} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb7a7cc3f88003887b1671261fa06e0995d5324f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.377ex; height:5.509ex;" alt="{\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {E} (\mathbf {x} )={1 \over \varepsilon _{0}}\rho (\mathbf {x} )}"></span></dd></dl> <p>som må gjelde i hvert punkt i rommet. Dette er den «lokale» eller differensielle formen av Gauss' lov. Den sier at hver feltlinje starter på en positiv ladning og ender på en tilsvarende negativ ladning.<sup id="cite_ref-HR_4-1" class="reference"><a href="#cite_note-HR-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Når feltet er radielt rettet i tre dimensjoner, må divergensen regnes ut i <a href="/wiki/Kulekoordinater" title="Kulekoordinater">kulekoordinater</a>. Da kan <a href="/wiki/Divergens" title="Divergens">divergensen</a> skrives som </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 {\boldsymbol {\nabla }}\cdot \mathbf {E} ={1 \over r^{2}}{\partial \over \partial r}(r^{2}E)={\partial E \over \partial r}+{2 \over r}E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>E</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>E</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>r</mi> </mfrac> </mrow> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {E} ={1 \over r^{2}}{\partial \over \partial r}(r^{2}E)={\partial E \over \partial r}+{2 \over r}E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a1f63bf0fbf6379bceeea3e63eeb48ad7b1413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:34.233ex; height:5.676ex;" alt="{\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {E} ={1 \over r^{2}}{\partial \over \partial r}(r^{2}E)={\partial E \over \partial r}+{2 \over r}E}"></span></dd></dl> <p>For den ladete kulen med <i>E</i> = <i>ρr</i>/3<i>ε</i><sub>0</sub>, blir da <b>∇ </b>⋅ <b>E</b> = <i>ρ</i>/<i>ε</i><sub>0</sub>  som forventet. Samme formel gir også at denne divergensen er null utenfor kulen hvor feltet varierer som <span class="nowrap">1/<i>r</i><sup> 2</sup></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Punktladning">Punktladning</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektrisk_felt&veaction=edit&section=6" title="Rediger avsnitt: Punktladning" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektrisk_felt&action=edit&section=6" title="Rediger kildekoden til seksjonen Punktladning"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En matematisk <a href="/wiki/Kontinuitetsligning#Punktpartikler" title="Kontinuitetsligning">punktladning</a> i punktet <b>r'</b> har en ladningstetthet som er uendelig stor i dette punktet og null utenfor slik at integralet over hele ladningsfordelingen gir en totalladning <i>q</i>. Dette tilsvarer definisjonen av <a href="/wiki/Diracs_deltafunksjon" title="Diracs deltafunksjon">Diracs deltafunksjon</a> som tillater å skrive en slik ladningstetthet som </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 \rho (\mathbf {r} )=q\delta (\mathbf {r} -\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> <mo>=</mo> <mi>q</mi> <mi>δ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>′</mo> </msup> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (\mathbf {r} )=q\delta (\mathbf {r} -\mathbf {r'} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5da0b626a4aec277fa19e6a95e8b3d2ec3cb6085" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.869ex; height:3.009ex;" alt="{\displaystyle \rho (\mathbf {r} )=q\delta (\mathbf {r} -\mathbf {r'} )}"></span></dd></dl> <p>Da det elektriske feltet fra denne ladningen er gitt ved Coulomb-feltet, gir Gauss' lov den matematiske sammenhengen </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 {\boldsymbol {\nabla }}\cdot {\mathbf {r} -\mathbf {r'} \over |\mathbf {r} -\mathbf {r'} |^{3}}=4\pi \delta (\mathbf {r} -\mathbf {r'} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>′</mo> </msup> </mrow> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>′</mo> </msup> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>4</mn> <mi>π</mi> <mi>δ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>′</mo> </msup> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nabla }}\cdot {\mathbf {r} -\mathbf {r'} \over |\mathbf {r} -\mathbf {r'} |^{3}}=4\pi \delta (\mathbf {r} -\mathbf {r'} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e117cd6841954d90f36f1d16ffbb364059f93068" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.999ex; height:6.676ex;" alt="{\displaystyle {\boldsymbol {\nabla }}\cdot {\mathbf {r} -\mathbf {r'} \over |\mathbf {r} -\mathbf {r'} |^{3}}=4\pi \delta (\mathbf {r} -\mathbf {r'} )}"></span></dd></dl> <p>Den opptrer i mange sammenhenger både i <a href="/wiki/Elektrostatikk" title="Elektrostatikk">elektrostatikken</a> og <a href="/wiki/Magnetostatikk" title="Magnetostatikk">magnetostatikken</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Ladninger_på_linjer_og_plan"><span id="Ladninger_p.C3.A5_linjer_og_plan"></span>Ladninger på linjer og plan</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektrisk_felt&veaction=edit&section=7" title="Rediger avsnitt: Ladninger på linjer og plan" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektrisk_felt&action=edit&section=7" title="Rediger kildekoden til seksjonen Ladninger på linjer og plan"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>I allminnelighet er det meget vanskelig å beregne det elektriske feltet nøyaktig for en generell ladningsfordeling. Men noen ganger lar det seg gjøre mer direkte. Det gjelder spesielt når problemet har en eller annen <a href="/wiki/Symmetri" title="Symmetri">symmetri</a> som forenkler oppgaven. For eksempel, for en sfærisk symmetrisk ladningsfordeling vil alle retninger være av samme betydning. Derav kan man med en gang si at det elektriske feltet må være radielt og ha samme størrelse i punkt med samme avstand fra fordelingens sentrum. Dette gjelder opplagt for en punktladning, men det gjelder like godt for en kuleformet, utstrakt fordeling hvor ladningen er uniformt fordelt. </p> <div class="mw-heading mw-heading3"><h3 id="Linjeladning">Linjeladning</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektrisk_felt&veaction=edit&section=8" title="Rediger avsnitt: Linjeladning" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektrisk_felt&action=edit&section=8" title="Rediger kildekoden til seksjonen Linjeladning"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Fil:Linjeladning.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Linjeladning.jpg/280px-Linjeladning.jpg" decoding="async" width="280" height="280" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Linjeladning.jpg/420px-Linjeladning.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/b/ba/Linjeladning.jpg 2x" data-file-width="535" data-file-height="535" /></a><figcaption>Feltet fra to motsatt plasserte linjeelement langs <i>y</i>-aksen gir et resulterende felt langs <i>x</i>-aksen.</figcaption></figure> <p>Feltet fra uendelig lang, rett linje med en konstant, lineær ladningstetthet <i>λ</i>, kan ikke variere med posisjonen langs linjen da alt må forbli uforandret ved en slik forflytning. Det kan derfor bare variere med avstanden fra linjen. </p><p>Plasseres linjeladningen langs <i>y</i>-aksen, vil et lite intervall <i>dy</i>  på denne i avstand +<i>y</i> fra origo ha ladningen <i>λdy</i>. Feltet fra dette intervallet har da størrelsen <span class="nowrap"><i>λdy</i>/4<i>πε</i><sub>0</sub><i>R</i><sup> 2</sup> </span> hvor <i>R</i> angir avstanden til feltpunktet. Har dette avstanden <i>x</i> fra <i>y</i>-aksen, er denne gitte ved <a href="/wiki/Pytagoras%E2%80%99_l%C3%A6resetning" title="Pytagoras’ læresetning">Pytagoras’ læresetning</a> som <span class="nowrap"><i>R</i><sup> 2</sup> = <i>x</i><sup> 2</sup> + <i>y</i><sup> 2</sup></span>. Da det også vil være et tilsvarende bidrag fra et tilsvarende intervall i punktet -<i>y</i>, vil summen av disse to gi en feltvektor normalt på <i>y</i>-aksen. Denne komponenten finnes ved å multiplisere hvert av disse bidragene med <i>x/R</i>. Det totale feltet i avstanden <i>x</i> fra linjen finnes nå ved å integrere opp alle disse bidragene fra hele linjen, </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 E={\lambda \over 4\pi \varepsilon _{0}}\int _{-\infty }^{\infty }{xdy \over (x^{2}+y^{2})^{3/2}}={\lambda \over 2\pi \varepsilon _{0}x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</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> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mi>d</mi> <mi>y</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mrow> <mn>2</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>x</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={\lambda \over 4\pi \varepsilon _{0}}\int _{-\infty }^{\infty }{xdy \over (x^{2}+y^{2})^{3/2}}={\lambda \over 2\pi \varepsilon _{0}x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53b6dfc7cfc8ee81c90efbd4cfaefcd097f8b291" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:38.242ex; height:6.343ex;" alt="{\displaystyle E={\lambda \over 4\pi \varepsilon _{0}}\int _{-\infty }^{\infty }{xdy \over (x^{2}+y^{2})^{3/2}}={\lambda \over 2\pi \varepsilon _{0}x}}"></span></dd></dl> <p>En mer direkte vei å finne dette resultatet følger fra <a href="/wiki/Gauss%27_lov" class="mw-redirect" title="Gauss' lov">Gauss' lov</a> ved å omslutte linjeladningen med en sylinderformet Gauss-flate. Ut fra symmetri er de elektriske feltvektorene rettet radielt utover og står derfor normalt på sylinderflaten. Har denne sylinderen radius <i>r</i> og høyde <i>h</i>, vil fluksen gjennom den være <span class="nowrap">2<i>πrhE</i></span>. Da flaten omslutter en total ladning <i>λh</i>, får man med en gang at <span class="nowrap"><i>E</i> = <i>λ</i>/2<i>πε</i><sub>0</sub><i>r</i> </span> hvor <i>r </i> igjen er avstanden til linjeladningen. </p><p>Dette resulatet er også omtrentlig riktig for en endelig lang linjeladning så lenge som man betrakter felt i nærheten av midten til linjen. Ved dens endepunkt er feltet ikke rettet radielt utover.