Elektromagnetisk felt

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toppen)</div> </a> </li> <li id="toc-Maxwells_ligninger" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Maxwells_ligninger"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Maxwells ligninger</span> </div> </a> <button aria-controls="toc-Maxwells_ligninger-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 Maxwells ligninger</span> </button> <ul id="toc-Maxwells_ligninger-sublist" class="vector-toc-list"> <li id="toc-Elektromagnetiske_potensial" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Elektromagnetiske_potensial"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Elektromagnetiske potensial</span> </div> </a> <ul id="toc-Elektromagnetiske_potensial-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Bølgeligninger" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bølgeligninger"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Bølgeligninger</span> </div> </a> <button aria-controls="toc-Bølgeligninger-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 Bølgeligninger</span> </button> <ul id="toc-Bølgeligninger-sublist" class="vector-toc-list"> <li id="toc-Retarderte_løsninger" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Retarderte_løsninger"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Retarderte løsninger</span> </div> </a> <ul id="toc-Retarderte_løsninger-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Coulomb-gauge" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Coulomb-gauge"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Coulomb-gauge</span> </div> </a> <ul id="toc-Coulomb-gauge-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Poyntings_teorem" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Poyntings_teorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Poyntings teorem</span> </div> </a> <button aria-controls="toc-Poyntings_teorem-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 Poyntings teorem</span> </button> <ul id="toc-Poyntings_teorem-sublist" class="vector-toc-list"> <li id="toc-Poyntings_vektor" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Poyntings_vektor"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Poyntings vektor</span> </div> </a> <ul id="toc-Poyntings_vektor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Elektromagnetisk_impulstetthet" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Elektromagnetisk_impulstetthet"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Elektromagnetisk impulstetthet</span> </div> </a> <ul id="toc-Elektromagnetisk_impulstetthet-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Kovariant_formulering" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kovariant_formulering"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Kovariant formulering</span> </div> </a> <button aria-controls="toc-Kovariant_formulering-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 Kovariant formulering</span> </button> <ul id="toc-Kovariant_formulering-sublist" class="vector-toc-list"> <li id="toc-Feltligninger" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Feltligninger"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Feltligninger</span> </div> </a> <ul id="toc-Feltligninger-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Plane_bølger" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Plane_bølger"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Plane bølger</span> </div> </a> <button aria-controls="toc-Plane_bølger-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 Plane bølger</span> </button> <ul id="toc-Plane_bølger-sublist" class="vector-toc-list"> <li id="toc-Energi_og_impuls" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Energi_og_impuls"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Energi og impuls</span> </div> </a> <ul id="toc-Energi_og_impuls-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polarisasjon" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polarisasjon"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Polarisasjon</span> </div> </a> <ul id="toc-Polarisasjon-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Elektromagnetisk_stråling" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Elektromagnetisk_stråling"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Elektromagnetisk stråling</span> </div> </a> <button aria-controls="toc-Elektromagnetisk_stråling-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 Elektromagnetisk stråling</span> </button> <ul id="toc-Elektromagnetisk_stråling-sublist" class="vector-toc-list"> <li id="toc-Stråling_fra_punktpartikkel" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stråling_fra_punktpartikkel"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Stråling fra punktpartikkel</span> </div> </a> <ul