<sup id="cite_ref-RM_6-0" class="reference"><a href="#cite_note-RM-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Ladet_plan">Ladet plan</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektrisk_felt&veaction=edit&section=9" title="Rediger avsnitt: Ladet plan" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektrisk_felt&action=edit&section=9" title="Rediger kildekoden til seksjonen Ladet plan"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For et plan med konstant flateladning <i>σ</i>  vil de elektiske feltvektorene stå normalt på planet, igjen ut fra symmetri. Hvis det ligger i <i>xy</i>-planet, kan man betrakte denne todimensjonale ladningsfordelingen som bestående av en uendelig rekke med parallelle linjeladninger som er parallelle med <i>y</i>-aksen, hver med en linjeladning <span class="nowrap"><i>λ = σdx</i></span>. Feltet fra en slik linje i avstand <i>x</i> fra <i>y</i>-aksen er fra det foregående gitt som <i>σdx</i>/2<i>πε</i><sub>0</sub><i>R</i> hvor nå <i>R</i> angir avstanden fra denne linjen til feltpunktet i avstand <i>z</i>  over origo, det vil si <span class="nowrap"><i>R</i><sup> 2</sup> = <i>x</i><sup> 2</sup> + <i>z</i><sup> 2</sup></span>. Men da det bare er komponentene normalt på planet som bidrar, må dette bidraget multipliseres med <i>z/R</i>. Det totale feltet i avstand <i>z</i> fra planet er derfor </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 E={\sigma \over 2\pi \varepsilon _{0}}\int _{-\infty }^{\infty }{zdx \over x^{2}+z^{2}}={\sigma \over 2\varepsilon _{0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>σ</mi> <mrow> <mn>2</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>z</mi> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>σ</mi> <mrow> <mn>2</mn> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={\sigma \over 2\pi \varepsilon _{0}}\int _{-\infty }^{\infty }{zdx \over x^{2}+z^{2}}={\sigma \over 2\varepsilon _{0}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e453307d77a323ba7312dd2d91e088c36891ea0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.003ex; height:6.009ex;" alt="{\displaystyle E={\sigma \over 2\pi \varepsilon _{0}}\int _{-\infty }^{\infty }{zdx \over x^{2}+z^{2}}={\sigma \over 2\varepsilon _{0}}}"></span></dd></dl> <p>At resultatet er det samme uavhengig av avstanden til planet, har mange viktige konsekvenser. Det følger også fra Gauss' lov ved å legge igjen en sylinderformet flate som står normalt på det og omslutter en liten del med areal <i>A</i>. Dette er også arealet til toppen og bunnen av sylinderen hvor fluksen går ut på begge sider av planet. I alt forlater derfor en fluks 2<i>EA</i> sylinderen som skyldes ladningen <i>σA</i>  innenfor. Dermed finner man igjen at <span class="nowrap"><i>E = σ</i>/2<i>ε</i><sub>0</sub></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Ladet_ring">Ladet ring</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektrisk_felt&veaction=edit&section=10" title="Rediger avsnitt: Ladet ring" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektrisk_felt&action=edit&section=10" title="Rediger kildekoden til seksjonen Ladet ring"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Fil:Ringladning.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Ringladning.jpg/280px-Ringladning.jpg" decoding="async" width="280" height="281" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Ringladning.jpg/420px-Ringladning.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/2/22/Ringladning.jpg 2x" data-file-width="451" data-file-height="452" /></a><figcaption>Det elektriske feltet i et punkt <i>P</i> på ringens akse fra linjeelementet <i>ds</i>.</figcaption></figure> <p>En annen, symmetrisk linjeladning er en ring med radius <i>a</i> og konstant ladningstetthet <i>λ</i>. Den totale ladningen på ringen er derfor <span class="nowrap"><i>q</i> = 2<i>πaλ</i></span>. Med en gang kan man da si at feltet er null ringens sentrum. Det skyldes at feltet fra ladningen i et punkt på ringen blir nøyaktig opphevet av feltet fra det diametralt motsatte punktet. </p><p>På samme måte har punktene langs en linje normalt på ringen og gjennom dens sentrum en spesielt symmetrisk plassering. Dette er ringens akse. Der vil bidragene fra to diametralt plasserte punkt på ringen gi et resulterende felt som peker langs <i>z</i>-aksen hvis ringen ligger i <i>xy</i>-planet. Et lite stykke av ringen med lengde <i>ds </i> vil gi et felt med størrelse <i>λds</i>/4<i>πε</i><sub>0</sub><i>R</i><sup> 2</sup>  hvor <i>R</i> angir avstanden fra denne ladningen til feltpunktet <i>P </i> i avstand <i>z</i>  over ringen. Her er nå <span class="nowrap"><i>R</i><sup> 2</sup> = <i>a</i><sup> 2</sup> + <i>z</i><sup> 2</sup></span>. Når vi summerer opp alle disse bidragene fra punkter langs ringen, er det bare brøkdelen <i>z/R</i>  av dette feltet langs ringens akse som vil bidra. I andre retninger vil komponentene kansellere ut. Da integralet av <i>ds </i> er omkretsen 2<i>π a</i>, blir dermed feltet langs <i>z</i>-aksen </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 E={qz \over 4\pi \varepsilon _{0}(a^{2}+z^{2})^{3/2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <mi>z</mi> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={qz \over 4\pi \varepsilon _{0}(a^{2}+z^{2})^{3/2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8303338168940ed9a38708a148fcc10e2626972" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:22.119ex; height:5.843ex;" alt="{\displaystyle E={qz \over 4\pi \varepsilon _{0}(a^{2}+z^{2})^{3/2}}}"></span></dd></dl> <p>I ringens sentrum er <i>z</i> = 0. Feltet er der null som forventet. Langt fra ringen hvor <i>z</i> >> <i>a</i>, er reduseres dette resultatet til <span class="nowrap"><i>E = q</i>/4<i>πε</i><sub>0</sub><i>z</i><sup>2</sup></span>. I sin slik posisjon ser ringen ut som en enkel punktladning i avstand <i>z</i>  og feltet er ganske enkelt gitt ved Coulombs lov.<sup id="cite_ref-YF_1-1" class="reference"><a href="#cite_note-YF-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Ladet_disk">Ladet disk</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektrisk_felt&veaction=edit&section=11" title="Rediger avsnitt: Ladet disk" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektrisk_felt&action=edit&section=11" title="Rediger kildekoden til seksjonen Ladet disk"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dette resultatet for en ladet ring, gjør det mulig også finne feltet på aksen til en ladet disk med konstant flateladningstetthet <i>σ</i>. Man betrakter da den som satt sammen av konsentriske ringer med variabel radius <i>r</i>. En slik ring med tykkelse <i>dr</i> har ladningen <i>dq</i> = 2<i>πrdr</i>. Har disken radius <i>R</i>, finner man ved integrasjon over alle ringene at feltet på aksen i avstand <i>z </i> blir </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 E={\sigma z \over 4\pi \varepsilon _{0}}\int _{-\infty }^{\infty }{2\pi rdr \over (r^{2}+z^{2})^{3/2}}={\sigma \over 2\varepsilon _{0}}\left(1-{z \over {\sqrt {z^{2}+R^{2}}}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>σ</mi> <mi>z</mi> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>r</mi> <mi>d</mi> <mi>r</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>σ</mi> <mrow> <mn>2</mn> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={\sigma z \over 4\pi \varepsilon _{0}}\int _{-\infty }^{\infty }{2\pi rdr \over (r^{2}+z^{2})^{3/2}}={\sigma \over 2\varepsilon _{0}}\left(1-{z \over {\sqrt {z^{2}+R^{2}}}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e14fdbb67edeba8a4991e224c1f8fcdf62f5949" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:54.264ex; height:7.509ex;" alt="{\displaystyle E={\sigma z \over 4\pi \varepsilon _{0}}\int _{-\infty }^{\infty }{2\pi rdr \over (r^{2}+z^{2})^{3/2}}={\sigma \over 2\varepsilon _{0}}\left(1-{z \over {\sqrt {z^{2}+R^{2}}}}\right)}"></span></dd></dl> <p>I grensen <i>R</i> → ∞ går disken over til å bli et uendelig plan hvor feltstyrken utenfor igjen sees å anta den konstante verdien <span class="nowrap"><i>E = σ</i>/2<i>ε</i><sub>0</sub></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Ledere_og_dielektrika">Ledere og dielektrika</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektrisk_felt&veaction=edit&section=12" title="Rediger avsnitt: Ledere og dielektrika" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektrisk_felt&action=edit&section=12" title="Rediger kildekoden til seksjonen Ledere og dielektrika"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Feltet utenfor en uniformt ladet plate er det samme uansett hva den er laget av. I et <a href="/wiki/Dielektrisk_materiale" title="Dielektrisk materiale">dielektrisk materiale</a> som er en <a href="/wiki/Isolator" class="mw-redirect" title="Isolator">isolator</a> kan ikke ladninger bevege seg fritt omkring, men kan plasseres i faste posisjoner. Har platen et areal <i>A</i>, vil en total ladning <i>Q </i> da kunne plasseres uniform inne i den slik at den har flatetettheten <span class="nowrap"><i>σ = Q/A</i> </span>. Feltet på hver side av platen har da den konstante verdien <span class="nowrap"><i>E = σ</i>/2<i>ε</i><sub>0</sub></span> = <i>Q</i>/2<i>Aε</i><sub>0</sub>. Derimot varierer feltet inni platen med verdien <span class="nowrap"><i>E</i> = 0 </span> i midten og er lineært voksende ut til overflaten hvor det tar den konstante verdien. </p><p>Er platen derimot en <a href="/wiki/Elektrisk_leder" title="Elektrisk leder">elektrisk leder</a>, vil ladningen <i>Q </i> fordele seg på overflaten av lederen slik at feltet inne i platen blir nøyaktig lik null. Hver av overflatene har da ladningstettheten <span class="nowrap"><i>σ' = Q/2A</i>  </span> og skaper feltene <i>E' </i> = <i>σ' </i>/2<i>ε</i><sub>0</sub>  som peker ut fra lederen og inn i metallet. I metallet virker disse to delfeltene i motsatt retning slik at der blir totaltfeltet <span class="nowrap"><i>E</i> = 0</span>  som det skal være. Men utenfor lederen adderer de seg opp slik at der blir feltet <span class="nowrap"><i>E</i> = 2<i>E' </i> = <i>σ' </i>/<i>ε</i><sub>0</sub> = <i>Q</i>/2<i>Aε</i><sub>0</sub></span>. </p><p>Dette resultatet kan også forstås med en sylindrisk Gauss-flate med topp og bunn parallelle til metalloverflaten med bunnen innfor og toppen utenfor. Da går det fluks bare ut gjennom toppflaten da feltet gjennom bunnen inni i lederen er null. I <a href="/wiki/Elektrostatikk" title="Elektrostatikk">elektrostatikken</a> er denne egenskapen til det elektriske feltet nær ledere av stor betydning. Denne oppførselen av det elektriske feltet forklarer også virkningen til et <a href="/wiki/Faradays_bur" title="Faradays bur">Faraday-bur</a>.<sup id="cite_ref-YF_1-2" class="reference"><a href="#cite_note-YF-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Elektrisk_potensial">Elektrisk potensial</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektrisk_felt&veaction=edit&section=13" title="Rediger avsnitt: Elektrisk potensial" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektrisk_felt&action=edit&section=13" title="Rediger kildekoden til seksjonen Elektrisk potensial"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Det elektriske feltet kan utledes fra et <a href="/wiki/Elektrisk_potensial" title="Elektrisk potensial">elektrisk potensial</a>. Ofte blir det betegnet som <i>U</i>, men det er også vanlig å angi det som <i>V</i>. I mer teoretiske arbeid blir også <i>Φ</i>  benyttet. </p><p>En enkel måte å vise hvordan det oppstår, er å gjøre bruk av resultatet </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 {\boldsymbol {\nabla }}{1 \over |\mathbf {r} -\mathbf {r} '|}=-{\mathbf {r} -\mathbf {r} ' \over |\mathbf {r} -\mathbf {r} '|^{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nabla }}{1 \over |\mathbf {r} -\mathbf {r} '|}=-{\mathbf {r} -\mathbf {r} ' \over |\mathbf {r} -\mathbf {r} '|^{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a0ec4e6f736ba712bcdc7ba7e6c8d95c0778a0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.905ex; height:6.676ex;" alt="{\displaystyle {\boldsymbol {\nabla }}{1 \over |\mathbf {r} -\mathbf {r} '|}=-{\mathbf {r} -\mathbf {r} ' \over |\mathbf {r} -\mathbf {r} '|^{3}}}"></span></dd></dl> <p>som kommer frem ved direkte <a href="/wiki/Derivasjon" title="Derivasjon">derivasjon</a>.<sup id="cite_ref-Griffiths_5-1" class="reference"><a href="#cite_note-Griffiths-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Feltet fra en kontinuerlig ladningsfordeling i rommet kan derfor skrives som <span class="nowrap"><b>E</b> = - <b>∇</b><i>V</i> </span> hvor det elektriske potensialet er </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 V(\mathbf {r} )=\int d^{3}x'{\rho (\mathbf {r} ') \over 4\pi \varepsilon _{0}|\mathbf {r} -\mathbf {r} '|}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>∫</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>x</mi> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(\mathbf {r} )=\int d^{3}x'{\rho (\mathbf {r} ') \over 4\pi \varepsilon _{0}|\mathbf {r} -\mathbf {r} '|}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/253105b328fe23179c5a664549bae2194e2e6b08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:27.803ex; height:6.509ex;" alt="{\displaystyle V(\mathbf {r} )=\int d^{3}x'{\rho (\mathbf {r} ') \over 4\pi \varepsilon _{0}|\mathbf {r} -\mathbf {r} '|}.}"></span></dd></dl> <p>En elektrisk ladning <i>q </i> i dette potensialet vil ha den <a href="/wiki/Potensiell_energi" title="Potensiell energi">potensielle energien</a> <i>U(<b>r</b>) = qV</i>(<b>r</b>). Denne varierer med posisjonen, noe som betyr at ladningen er utsatt for en kraft <span class="nowrap"><b>F</b> = -<b>∇</b> <i>U</i></span>. Dette er akkurat kraften <span class="nowrap"><b>F</b> = <i>q</i> <b>E</b> </span> fra det elektriske feltet. </p><p>Ved mange beregninger av elektriske felt er det ofte enklere å først regne ut potensialet som er en <a href="/wiki/Skalar" title="Skalar">skalar</a> størrelse. At feltet er gitt som <a href="/wiki/Gradient" title="Gradient">gradienten</a> av potensialet, betyr at det er <a href="/wiki/Potensiell_energi#Konservative_krefter" title="Potensiell energi">konservativt</a> og oppfyller <span class="nowrap"><b>∇</b> × <b>E</b> = 0</span>. Dette er <a href="/wiki/Maxwells_ligninger" class="mw-redirect" title="Maxwells ligninger">Maxwells tredje ligning</a> for det elektrostatiske feltet. </p><p>Gauss' lov <b>∇ </b>⋅ <b>E</b> = <i>ρ</i>/<i>ε</i><sub>0</sub>  uttrykt ved potensialet gir </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}V(\mathbf {x} )=-{1 \over \varepsilon _{0}}\rho (\mathbf {x} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>V</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}V(\mathbf {x} )=-{1 \over \varepsilon _{0}}\rho (\mathbf {x} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a391bf7d8f86cf669a9eb66100273c5df3f64f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.3ex; height:5.509ex;" alt="{\displaystyle \nabla ^{2}V(\mathbf {x} )=-{1 \over \varepsilon _{0}}\rho (\mathbf {x} )}"></span></dd></dl> <p>som er <a href="/wiki/Poissons_ligning" class="mw-redirect" title="Poissons ligning">Poissons ligning</a>. Den er en andreordens, <a href="/wiki/Partiell_differensialligning" class="mw-redirect" title="Partiell differensialligning">partiell differensialligning</a>. For en gitt ladningsfordeling kan den i prinsippet løses og gi potensialet overalt. Der det ikke finnes ladninger, er <i>ρ</i> = 0  og ligningen reduseres til <a href="/wiki/Laplace-ligning" class="mw-redirect" title="Laplace-ligning">Laplace-ligningen</a>. Kjenner man potensialet overalt, kan det elektriske feltet beregnes ved en enkel derivasjon. Dette er ofte mye enklere enn å beregne feltet direkte fra ladningene da potensialet er en <a href="/wiki/Skalar" title="Skalar">skalar</a> størrelse.<sup id="cite_ref-RM_6-1" class="reference"><a href="#cite_note-RM-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Kondensator">Kondensator</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektrisk_felt&veaction=edit&section=14" title="Rediger avsnitt: Kondensator" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektrisk_felt&action=edit&section=14" title="Rediger kildekoden til seksjonen Kondensator"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:PlattenkondensatorFeld.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/PlattenkondensatorFeld.svg/250px-PlattenkondensatorFeld.svg.png" decoding="async" width="250" height="219" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/PlattenkondensatorFeld.svg/375px-PlattenkondensatorFeld.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/PlattenkondensatorFeld.svg/500px-PlattenkondensatorFeld.svg.png 2x" data-file-width="400" data-file-height="350" /></a><figcaption>Elektriske feltlinjer i en oppladet plate-kondensator.</figcaption></figure> <p>En elektrisk <a href="/wiki/Kondensator_(elektrisk)" title="Kondensator (elektrisk)">kondensator</a> kan lagre elektrisk ladning og er en videreføring av den opprinnelige <a href="/wiki/Leidnerflaske" title="Leidnerflaske">Leidnerflasken</a>. Den enkleste utgaven er en «platekondensator» som består av to parallelle, ledende plater som holdes i en viss avstand <i>d </i> fra hverandre. Fra en ytre spenningskilde kan disse lades opp slik at de bærer motsatte ladninger ±<i>Q</i>. Hvis hver plate antas å ha arealet <i>A</i>  og deres avstand <i>d </i> er tilstrekkelig liten, er det elektriske feltet fra hver av dem det samme som om de var uendelig store. Feltvektorene vil da overalt stå normalt på platene og ha størrelse <i>Q</i>/2<i>Aε</i><sub>0</sub> fra hver av dem når man antar at det er luft eller vakum mellom platene. Hvis det er et annet, isolerende materiale mellom platene enn luft, må <i>ε</i><sub>0</sub>  erstattes med materialets <a href="/wiki/Dielektrisitetskonstant" class="mw-redirect" title="Dielektrisitetskonstant">dielektrisitetskonstant</a> <i>ε</i>.<sup id="cite_ref-HR_4-2" class="reference"><a href="#cite_note-HR-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Utenfor platene ville de to feltene kansellere hverandre, mens mellom platene peker de i samme retning og gir totalfeltet <span class="nowrap"><i>E</i> = <i>Q</i>/<i>Aε</i><sub>0</sub></span>. Dette er konstant slik at den <a href="/wiki/Elektrisk_spenning" title="Elektrisk spenning">elektriske spenningen</a> eller potensialet mellom platene er <span class="nowrap"><i>V = Ed</i>.</span> Ladningen på platene kan da uttrykkes ved spenningen som <span class="nowrap"><i>Q = CV</i> </span> hvor </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 C=\varepsilon _{0}{A \over d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mi>d</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=\varepsilon _{0}{A \over d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df92b2043c1c2601399aee22329fa2e0e638cfb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.582ex; height:5.509ex;" alt="{\displaystyle C=\varepsilon _{0}{A \over d}}"></span></dd></dl> <p>er konsensatorens <a href="/wiki/Kapasitans" title="Kapasitans">kapasitans</a>. Den sier hvor mye ladning den kan lagre under en gitt spenning. </p> <div class="mw-heading mw-heading3"><h3 id="Elektrisk_feltenergi">Elektrisk feltenergi</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektrisk_felt&veaction=edit&section=15" title="Rediger avsnitt: Elektrisk feltenergi" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektrisk_felt&action=edit&section=15" title="Rediger kildekoden til seksjonen Elektrisk feltenergi"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mellom de to platene virker det en kraft som trekker dem mot hverandre og som vanligvis blir nøytralisert på mekanisk vis. Denne elektriske kraften er et uttrykk for at kondensatoren inneholder en elektrisk energi som ligger lagret i feltet mellom platene. Denne energien er tilført via arbeidet som må utføres for å lade kondensatoren opp. Man tenker seg at man starter med en uladet kondensator med null spenning. Så gjennomfører man en serie med flytninger av små ladninger <i>dq </i> fra den ene platene til den andre til man til slutt har ladningene ±<i>Q</i>  på hver av dem. Før denne tilstanden er nådd, vil de ha ladningene ±<i>q</i>  med