id="toc-Stråling_fra_punktpartikkel-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Elektrisk_strålingsfelt" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Elektrisk_strålingsfelt"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Elektrisk strålingsfelt</span> </div> </a> <ul id="toc-Elektrisk_strålingsfelt-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Larmors_formel" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Larmors_formel"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Larmors formel</span> </div> </a> <ul id="toc-Larmors_formel-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sirkulær_bevegelse" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sirkulær_bevegelse"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Sirkulær bevegelse</span> </div> </a> <ul id="toc-Sirkulær_bevegelse-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Se_også" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Se_også"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Referanser"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Eksterne_lenker"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</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">Elektromagnetisk 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 72 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-72" 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">72 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/Elektromagnetisk_felt" title="Elektromagnetisk felt – norsk nynorsk" lang="nn" hreflang="nn" data-title="Elektromagnetisk 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/Elektromagnetisk_felt" title="Elektromagnetisk felt – dansk" lang="da" hreflang="da" data-title="Elektromagnetisk 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/Elektromagnetiskt_f%C3%A4lt" title="Elektromagnetiskt fält – svensk" lang="sv" hreflang="sv" data-title="Elektromagnetiskt 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/Rafsegulsvi%C3%B0" title="Rafsegulsvið – islandsk" lang="is" hreflang="is" data-title="Rafsegulsvið" data-language-autonym="Íslenska" data-language-local-name="islandsk" class="interlanguage-link-target"><span>Íslenska</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%B7%D9%8A%D8%B3%D9%8A" title="حقل كهرطيسي – arabisk" lang="ar" hreflang="ar" data-title="حقل كهرطيسي" data-language-autonym="العربية" data-language-local-name="arabisk" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Campu_electromagn%C3%A9ticu" title="Campu electromagnéticu – asturisk" lang="ast" hreflang="ast" data-title="Campu electromagnéticu" 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/Elektromaqnit_sah%C9%99si" title="Elektromaqnit sahəsi – aserbajdsjansk" lang="az" hreflang="az" data-title="Elektromaqnit 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%D9%88%D9%85%D8%BA%D9%86%D8%A7%D8%B7%DB%8C%D8%B3_%D9%85%D8%A6%DB%8C%D8%AF%D8%A7%D9%86%DB%8C" title="الکترومغناطیس مئیدانی – søraserbajdsjansk" lang="azb" hreflang="azb" data-title="الکترومغناطیس 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%9A%E0%A7%81%E0%A6%AE%E0%A7%8D%E0%A6%AC%E0%A6%95%E0%A7%80%E0%A6%AF%E0%A6%BC_%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A7%87%E0%A6%A4%E0%A7%8D%E0%A6%B0" title="তড়িৎচুম্বকীয় ক্ষেত্র – bengali" lang="bn" hreflang="bn" data-title="তড়িৎচুম্বকীয় 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-ch%C3%BB-ti%C3%BB%E2%81%BF" title="Tiān-chû-tiûⁿ – minnan" lang="nan" hreflang="nan" data-title="Tiān-chû-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%D0%B0%D0%BC%D0%B0%D0%B3%D0%BD%D1%96%D1%82%D0%BD%D0%B0%D0%B5_%D0%BF%D0%BE%D0%BB%D0%B5" title="Электрамагнітнае поле – belarusisk" lang="be" hreflang="be" data-title="Электрамагнітнае 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%D0%B0%D0%BC%D0%B0%D0%B3%D0%BD%D1%96%D1%82%D0%BD%D0%B0%D0%B5_%D0%BF%D0%BE%D0%BB%D0%B5" title="Электрамагнітнае поле – belarusisk (klassisk ortografi)" lang="be-tarask" hreflang="be-tarask" data-title="Электрамагнітнае 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-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%95%D0%BB%D0%B5%D0%BA%D1%82%D1%80%D0%BE%D0%BC%D0%B0%D0%B3%D0%BD%D0%B8%D1%82%D0%BD%D0%BE_%D0%BF%D0%BE%D0%BB%D0%B5" title="Електромагнитно поле – bulgarsk" lang="bg" hreflang="bg" data-title="Електромагнитно 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/Elektromagnetno_polje" title="Elektromagnetno polje – bosnisk" lang="bs" hreflang="bs" data-title="Elektromagnetno 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_electromagn%C3%A8tic" title="Camp electromagnètic – katalansk" lang="ca" hreflang="ca" data-title="Camp electromagnètic" 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%D0%BC%D0%B0%D0%B3%D0%BD%D0%B8%D1%82%D0%BB%D0%B0_%D1%83%D0%B9" title="Электромагнитла уй – tsjuvasjisk" lang="cv" hreflang="cv" data-title="Электромагнитла 