spenningen <i>V = q/C</i>. Flytter man så en ny ladning <i>dq</i>, utføres et lite arbeid slik at energien til kondensatoren øker med <span class="nowrap"><i>dU = V dq</i></span>. Totalenergien ved full oppladning blir dermed </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_{E}={1 \over C}\int _{0}^{Q}qdq={Q^{2} \over 2C}={1 \over 2}QV={1 \over 2}CV^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>C</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> </mrow> </msubsup> <mi>q</mi> <mi>d</mi> <mi>q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>C</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>Q</mi> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>C</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{E}={1 \over C}\int _{0}^{Q}qdq={Q^{2} \over 2C}={1 \over 2}QV={1 \over 2}CV^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/775c17c02a7b7e2e6a17ead556f0859093497e14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:42.322ex; height:6.343ex;" alt="{\displaystyle U_{E}={1 \over C}\int _{0}^{Q}qdq={Q^{2} \over 2C}={1 \over 2}QV={1 \over 2}CV^{2}}"></span></dd></dl> <p>Uttrykt ved feltet kan dette skrives som </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_{E}={1 \over 2}\varepsilon _{0}AdE^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>=</mo> <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> <mi>A</mi> <mi>d</mi> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{E}={1 \over 2}\varepsilon _{0}AdE^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/422de022de6866755636a1b5fb6f270acea6d272" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.117ex; height:5.176ex;" alt="{\displaystyle U_{E}={1 \over 2}\varepsilon _{0}AdE^{2}}"></span></dd></dl> <p>Men her er <i>Ad </i> volumet av rommet mellom platene, det vil si volumet til det elektriske feltet. Det kan derfor tilskrives en «elektrisk feltenergitetthet» <i>U/Ad </i> som er </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_{E}={1 \over 2}\varepsilon _{0}E^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>=</mo> <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">{\displaystyle u_{E}={1 \over 2}\varepsilon _{0}E^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17c9b373f6b085a032d7b387a617c4c7b979f0a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.9ex; height:5.176ex;" alt="{\displaystyle u_{E}={1 \over 2}\varepsilon _{0}E^{2}}"></span></dd></dl> <p>Her er denne konstant mellom platene. Men en mer generell utledning i <a href="/wiki/Elektrostatikk" title="Elektrostatikk">elektrostatikken</a> viser at dette er det generelle uttrykket for den elektriske energitettheten også for et felt som varierer i tid og rom. Et tilsvarende resultat gjelder for energitettheten i det <a href="/wiki/Magnetisk_felt" class="mw-redirect" title="Magnetisk felt">magnetiske feltet</a>.<sup id="cite_ref-HLL_7-0" class="reference"><a href="#cite_note-HLL-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Dipolfeltet">Dipolfeltet</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektrisk_felt&veaction=edit&section=16" title="Rediger avsnitt: Dipolfeltet" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektrisk_felt&action=edit&section=16" title="Rediger kildekoden til seksjonen Dipolfeltet"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Fil:EfieldTwoOppositePointCharges.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/EfieldTwoOppositePointCharges.svg/280px-EfieldTwoOppositePointCharges.svg.png" decoding="async" width="280" height="198" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/EfieldTwoOppositePointCharges.svg/420px-EfieldTwoOppositePointCharges.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/EfieldTwoOppositePointCharges.svg/560px-EfieldTwoOppositePointCharges.svg.png 2x" data-file-width="619" data-file-height="438" /></a><figcaption>Illustrasjon av det elektriske feltet rundt en positiv (rød) og en negativ (grønn) ladning.</figcaption></figure> <p>Det elektrisk feltet fra en positiv og en like stor, men negativ ladning, kalles et «dipolfelt» i grensen hvor avstanden mellom dem blir tilstrekkelig liten. Betrakter man en Gauss-flate som omslutter begge ladningene, er da totalladningen innenfor flaten null. Like mye fluks må derfor gå ut av flaten som går inn i den. Derfor vil det elektriske feltet i stor avstand fra ladningene avta raskere enn 1/<i>r</i><sup>2</sup>. </p><p>For å beregne feltet fra <a href="/wiki/Dipol" title="Dipol">dipolen</a> antar man at de er skapt av to punktladninger <i>q</i> og <i>-q</i> som er separert med avstandsvektoren <b>d</b> som går fra den negative til den positive ladningen. Plasseres disse symmetrisk om origo, vil feltet i et punkt <b>r</b> består da av Coulomb-feltet fra den positive ladningen på stedet <b>d</b>/2  pluss feltet fra den negative ladningen i posisjon -<b>d</b>/2, det vil si </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} )={q \over 4\pi \varepsilon _{0}}\left({\mathbf {r} -\mathbf {d} /2 \over |\mathbf {r} -\mathbf {d} /2|^{3}}-{\mathbf {r} +\mathbf {d} /2 \over |\mathbf {r} +\mathbf {d} /2|^{3}}\right)}"> <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> <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> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} (\mathbf {r} )={q \over 4\pi \varepsilon _{0}}\left({\mathbf {r} -\mathbf {d} /2 \over |\mathbf {r} -\mathbf {d} /2|^{3}}-{\mathbf {r} +\mathbf {d} /2 \over |\mathbf {r} +\mathbf {d} /2|^{3}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38ef33134010f1ff670d0256927927b571cb1d7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.017ex; height:7.509ex;" alt="{\displaystyle \mathbf {E} (\mathbf {r} )={q \over 4\pi \varepsilon _{0}}\left({\mathbf {r} -\mathbf {d} /2 \over |\mathbf {r} -\mathbf {d} /2|^{3}}-{\mathbf {r} +\mathbf {d} /2 \over |\mathbf {r} +\mathbf {d} /2|^{3}}\right)}"></span></dd></dl> <p>I dipolgrensen hvor avstanden mellom ladningene <i>d</i> = |<b>d</b>| << |<b>r</b>| = <i>r</i>, kan man skrive </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 {r} \pm \mathbf {d} /2|^{3}=r^{2}\pm \mathbf {r} \cdot \mathbf {d} +d^{2}/4)^{3/2}=r(r^{2}\pm {3 \over 2}\mathbf {r} \cdot \mathbf {d} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mo>+</mo> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>r</mi> <mo stretchy="false">(</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>±</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathbf {r} \pm \mathbf {d} /2|^{3}=r^{2}\pm \mathbf {r} \cdot \mathbf {d} +d^{2}/4)^{3/2}=r(r^{2}\pm {3 \over 2}\mathbf {r} \cdot \mathbf {d} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc96a38d99e9a26d6ede5ce1d8d77fe3397413b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:50.614ex; height:5.176ex;" alt="{\displaystyle |\mathbf {r} \pm \mathbf {d} /2|^{3}=r^{2}\pm \mathbf {r} \cdot \mathbf {d} +d^{2}/4)^{3/2}=r(r^{2}\pm {3 \over 2}\mathbf {r} \cdot \mathbf {d} )}"></span></dd></dl> <p>når man ser bort fra høyere ordens ledd. Innsatt gir dette totalfeltet fra dipolen </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}r^{3}}{\Big [}3(\mathbf {p} \cdot {\hat {\mathbf {r} }}){\hat {\mathbf {r} }}-\mathbf {p} {\Big ]}}"> <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> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">[</mo> </mrow> </mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <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> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} (\mathbf {r} )={1 \over 4\pi \varepsilon _{0}r^{3}}{\Big [}3(\mathbf {p} \cdot {\hat {\mathbf {r} }}){\hat {\mathbf {r} }}-\mathbf {p} {\Big ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0be5155438607523822f5c11d578f3a6a9d119d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.32ex; height:5.843ex;" alt="{\displaystyle \mathbf {E} (\mathbf {r} )={1 \over 4\pi \varepsilon _{0}r^{3}}{\Big [}3(\mathbf {p} \cdot {\hat {\mathbf {r} }}){\hat {\mathbf {r} }}-\mathbf {p} {\Big ]}}"></span></dd></dl> <p>uttrykt ved enhetsvektoren <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 {\hat {\mathbf {r} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {r} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9740464b71653e12932278ee944540be8caa5b96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.343ex;" alt="{\displaystyle {\hat {\mathbf {r} }}}"></span> = <b>r</b>/<i>r</i> og det <b>elektriske dipolmomentet</b> <b>p</b> = <i>q</i> <b>d</b>  for de to ladningene. Det er en vektor med retning fra den negative til den positive ladningen. Dipolfeltet er symmetrisk om denne retningen. </p><p>Som et eksempel på bruk av dette resultatet, kan man betrakte en dipol i origo som ligger langs den positive <i>x</i>-aksen. Da blir feltet lenger ut til høyre på denne aksen <span class="nowrap"><b>E</b> = 2<i>p</i> <b>e</b><sub><i>x</i></sub>/4<i>πε</i><sub>0</sub><i>x</i><sup> 3</sup></span>, mens det i et punkt på <i>y</i>-aksen er <span class="nowrap"><b>E</b> = <i>-p</i> <b>e</b><sub><i>x</i></sub>/4<i>πε</i><sub>0</sub><i>y</i><sup> 3</sup></span>. </p><p>I stedet for å beregne komponentene til dipolfeltet i <a href="/wiki/Kartesisk_koordinatsystem" title="Kartesisk koordinatsystem">kartesiske koordinater</a>, kan man benytte <a href="/wiki/Polarkoordinatsystem" title="Polarkoordinatsystem">polarkoordinater</a> (<i>r,θ</i>). Feltvektoren har da de to komponentene<sup id="cite_ref-Griffiths_5-2" class="reference"><a href="#cite_note-Griffiths-5"><span class="cite-bracket">[</span>5<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 E_{r}={2p \over 4\pi \varepsilon _{0}r^{3}}\cos \theta ,\;\;\;E_{\theta }={p \over 4\pi \varepsilon _{0}r^{3}}\sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>p</mi> </mrow> <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>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{r}={2p \over 4\pi \varepsilon _{0}r^{3}}\cos \theta ,\;\;\;E_{\theta }={p \over 4\pi \varepsilon _{0}r^{3}}\sin \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52ca85dd374e8eb2e7390029d40810de11d31def" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.412ex; height:6.009ex;" alt="{\displaystyle E_{r}={2p \over 4\pi \varepsilon _{0}r^{3}}\cos \theta ,\;\;\;E_{\theta }={p \over 4\pi \varepsilon _{0}r^{3}}\sin \theta }"></span></dd></dl> <p>For punkter på <i>x</i>-aksen, henholdsvis <i>y</i>-aksen, er disse i overensstemmelse med resultatet for de kartesiske komponentene. </p> <div class="mw-heading mw-heading3"><h3 id="Dipolpotensialet">Dipolpotensialet</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektrisk_felt&veaction=edit&section=17" title="Rediger avsnitt: Dipolpotensialet" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektrisk_felt&action=edit&section=17" title="Rediger kildekoden til seksjonen Dipolpotensialet"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Fil:Electric_dipole_field_lines.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Electric_dipole_field_lines.svg/210px-Electric_dipole_field_lines.svg.png" decoding="async" width="210" height="202" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Electric_dipole_field_lines.svg/315px-Electric_dipole_field_lines.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/42/Electric_dipole_field_lines.svg/420px-Electric_dipole_field_lines.svg.png 2x" data-file-width="454" data-file-height="436" /></a><figcaption><a href="/wiki/Feltlinje" title="Feltlinje">Feltlinjer</a> fra en positiv og en negativ ladning som tilsammen utgjør en elektrisk dipol.</figcaption></figure> <p>Det elektriske potensialet til en dipol følger direkte fra det mindre kompliserte uttrykket </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 V(\mathbf {r} )={q \over 4\pi \varepsilon _{0}}\left({1 \over |\mathbf {r} -\mathbf {d} /2|}-{1 \over |\mathbf {r} +\mathbf {d} /2|}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</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> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(\mathbf {r} )={q \over 4\pi \varepsilon _{0}}\left({1 \over |\mathbf {r} -\mathbf {d} /2|}-{1 \over |\mathbf {r} +\mathbf {d} /2|}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc6569449677fe4c5facd8245cd5219c3b51c8b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.679ex; height:6.343ex;" alt="{\displaystyle V(\mathbf {r} )={q \over 4\pi \varepsilon _{0}}\left({1 \over |\mathbf {r} -\mathbf {d} /2|}-{1 \over |\mathbf {r} +\mathbf {d} /2|}\right)}"></span></dd></dl> <p>For store avstander kan dette forenkles på samme måte som for beregning av feltet. Resultatet kan skrives på den kompakte formen </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 V(\mathbf {r} )={\mathbf {p} \cdot \mathbf {r} \over 4\pi \varepsilon _{0}r^{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</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> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <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>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(\mathbf {r} )={\mathbf {p} \cdot \mathbf {r} \over 4\pi \varepsilon _{0}r^{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/638250f3a1e0b7b8a7f2ffaa35608d85455e9945" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.368ex; height:5.509ex;" alt="{\displaystyle V(\mathbf {r} )={\mathbf {p} \cdot \mathbf {r} \over 4\pi \varepsilon _{0}r^{3}}}"></span></dd></dl> <p>Dipolfeltet kan nå gjenfinnes fra <span class="nowrap"><b>E</b> = - <b>∇</b><i>V</i> </span>. Det står overalt <a href="/wiki/Vinkelrett" title="Vinkelrett">vinkelrett</a> på flater gitt ved en konstant verdi av potensialet, det vil si det som kalles <a href="/wiki/Ekvipotensialflate" title="Ekvipotensialflate">ekvipotensialflater</a>. </p><p>Ved bruk av polarkoordinater tar dipolpotensialet formen </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 V(\mathbf {r} )={p\cos \theta \over 4\pi \varepsilon _{0}r^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</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> <mrow> <mi>p</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mrow> <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> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(\mathbf {r} )={p\cos \theta \over 4\pi \varepsilon _{0}r^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6705eceda42cf7c63fa96a91c23398d8844cd8d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.368ex; height:6.009ex;" alt="{\displaystyle V(\mathbf {r} )={p\cos \theta \over 4\pi \varepsilon _{0}r^{2}}}"></span></dd></dl> <p>når dipolen ligger langs den positive <i>x</i>-aksen slik at <b>p</b>⋅<b>r</b> = <i>pr</i> cos<i>θ</i>. De to komponentene finnes nå lett fra <a href="/wiki/Gradient" title="Gradient">gradienten</a> i dette koordinatsystemet som gir <span class="nowrap"><i>E<sub>r</sub></i> = - ∂<i>V</i>/∂<i>r</i> </span> og <span class="nowrap"><i>E<sub>θ</sub></i> = - (1/<i>r</i>)∂<i>V</i>/∂<i>θ</i></span>. Det gir det samme resultatet for disse komponentene på en mer direkte måte. </p> <div class="mw-heading mw-heading3"><h3 id="Multipoler">Multipoler</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektrisk_felt&veaction=edit&section=18" title="Rediger avsnitt: Multipoler" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektrisk_felt&action=edit&section=18" title="Rediger kildekoden til seksjonen Multipoler"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Et dipolpotensial kan oppstå fra ladningsfordelinger med mer enn to partikler. Har man like mange positive som negative ladninger <i>q<sub>i</sub> </i> i posisjoner <b>r</b><sub><i>i</i></sub>, defineres deres dipolmoment som </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 {p} =\sum _{i}q_{i}\mathbf {r} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <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 {p} =\sum _{i}q_{i}\mathbf {r} _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ff57930d6f046db21bbf7a720c33ac1ecb52b98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.064ex; height:5.509ex;" alt="{\displaystyle \mathbf {p} =\sum _{i}q_{i}\mathbf {r} _{i}}"></span></dd></dl> <p>Tar man med høyere ordens ledd i utviklingen av potensialet fra hver av ladningene, vil det resulterende potensialet inneholde ledd som avtar med avstanden raskere enn dipolpotensialet. Dette kalles en <a href="/w/index.php?title=Multipolutvikling&action=edit&redlink=1" class="new" title="Multipolutvikling (ikke skrevet ennå)">multipolutvikling</a>. For eksempel kan to positive og to negative ladninger som alle ligger nær hverandre, betraktes som to dipoler som ligger nær hverandre. Avhengig av deres relative plassering, kan de da gi opphav til et «kvadrupolpotensial». Dette kan regnes ut fra dipolpotensialet på samme måte som at dette kan regnes ut fra Coulomb-potensialet for to motsatte ladninger som ligger nær hverandre. </p> <div class="mw-heading mw-heading2"><h2 id="Dipol_i_ytre_felt">Dipol i ytre felt</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektrisk_felt&veaction=edit&section=19" title="Rediger avsnitt: Dipol i ytre felt" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektrisk_felt&action=edit&section=19" title="Rediger kildekoden til seksjonen Dipol i ytre felt"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:ElektriskDipol.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/ElektriskDipol.jpg/250px-ElektriskDipol.jpg" decoding="async" width="250" height="263" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ef/ElektriskDipol.jpg/375px-ElektriskDipol.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/e/ef/ElektriskDipol.jpg 2x" data-file-width="407" data-file-height="428" /></a><figcaption>Elektrisk dipol <b>p</b>  danner vinkel <i>θ </i> med ytre felt <b>E</b>.</figcaption></figure> <p>Dipolpotensialet må ikke forveksles med den <a href="/wiki/Potensiell_energi" title="Potensiell energi">potensielle energien</a> for en dipol i et ytre, elektriske felt. Et slikt felt virker på hver av ladningene som utgjør dipolen. Resultatet er at kreftene som dette ytre feltet forårsaker, vil dreie dipolen slik at den peker mest mulig langs feltet. Den har da minimal, potensiell energi. </p><p>En dipole består av en ladning <i>q</i><sub>1</sub> = <i>q </i> i posisjon <b>r</b><sub>1</sub>  og en ladning <i>q</i><sub>2</sub> = -<i>q </i> i posisjon <b>r</b><sub>2</sub>  med gjensidig avstand <span class="nowrap"><b>d</b> = <b>r</b><sub>1</sub> - <b>r</b><sub>2</sub></span>. Befinner den seg i et ytre, elektrisk felt <b>E</b>(<b>r</b>), er den totale kraften som virker på den </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 {F} =q_{1}\mathbf {E} (\mathbf {r} _{1})+q_{2}\mathbf {E} (\mathbf {r} _{2})=q(\mathbf {E} (\mathbf {r} _{1})-\mathbf {E} (\mathbf {r} _{2}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =q_{1}\mathbf {E} (\mathbf {r} _{1})+q_{2}\mathbf {E} (\mathbf {r} _{2})=q(\mathbf {E} (\mathbf {r} _{1})-\mathbf {E} (\mathbf {r} _{2}))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cd7a0c1340191b34d41b89c57f3abe6f2918d72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.512ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} =q_{1}\mathbf {E} (\mathbf {r} _{1})+q_{2}\mathbf {E} (\mathbf {r} _{2})=q(\mathbf {E} (\mathbf {r} _{1})-\mathbf {E} (\mathbf {r} _{2}))}"></span></dd></dl> <p>Er feltet det samme overalt, vil differansen i parentesen være null og totalkraften er null. Kraften som virker på den ene ladningen blir opphevet av kraften på den andre ladningen. Tyngdepunktet til dipolen vil derfor ikke flytte seg i dette feltet. </p><p>Men disse to, motsatte rettete kreftene <b>F</b><sub>1</sub>  og <b>F</b><sub>2</sub>  utgjør et <a href="/wiki/Dreiemoment" title="Dreiemoment">dreiemoment</a>. Når dette beregnes om dipolens tyngdepunkt, har det størrelsen </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 {\begin{aligned}\mathbf {T} &={1 \over 2}\mathbf {d} \times \mathbf {F} _{1}-{1 \over 2}\mathbf {d} \times \mathbf {F} _{2}={1 \over 2}\mathbf {d} \times q\mathbf {E} +{1 \over 2}\mathbf {d} \times q\mathbf {E} \\&=\mathbf {p} \times \mathbf {E} \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mo>×</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mo>×</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {T} &={1 \over 2}\mathbf {d} \times \mathbf {F} _{1}-{1 \over 2}\mathbf {d} \times \mathbf {F} _{2}={1 \over 2}\mathbf {d} \times q\mathbf {E} +{1 \over 2}\mathbf {d} \times q\mathbf {E} \\&=\mathbf {p} \times \mathbf {E} \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1ebf956f993595d8aeec4063b15139d4b6a16bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:50.913ex; height:8.176ex;" alt="{\displaystyle {\begin{aligned}\mathbf {T} &={1 \over 2}\mathbf {d} \times \mathbf {F} _{1}-{1 \over 2}\mathbf {d} \times \mathbf {F} _{2}={1 \over 2}\mathbf {d} \times q\mathbf {E} +{1 \over 2}\mathbf {d} \times q\mathbf {E} \\&=\mathbf {p} \times \mathbf {E} \end{aligned}}}"></span></dd></dl> <p>hvor dipolmomentet er <b>p</b> = <i>q</i> <b>d</b>. Størrelsen til dreiemomentet kan skrives som <i>pE</i> sin<i>θ</i>  når man innfører vinkelen <i>θ</i>  mellom feltvektoren og dipolens retning. Det er derfor forskjellig fra null bortsett fra når dipolen har samme retning som det ytre feltet. </p> <div class="mw-heading mw-heading3"><h3 id="Potensiell_energi">Potensiell energi</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektrisk_felt&veaction=edit&section=20" title="Rediger avsnitt: Potensiell energi" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektrisk_felt&action=edit&section=20" title="Rediger kildekoden til seksjonen Potensiell energi"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dreiemomentet er et uttrykk for at dipolen kan utføre et <a href="/wiki/Arbeid_(fysikk)" title="Arbeid (fysikk)">arbeid</a> ved rotasjon i feltet. Den har derfor en <a href="/wiki/Potensiell_energi" title="Potensiell energi">potensiell energi</a> som kan relateres til potensialet <i>V </i> for det ytre feltet <span class="nowrap"><b>E</b> = - <b>∇</b> <i>V</i> </span> ut fra definisjonen som </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 {\begin{aligned}U&=q_{1}V(\mathbf {r} _{1})+q_{2}V(\mathbf {r} _{2})=q(V(\mathbf {r} _{1})-V(\mathbf {r} _{2}))=-q\int _{2}^{1}d\mathbf {r} \cdot \mathbf {E} =-q(\mathbf {r} _{1}-\mathbf {r} _{2})\cdot \mathbf {E} \\&=-q\,\mathbf {d} \cdot \mathbf {E} =-\mathbf {p} \cdot \mathbf {E} \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>U</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>V</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>V</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>−</mo> <mi>V</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>q</mi> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mo>−</mo> <mi>q</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi>q</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}U&=q_{1}V(\mathbf {r} _{1})+q_{2}V(\mathbf {r} _{2})=q(V(\mathbf {r} _{1})-V(\mathbf {r} _{2}))=-q\int _{2}^{1}d\mathbf {r} \cdot \mathbf {E} =-q(\mathbf {r} _{1}-\mathbf {r} _{2})\cdot \mathbf {E} \\&=-q\,\mathbf {d} \cdot \mathbf {E} =-\mathbf {p} \cdot \mathbf {E} \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e05a076fef1201609e47932346e1e8ce64331ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:78.882ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}U&=q_{1}V(\mathbf {r} _{1})+q_{2}V(\mathbf {r} _{2})=q(V(\mathbf {r} _{1})-V(\mathbf {r} _{2}))=-q\int _{2}^{1}d\mathbf {r} \cdot \mathbf {E} =-q(\mathbf {r} _{1}-\mathbf {r} _{2})\cdot \mathbf {E} \\&=-q\,\mathbf {d} \cdot \mathbf {E} =-\mathbf {p} \cdot \mathbf {E} \end{aligned}}}"></span></dd></dl> <p>Da størrelsen til denne energien er <i>-pE</i> cos<i>θ</i>, vil dipolen ha minst energi <i>-pE </i> når den peker langs det elektriske feltet slik at vinkelen <i>θ</i> = 0. Peker den i motsatt retning, vil derimot energien være maksimal med størrelse <i>pE </i>. Selv om dreiemomentet som da virker på den er null, vil den minste forstyrrelse fra denne retningen forårsake at dipolen prøver å rotere inn i likevektsstillingen med lavest energi. Disse to størrelsene er forbundet ved relasjonen <span class="nowrap"><i>T</i> = - ∂<i>U</i>/∂<i>θ</i> </span> som er typisk for sammenhengen mellom arbeid og energi. </p> <div class="mw-heading mw-heading3"><h3 id="Inhomogent_felt">Inhomogent felt</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektrisk_felt&veaction=edit&section=21" title="Rediger avsnitt: Inhomogent felt" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektrisk_felt&action=edit&section=21" title="Rediger kildekoden til seksjonen Inhomogent felt"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Når en dipole befinner seg i et felt som ikke er konstant, vil kreftene på de to ladningene ikke lenger oppheve hverandre. I stedet virker det en resulterende kraft på den som er </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 {\begin{aligned}\mathbf {F} &=q\mathbf {E} (\mathbf {r} +\mathbf {d} /2)-q\mathbf {E} (\mathbf {r} -\mathbf {d} /2)=q\mathbf {E} +q(\mathbf {d} /2\cdot {\boldsymbol {\nabla }})\mathbf {E} -q\mathbf {E} +q(\mathbf {d} /2\cdot {\boldsymbol {\nabla }})\mathbf {E} \\&=(\mathbf {p} \cdot {\boldsymbol {\nabla }})\mathbf {E} =p_{x}{\partial \mathbf {E} \over \partial x}+p_{y}{\partial \mathbf {E} \over \partial y}+p_{z}{\partial \mathbf {E} \over \partial z}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>q</mi> <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>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>q</mi> <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>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>+</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>−</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>+</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {F} &=q\mathbf {E} (\mathbf {r} +\mathbf {d} /2)-q\mathbf {E} (\mathbf {r} -\mathbf {d} /2)=q\mathbf {E} +q(\mathbf {d} /2\cdot {\boldsymbol {\nabla }})\mathbf {E} -q\mathbf {E} +q(\mathbf {d} /2\cdot {\boldsymbol {\nabla }})\mathbf {E} \\&=(\mathbf {p} \cdot {\boldsymbol {\nabla }})\mathbf {E} =p_{x}{\partial \mathbf {E} \over \partial x}+p_{y}{\partial \mathbf {E} \over \partial y}+p_{z}{\partial \mathbf {E} \over \partial z}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a510f235305a9f1d48affc518e89e6b018ade9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.699ex; margin-bottom: -0.306ex; width:75.126ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}\mathbf {F} &=q\mathbf {E} (\mathbf {r} +\mathbf {d} /2)-q\mathbf {E} (\mathbf {r} -\mathbf {d} /2)=q\mathbf {E} +q(\mathbf {d} /2\cdot {\boldsymbol {\nabla }})\mathbf {E} -q\mathbf {E} +q(\mathbf {d} /2\cdot {\boldsymbol {\nabla }})\mathbf {E} \\&=(\mathbf {p} \cdot {\boldsymbol {\nabla }})\mathbf {E} =p_{x}{\partial \mathbf {E} \over \partial x}+p_{y}{\partial \mathbf {E} \over \partial y}+p_{z}{\partial \mathbf {E} \over \partial z}\end{aligned}}}"></span></dd></dl> <p>Det ytre feltet vil nå påvirke dipolen med et dreiemoment som vil forsøke å rotere den samt en ekstra kraft som vil forsøke å flytte den i rommet. Retningen til denne forflytningen er i alminnelighet ikke i samme retning som det ytre feltet. </p><p>Egenskaper til <a href="/wiki/Magnetisk_moment" title="Magnetisk moment">magnetiske dipoler</a> i et ytre, <a href="/wiki/Magnetisk_felt" class="mw-redirect" title="Magnetisk felt">magnetisk felt</a> er beskrevet ved helt tilsvarende ligninger som for elektriske dipoler her. Grunnen er at en <a href="/wiki/Magnetisk_dipol" title="Magnetisk dipol">magnetisk dipol</a> kan tenkes oppbygd av to motsatt ladete, <a href="/wiki/Magnetisk_monopol" title="Magnetisk monopol">magnetiske monopoler</a>. Selv om disse monopolene ikke finnes, gir et slikt bilde likevel en riktig beskrivelse. Det sies å være en brukbar «modell» for en magnetisk dipol.<sup id="cite_ref-HR_4-3" class="reference"><a href="#cite_note-HR-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Elektrisk_polarisasjon">Elektrisk polarisasjon</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektrisk_felt&veaction=edit&section=22" title="Rediger avsnitt: Elektrisk polarisasjon" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektrisk_felt&action=edit&section=22" title="Rediger kildekoden til seksjonen Elektrisk polarisasjon"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fil:Capacitor_schematic_with_dielectric.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Capacitor_schematic_with_dielectric.svg/250px-Capacitor_schematic_with_dielectric.svg.png" decoding="async" width="250" height="275" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Capacitor_schematic_with_dielectric.svg/375px-Capacitor_schematic_with_dielectric.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Capacitor_schematic_with_dielectric.svg/500px-Capacitor_schematic_with_dielectric.svg.png 2x" data-file-width="430" data-file-height="473" /></a><figcaption>Polarisasjon av det dielektriske materialet i en platekondensator.</figcaption></figure> <p>Et <a href="/wiki/Dielektrisk_materiale" title="Dielektrisk materiale">dielektrisk materiale</a> er en <a href="/wiki/Elektrisk_isolator" title="Elektrisk isolator">elektrisk isolator</a> som består av <a href="/wiki/Atom" title="Atom">atomer</a> eller <a href="/wiki/Molekyl" title="Molekyl">molekyler</a> som kan «polariseres» under påvirkning av et elektrisk felt. Det betyr at de har et elektrisk dipolmoment som vil forsøke å rette seg inn langs det ytre feltet. Brukes et slikt materiale i en <a href="/wiki/Kondensator_(elektrisk)" title="Kondensator (elektrisk)">kondensator</a> i stedet for luft, vil det elektriske feltet mellom de metalliske platene reduseres. For eksempel, tett opp til den positive platen vil de negative endene til de nærmeste dipolene i materialet tiltrekkes mot platen og effektivt redusere ladningen på denne selv om dipolladningene fremdeles sitter fast i materialet. Og tilsvarende vil den negative platen få sin ladningstetthet effektivt redusert. </p><p>Før det dielektriske materialet blir satt inn mellom platene, har hver ladningstettheten <span class="nowrap"><i>σ = Q/A </i></span> slik at feltet mellom dem er <i>σ/ε</i><sub>0</sub>. Etter at materialet er innsatt, er den effektive ladningstettheten ved hver plate redusert som igjen resulterer i et mindre, elektrisk felt <span class="nowrap"><i>E</i> = (<i>σ - σ<sub>b</sub></i>)/<i>ε</i><sub>0</sub> </span> der reduksjonen skyldes dipolladningene som har vendt seg langs det påtrykte feltet. Denne induserte ladningstettheten kan uttrykkes ved <a href="/wiki/Dielektrisk_materiale#Elektrisk_polarisasjon" title="Dielektrisk materiale">polarisasjonen</a> <i>P </i> til materialet som <i>σ<sub>b</sub> = P</i>. Dimensjonen til denne nye størrelsen er derfor C/m<sup>2</sup>  og lik med dimensjonen til flateladningstettheten. </p><p>Det er vanlig å definere det elektriske feltet i et materiale som skyldes de frie ladningene ±<i>Q</i>  på platene som <a href="/wiki/Dielektrisk_materiale#Forskyvningsfeltet" title="Dielektrisk materiale">forskyvningsfeltet</a> <span class="nowrap"><i>D</i> = <i>σ</i></span>. Dette er tilstede sammen med det vanlige, elektriske feltet <i>E</i>  inni materialet. Sammenhengen mellom disse to i en platekondensator sees nå å være <span class="nowrap"><i>D</i> = <i>ε</i><sub>0</sub><i>E</i> + <i>P</i></span>. Men disse tre størrelsene kan ikke være uavhengige av hverandre da polarisasjonen i allminnelighet avhenger av det elektriske feltet.