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/Elektromagnetick%C3%A9_pole" title="Elektromagnetické pole – tsjekkisk" lang="cs" hreflang="cs" data-title="Elektromagnetické pole" data-language-autonym="Čeština" data-language-local-name="tsjekkisk" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Elektromagnetisches_Feld" title="Elektromagnetisches Feld – tysk" lang="de" hreflang="de" data-title="Elektromagnetisches Feld" data-language-autonym="Deutsch" data-language-local-name="tysk" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%97%CE%BB%CE%B5%CE%BA%CF%84%CF%81%CE%BF%CE%BC%CE%B1%CE%B3%CE%BD%CE%B7%CF%84%CE%B9%CE%BA%CF%8C_%CF%80%CE%B5%CE%B4%CE%AF%CE%BF" title="Ηλεκτρομαγνητικό πεδίο – gresk" lang="el" hreflang="el" data-title="Ηλεκτρομαγνητικό 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/Electromagnetic_field" title="Electromagnetic field – engelsk" lang="en" hreflang="en" data-title="Electromagnetic 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_electromagn%C3%A9tico" title="Campo electromagnético – spansk" lang="es" hreflang="es" data-title="Campo electromagnético" 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/Elektromagneta_kampo" title="Elektromagneta kampo – esperanto" lang="eo" hreflang="eo" data-title="Elektromagneta kampo" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Eremu_elektromagnetiko" title="Eremu elektromagnetiko – baskisk" lang="eu" hreflang="eu" data-title="Eremu elektromagnetiko" 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%D9%88%D9%85%D8%BA%D9%86%D8%A7%D8%B7%DB%8C%D8%B3%DB%8C" title="میدان الکترومغناطیسی – persisk" lang="fa" hreflang="fa" data-title="میدان الکترومغناطیسی" 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%A9lectromagn%C3%A9tique" title="Champ électromagnétique – fransk" lang="fr" hreflang="fr" data-title="Champ électromagnétique" data-language-autonym="Français" data-language-local-name="fransk" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Campo_electromagn%C3%A9tico" title="Campo electromagnético – galisisk" lang="gl" hreflang="gl" data-title="Campo electromagnético" 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%EC%9E%90%EA%B8%B0%EC%9E%A5" title="전자기장 – koreansk" lang="ko" hreflang="ko" data-title="전자기장" 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%B4%D5%A1%D5%A3%D5%B6%D5%AB%D5%BD%D5%A1%D5%AF%D5%A1%D5%B6_%D5%A4%D5%A1%D5%B7%D5%BF" title="Էլեկտրամագնիսական դաշտ – armensk" lang="hy" hreflang="hy" data-title="Էլեկտրամագնիսական 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%A4%9A%E0%A5%81%E0%A4%AE%E0%A5%8D%E0%A4%AC%E0%A4%95%E0%A5%80%E0%A4%AF_%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0" title="विद्युतचुम्बकीय क्षेत्र – hindi" lang="hi" hreflang="hi" data-title="विद्युतचुम्बकीय 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/Elektromagnetsko_polje" title="Elektromagnetsko polje – kroatisk" lang="hr" hreflang="hr" data-title="Elektromagnetsko polje" data-language-autonym="Hrvatski" data-language-local-name="kroatisk" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Medan_elektromagnetik" title="Medan elektromagnetik – indonesisk" lang="id" hreflang="id" data-title="Medan elektromagnetik" 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-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Campo_elettromagnetico" title="Campo elettromagnetico – italiensk" lang="it" hreflang="it" data-title="Campo elettromagnetico" 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%90%D7%9C%D7%A7%D7%98%D7%A8%D7%95%D7%9E%D7%92%D7%A0%D7%98%D7%99" title="שדה אלקטרומגנטי – hebraisk" lang="he" hreflang="he" data-title="שדה אלקטרומגנטי" data-language-autonym="עברית" data-language-local-name="hebraisk" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%94%E1%83%9A%E1%83%94%E1%83%A5%E1%83%A2%E1%83%A0%E1%83%9D%E1%83%9B%E1%83%90%E1%83%92%E1%83%9C%E1%83%98%E1%83%A2%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%95%E1%83%94%E1%83%9A%E1%83%98" title="ელექტრომაგნიტური ველი – georgisk" lang="ka" hreflang="ka" data-title="ელექტრომაგნიტური 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%D0%BC%D0%B0%D0%B3%D0%BD%D0%B8%D1%82%D1%82%D1%96%D0%BA_%D3%A9%D1%80%D1%96%D1%81" title="Электрмагниттік өріс – kasakhisk" lang="kk" hreflang="kk" data-title="Электрмагниттік өріс" data-language-autonym="Қазақша" data-language-local-name="kasakhisk" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Elektromagn%C4%93tiskais_lauks" title="Elektromagnētiskais lauks – latvisk" lang="lv" hreflang="lv" data-title="Elektromagnētiskais 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/Elektromagnetinis_laukas" title="Elektromagnetinis laukas – litauisk" lang="lt" hreflang="lt" data-title="Elektromagnetinis 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/Lektromagnetisch_veldj" title="Lektromagnetisch veldj – limburgsk" lang="li" hreflang="li" data-title="Lektromagnetisch