<sup id="cite_ref-HLL_7-1" class="reference"><a href="#cite_note-HLL-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Lineært_materiale"><span id="Line.C3.A6rt_materiale"></span>Lineært materiale</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektrisk_felt&veaction=edit&section=23" title="Rediger avsnitt: Lineært materiale" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektrisk_felt&action=edit&section=23" title="Rediger kildekoden til seksjonen Lineært materiale"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Vanligvis øker polarisasjonen proporsjonalt med det elektriske feltet. Man har da den lineære sammenhengen <span class="nowrap"><i>P</i> = <i>ε</i><sub>0</sub><i>χ<sub>e</sub>E</i></span>  hvor proporsjonalitetskonstanten <i>χ<sub>e</sub> </i> kalles den <a href="/wiki/Dielektrisk_materiale#Eleketrisk_polarisasjont" title="Dielektrisk materiale">elektriske susceptibiliteten</a> til materialet og er et dimensjonsløst tall som kan forventes å være postivt. Da kan det elektriske feltet i platekondensatoren skrives som <span class="nowrap"><i>E = E</i><sub>0</sub>/(1 + <i>χ<sub>e</sub></i>) </span> uttrykt ved feltet <i>E</i><sub>0</sub> = <i>σ/ε</i><sub>0</sub>  mellom flatene før materialet ble innsatt. Det er derfor blitt redusert med faktoren <span class="nowrap"><i>ε<sub>r</sub></i> = 1 + <i>χ<sub>e</sub></i> </span> som er den <a href="/wiki/Permittivitet" title="Permittivitet">relative permittivitet</a> til materialet. Da <span class="nowrap"><i>σ = Q/A </i></span>, betyr det at kapasiteten til platekondensateren dermed er øket til </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 C=\varepsilon _{0}(1+\chi _{e}){A \over d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mi>d</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=\varepsilon _{0}(1+\chi _{e}){A \over d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9000b6ed2c86068c5c0bbb5068acaf97b83b4b34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.847ex; height:5.509ex;" alt="{\displaystyle C=\varepsilon _{0}(1+\chi _{e}){A \over d}}"></span></dd></dl> <p>når avstanden mellom platene er <i>d</i>. En kondensator med et dielektrikum kan derfor lagre en mye større ladning for en gitt spenning. </p><p>Det elektriske feltet er en vektoriell størrelse. Derfor vil den elektriske polarisasjonen også være beskrevet ved et vektorfelt som i et lineært og homogent medium kan skrives som <span class="nowrap"><b>P</b> = <i>ε</i><sub>0</sub><i>χ<sub>e</sub></i> <b>E</b></span>. På overflaten av materialet vil det da formes en indusert ladningstetthet </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 \sigma _{b}=\mathbf {P} \cdot {\hat {\mathbf {n} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{b}=\mathbf {P} \cdot {\hat {\mathbf {n} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3455318c7540f3c8c38e9027d71197f57b073d2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.355ex; height:2.676ex;" alt="{\displaystyle \sigma _{b}=\mathbf {P} \cdot {\hat {\mathbf {n} }}}"></span></dd></dl> <p>hvor enhetsvektoren <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 {\hat {\mathbf {n} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {n} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aae87b164ba005e99b51066c46d1eacc7f56564a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.343ex;" alt="{\displaystyle {\hat {\mathbf {n} }}}"></span> står normalt på flaten. Men hvis polarisasjonen ikke er konstant, vil den også indusere en romlig ladningstetthet </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 \rho _{b}=-{\boldsymbol {\nabla }}\cdot \mathbf {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>b</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{b}=-{\boldsymbol {\nabla }}\cdot \mathbf {P} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c1a8591db59009b3943f5f4c6d9fefb1e468a07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.778ex; height:2.676ex;" alt="{\displaystyle \rho _{b}=-{\boldsymbol {\nabla }}\cdot \mathbf {P} }"></span></dd></dl> <p>som består av bundne dipolladninger inni materialet som er forskjøvet litt i forhold til hverandre. </p><p>Forskyvningsfeltet vil også være en vektor som nå blir </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 {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mo>=</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3061b05e80b3eee6d58c6aec1fc14e068bf0115c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.71ex; height:2.509ex;" alt="{\displaystyle \mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} }"></span></dd></dl> <p>Da det elektriske feltet i hvert punkt i materialet skyldes et samspill av all ladninger, fri og bundne, må det alltid oppfylle <a href="/wiki/Gauss%27_lov" class="mw-redirect" title="Gauss' lov">Gauss' lov</a>. Lokalt i materialet tar den da formen <b>∇</b>⋅<b>E</b> = (<i>ρ + ρ<sub>b</sub></i>)/<i>ε</i><sub>0</sub>. Det betyr at forskyvningsfeltet oppfyller Gauss' lov på den mer generelle formen </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 {\boldsymbol {\nabla }}\cdot \mathbf {D} =\rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mo>=</mo> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {D} =\rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73967793069f1f02f3f6f18078ddea3fdcb32f07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.255ex; height:2.676ex;" alt="{\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {D} =\rho }"></span></dd></dl> <p>hvor forskyvningsfeltet nå kan skrives som <span class="nowrap"><b>D</b> = <i>ε</i> <b>E</b></span> med <a href="/wiki/Permittivitet" title="Permittivitet">permittiviteten</a> <span class="nowrap"><i>ε</i> = <i>ε</i><sub>0</sub>(1 + <i>χ<sub>e</sub></i>)</span>. Dette feltet kan derfor tilskrives fri ladninger alene. Dette er <a href="/wiki/Maxwells_ligninger" class="mw-redirect" title="Maxwells ligninger">Maxwells første ligning</a> på sin mest generelle form. </p> <div class="mw-heading mw-heading2"><h2 id="Elektrodynamikk">Elektrodynamikk</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektrisk_felt&veaction=edit&section=24" title="Rediger avsnitt: Elektrodynamikk" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektrisk_felt&action=edit&section=24" title="Rediger kildekoden til seksjonen Elektrodynamikk"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Elektriske ladninger som beveger seg, skaper et <a href="/wiki/Magnetisk_felt" class="mw-redirect" title="Magnetisk felt">magnetisk felt</a> som kan beregnes fra <a href="/wiki/Biot-Savarts_lov" title="Biot-Savarts lov">Biot-Savarts lov</a>. Dette vil opptre sammen med det elektriske feltet fra ladningene. Den elektrostatiske beskrivelsen må da erstattes av en mer generell <a href="/wiki/Elektrodynamikk" title="Elektrodynamikk">elektrodynamikk</a> som omhandler både elektriske og magnetiske felt skapt av ladninger i bevegelse. Disse feltene vil i allminnelighet variere med tiden og må beskrives ved å ta alle <a href="/wiki/Maxwells_ligninger" class="mw-redirect" title="Maxwells ligninger">Maxwells ligninger</a> i bruk.<sup id="cite_ref-RM_6-2" class="reference"><a href="#cite_note-RM-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>På samme måte som det elektriske feltet i det statiske tilfellet er gitt som <a href="/wiki/Gradient" title="Gradient">gradienten</a> av et skalart potensial <i>V</i>, er det magnetisk feltet alltid gitt som <a href="/wiki/Curl" title="Curl">curl</a> av et <a href="/wiki/Magnetisk_felt" class="mw-redirect" title="Magnetisk felt">vektorpotensial</a> <b>A</b>, </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 {B} ={\boldsymbol {\nabla }}\times \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} ={\boldsymbol {\nabla }}\times \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/142c95e056cf9a3de7d48ae1fcc7c945189c51fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.086ex; height:2.176ex;" alt="{\displaystyle \mathbf {B} ={\boldsymbol {\nabla }}\times \mathbf {A} }"></span></dd></dl> <p>Dette gjelder også når feltene varierer med tiden. <a href="/wiki/Faradays_induksjonslov" title="Faradays induksjonslov">Faradays induksjonslov</a> <span class="nowrap"><b>∇</b> × <b>E</b> = - ∂<b>B</b>/∂<i>t</i> </span> kan da skrives som </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 {\boldsymbol {\nabla }}\times {\Big (}\mathbf {E} +{\partial \mathbf {A} \over \partial t}{\Big )}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nabla }}\times {\Big (}\mathbf {E} +{\partial \mathbf {A} \over \partial t}{\Big )}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f81a29e232c0ba54056cd9d99a40920cdf97cbc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.874ex; height:5.509ex;" alt="{\displaystyle {\boldsymbol {\nabla }}\times {\Big (}\mathbf {E} +{\partial \mathbf {A} \over \partial t}{\Big )}=0}"></span></dd></dl> <p>Innholdet i parentesen må derfor være en gradient for at curl av innholdet alltid skal gi null. Og den gradienten må stemme overens med uttrykket for det elektriske feltet i det statiske tilfellet. Derfor er dette feltet generelt gitt ved de to potensialene som </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} =-{\boldsymbol {\nabla }}V-{\partial \mathbf {A} \over \partial t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mi>V</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} =-{\boldsymbol {\nabla }}V-{\partial \mathbf {A} \over \partial t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aea343d437f18b7eee61750b677af0dcf8f76e75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.691ex; height:5.509ex;" alt="{\displaystyle \mathbf {E} =-{\boldsymbol {\nabla }}V-{\partial \mathbf {A} \over \partial t}}"></span></dd></dl> <p>Dette siste leddet er i mange sammenhenger helt avgjørende for beskrivelsen av <a href="/wiki/Elektromagnetisme" title="Elektromagnetisme">elektromagnetiske fenomen</a>. Det følger også direkte fra <a href="/wiki/Kovariant_relativitetsteori#Ladet_partikkel_i_elektromagnetisk_felt" title="Kovariant relativitetsteori">den spesielle relativitetsteorien</a>.<sup id="cite_ref-RPF_8-0" class="reference"><a href="#cite_note-RPF-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Kvanteelektrodynamikk">Kvanteelektrodynamikk</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektrisk_felt&veaction=edit&section=25" title="Rediger avsnitt: Kvanteelektrodynamikk" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektrisk_felt&action=edit&section=25" title="Rediger kildekoden til seksjonen Kvanteelektrodynamikk"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>I elektrodynamikk beskrives det elektriske feltet ved <a href="/wiki/Maxwells_ligninger" class="mw-redirect" title="Maxwells ligninger">Maxwells ligninger</a>, mens de ladete partiklene beskrives ved <a href="/wiki/Kovariant_relativitetsteori#Ladet_partikkel_i_elektromagnetisk_felt" title="Kovariant relativitetsteori">relativistisk mekanikk</a>. Når denne klassiske beskrivelsen utvides ved bruk av <a href="/wiki/Kvantemekanikk" title="Kvantemekanikk">kvantemekanikk</a>, fremkommer <a href="/wiki/Kvanteelektrodynamikk" title="Kvanteelektrodynamikk">kvanteelektrodynamikk</a>. Det elektromagnetiske feltet må da forstås som et <a href="/wiki/Kvantefeltteori" title="Kvantefeltteori">kvantefelt</a> hvor <a href="/wiki/Kvant" title="Kvant">kvantene</a> er <a href="/wiki/Foton" title="Foton">fotoner</a> som kan absorberes og emitteres som partikler. <a href="/wiki/Coulombs_lov" title="Coulombs lov">Coulomb-kraften</a> fremkommer ved utveksling av «virtuelle» fotoner mellom partikler som har <a href="/wiki/Elektrisk_ladning" title="Elektrisk ladning">elektrisk ladning</a>. Disse er vanligvis <a href="/wiki/Elektron" title="Elektron">elektroner</a> som også må beskrives ved bruk av kvantefeltteori. Det tilsvarende <a href="/wiki/Felt_(fysikk)" title="Felt (fysikk)">feltet</a> for elektronene er styrt av <a href="/wiki/Dirac-ligning" title="Dirac-ligning">Dirac-ligningen</a> som automatisk forklarer eksistensen av deres <a href="/wiki/Antipartikkel" title="Antipartikkel">antipartikler</a> som er <a href="/wiki/Positron" title="Positron">positroner</a>. </p><p>Det klassiske, tomme rommet eller <a href="/wiki/Vakuum" title="Vakuum">vakuum</a> vil i kvantelektrodynamikken bestå av en kokende <i>suppe</i> av virtuelle fotoner og virtuelle elektron-positron par. Avstanden mellom de to partiklene i et slikt par er gitt ved <a href="/wiki/Compton-effekt#Compton-bølgelengden" title="Compton-effekt">Compton-bølgelengden</a> <span class="nowrap"><i>λ<sub>e</sub> = ħ/m<sub>e</sub>c </i></span>. Hvis det er et elektrisk felt <i>E </i> i dette rommet, vil elektron-positron paret ha en elektrisk, potensiell energi <i>eEλ<sub>e</sub></i>. Er denne større enn hvileenergien 2<i>m<sub>e</sub>c</i><sup>2</sup>  til de to virtuelle partiklene, vil de opptre som reelle og dermed frigjøres i feltet. Dette forklarer eksistensen av det kritiske, elektriske feltet <i>E<sub>c</sub> </i> som ikke kan overstiges fordi vakuumet da brytes ned i <a href="/wiki/Pardannelse" title="Pardannelse">pardannelse</a> av elektroner og positroner.<sup id="cite_ref-IZ_2-1" class="reference"><a href="#cite_note-IZ-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Se_også"><span id="Se_ogs.C3.A5"></span>Se også</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektrisk_felt&veaction=edit&section=26" title="Rediger avsnitt: Se også" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektrisk_felt&action=edit&section=26" title="Rediger kildekoden til seksjonen Se også"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Coulombs_lov" title="Coulombs lov">Coulombs lov</a></li> <li><a href="/wiki/Elektrisk_potensial" title="Elektrisk potensial">Elektrisk potensial</a></li> <li><a href="/wiki/Elektrisk_spenning" title="Elektrisk spenning">Elektrisk spenning</a></li> <li><a href="/wiki/Elektrostatikk" title="Elektrostatikk">Elektrostatikk</a></li> <li><a href="/wiki/Elektrisk_str%C3%B8m" title="Elektrisk strøm">Elektrisk strøm</a></li> <li><a href="/wiki/Magnetisk_felt" class="mw-redirect" title="Magnetisk felt">Magnetisk felt</a></li> <li><a href="/wiki/Maxwells_likninger" title="Maxwells likninger">Maxwells likninger</a></li> <li><a href="/wiki/Elektromagnetisk_felt" title="Elektromagnetisk felt">Elektromagnetisk felt</a></li> <li><a href="/wiki/Elektromagnetisme" title="Elektromagnetisme">Elektromagnetisme</a></li> <li><a href="/wiki/Elektrodynamikk" title="Elektrodynamikk">Elektrodynamikk</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Referanser">Referanser</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektrisk_felt&veaction=edit&section=27" title="Rediger avsnitt: Referanser" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektrisk_felt&action=edit&section=27" title="Rediger kildekoden til seksjonen Referanser"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-YF-1"><b>^</b> <a href="#cite_ref-YF_1-0"><sup>a</sup></a> <a href="#cite_ref-YF_1-1"><sup>b</sup></a> <a href="#cite_ref-YF_1-2"><sup>c</sup></a> <span class="reference-text"> H.D. Young og R.A. Freedman, <i>University Physics</i>, Addison-Wesley, New York (2008). <a href="/wiki/Spesial:Bokkilder/9780321501301" class="internal mw-magiclink-isbn">ISBN 978-0-321-50130-1</a>.</span> </li> <li id="cite_note-IZ-2"><b>^</b> <a href="#cite_ref-IZ_2-0"><sup>a</sup></a> <a href="#cite_ref-IZ_2-1"><sup>b</sup></a> <span class="reference-text"> C. Itzykson and J-B. Zuber, <i>Quantum Field Theory</i>, McGraw-Hill, New York (1980). <a href="/wiki/Spesial:Bokkilder/0070320713" class="internal mw-magiclink-isbn">ISBN 0-07-032071-3</a>.</span> </li> <li id="cite_note-3"><b><a href="#cite_ref-3">^</a></b> <span class="reference-text"> A. Hansen and F. Ravndal, <i>Klein’s Paradox and its Resolution,</i> Physica Scripta <b>23</b>, 1030–1042 (1981).</span> </li> <li id="cite_note-HR-4"><b>^</b> <a href="#cite_ref-HR_4-0"><sup>a</sup></a> <a href="#cite_ref-HR_4-1"><sup>b</sup></a> <a href="#cite_ref-HR_4-2"><sup>c</sup></a> <a href="#cite_ref-HR_4-3"><sup>d</sup></a> <span class="reference-text"> D. Halliday and R. Resnick, <i>Fundamentals of Physics</i>, John Wiley & Sons, New York (1988). <a href="/wiki/Spesial:Bokkilder/047163736X" class="internal mw-magiclink-isbn">ISBN 0-471-63736-X</a>.</span> </li> <li id="cite_note-Griffiths-5"><b>^</b> <a href="#cite_ref-Griffiths_5-0"><sup>a</sup></a> <a href="#cite_ref-Griffiths_5-1"><sup>b</sup></a> <a href="#cite_ref-Griffiths_5-2"><sup>c</sup></a> <span class="reference-text"> D.J. Griffiths, <i>Introduction to Electrodynamics</i>, Prentice Hall, New Jersey (1999). <a href="/wiki/Spesial:Bokkilder/013805326X" class="internal mw-magiclink-isbn">ISBN 0-13-805326-X</a>.</span> </li> <li id="cite_note-RM-6"><b>^</b> <a href="#cite_ref-RM_6-0"><sup>a</sup></a> <a href="#cite_ref-RM_6-1"><sup>b</sup></a> <a href="#cite_ref-RM_6-2"><sup>c</sup></a> <span class="reference-text"> J.R. Reitz and F.J. Milford, <i>Foundations of Electromagnetic Theory</i>, Addison-Wesley Publishing Company, Reading (1960).</span> </li> <li id="cite_note-HLL-7"><b>^</b> <a href="#cite_ref-HLL_7-0"><sup>a</sup></a> <a href="#cite_ref-HLL_7-1"><sup>b</sup></a> <span class="reference-text">O. Hunderi, J.R. Lien og G. Løvhøiden, <i>Generell fysikk for universiteter og høgskoler</i>, Bind 2, Universitetsforlaget, Oslo (2001). <a href="/wiki/Spesial:Bokkilder/9788215000060" class="internal mw-magiclink-isbn">ISBN 978-82-1500-006-0</a>.</span> </li> <li id="cite_note-RPF-8"><b><a href="#cite_ref-RPF_8-0">^</a></b> <span class="reference-text"> R.P. Feynman, <a rel="nofollow" class="external text" href="http://www.feynmanlectures.caltech.edu/II_26.html"><i>The Feynman Lectures on Physics, Vol II</i></a>, Addison-Wesley, Longman. (1970). <a href="/wiki/Spesial:Bokkilder/9780201021158" class="internal mw-magiclink-isbn">ISBN 978-0-201-02115-8</a>.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Eksterne_lenker">Eksterne lenker</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektrisk_felt&veaction=edit&section=28" title="Rediger avsnitt: Eksterne lenker" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektrisk_felt&action=edit&section=28" title="Rediger kildekoden til seksjonen Eksterne lenker"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Johannes Skaar, NTNU, <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=WptLk0fU7KY&index=4&list=PLUHTGp7T4Zn_q2Ic5R3kOaB3xxXR2TlrQ">Forelesninger TFE4120: Elektromagnetisme</a>, Youtube (2017).</li> <li>Walter Levin, MIT, <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=JhV-GOS4y8g">Lectures on Electricity and Magnetism</a>, Youtube (2015).</li> <li>Hyperphysics, <a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elelin.html#c1">Electric Field of Line Charge</a>.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r23230704">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist 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Elektrisk felt – Wikipedia