veldj" 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/Elektrom%C3%A1gneses_mez%C5%91" title="Elektromágneses mező – ungarsk" lang="hu" hreflang="hu" data-title="Elektromágneses 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%BE%D0%BC%D0%B0%D0%B3%D0%BD%D0%B5%D1%82%D0%BD%D0%BE_%D0%BF%D0%BE%D0%BB%D0%B5" title="Електромагнетно поле – makedonsk" lang="mk" hreflang="mk" data-title="Електромагнетно поле" data-language-autonym="Македонски" data-language-local-name="makedonsk" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Anjanandrianaratra" title="Anjanandrianaratra – gassisk" lang="mg" hreflang="mg" data-title="Anjanandrianaratra" data-language-autonym="Malagasy" data-language-local-name="gassisk" class="interlanguage-link-target"><span>Malagasy</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%B4%BE%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B4%BF%E0%B4%95%E0%B4%AE%E0%B4%A3%E0%B5%8D%E0%B4%A1%E0%B4%B2%E0%B4%82" title="വൈദ്യുതകാന്തികമണ്ഡലം – malayalam" lang="ml" hreflang="ml" data-title="വൈദ്യുതകാന്തികമണ്ഡലം" data-language-autonym="മലയാളം" data-language-local-name="malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Medan_elektromagnet" title="Medan elektromagnet – malayisk" lang="ms" hreflang="ms" data-title="Medan elektromagnet" 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_%D1%81%D0%BE%D1%80%D0%BE%D0%BD%D0%B7%D0%BE%D0%BD_%D0%BE%D1%80%D0%BE%D0%BD" title="Цахилгаан соронзон орон – mongolsk" lang="mn" hreflang="mn" data-title="Цахилгаан соронзон орон" data-language-autonym="Монгол" data-language-local-name="mongolsk" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Elektromagnetisch_veld" title="Elektromagnetisch veld – nederlandsk" lang="nl" hreflang="nl" data-title="Elektromagnetisch 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%E7%A3%81%E5%A0%B4" title="電磁場 – japansk" lang="ja" hreflang="ja" data-title="電磁場" 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-nap mw-list-item"><a href="https://nap.wikipedia.org/wiki/Campo_elettromagnetico" title="Campo elettromagnetico – napolitansk" lang="nap" hreflang="nap" data-title="Campo elettromagnetico" data-language-autonym="Napulitano" data-language-local-name="napolitansk" class="interlanguage-link-target"><span>Napulitano</span></a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Camp_electromagnetic" title="Camp electromagnetic – oksitansk" lang="oc" hreflang="oc" data-title="Camp electromagnetic" data-language-autonym="Occitan" data-language-local-name="oksitansk" class="interlanguage-link-target"><span>Occitan</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Elektromagnit_maydon" title="Elektromagnit maydon – usbekisk" lang="uz" hreflang="uz" data-title="Elektromagnit 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%A9%8B%E0%A8%AE%E0%A9%88%E0%A8%97%E0%A8%A8%E0%A9%88%E0%A8%9F%E0%A8%BF%E0%A8%95_%E0%A8%AB%E0%A9%80%E0%A8%B2%E0%A8%A1" title="ਇਲੈਕਟ੍ਰੋਮੈਗਨੈਟਿਕ ਫੀਲਡ – panjabi" lang="pa" hreflang="pa" data-title="ਇਲੈਕਟ੍ਰੋਮੈਗਨੈਟਿਕ ਫੀਲਡ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="panjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</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_%D9%85%D9%82%D9%86%D8%A7%D8%B7%D9%8A%D8%B3%D9%8A_%D8%B3%D8%A7%D8%AD%D9%87" title="برېښنايي مقناطيسي ساحه – pashto" lang="ps" hreflang="ps" data-title="برېښنايي مقناطيسي ساحه" data-language-autonym="پښتو" data-language-local-name="pashto" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Pole_elektromagnetyczne" title="Pole elektromagnetyczne – polsk" lang="pl" hreflang="pl" data-title="Pole elektromagnetyczne" 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_eletromagn%C3%A9tico" title="Campo eletromagnético – portugisisk" lang="pt" hreflang="pt" data-title="Campo eletromagnético" 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_electromagnetic" title="Câmp electromagnetic – rumensk" lang="ro" hreflang="ro" data-title="Câmp electromagnetic" 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%BE%D0%BC%D0%B0%D0%B3%D0%BD%D0%B8%D1%82%D0%BD%D0%BE%D0%B5_%D0%BF%D0%BE%D0%BB%D0%B5" title="Электромагнитное поле – russisk" lang="ru" hreflang="ru" data-title="Электромагнитное 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_elektromagnetike" title="Fusha elektromagnetike – albansk" lang="sq" hreflang="sq" data-title="Fusha elektromagnetike" data-language-autonym="Shqip" data-language-local-name="albansk" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Elektromagnetick%C3%A9_pole" title="Elektromagnetické pole – slovakisk" lang="sk" hreflang="sk" data-title="Elektromagnetické 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/Elektromagnetno_polje" title="Elektromagnetno polje – slovensk" lang="sl" hreflang="sl" data-title="Elektromagnetno 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%86%D9%85%D9%88%DA%AF%D9%86%D8%A7%D8%AA%DB%8C%D8%B3%DB%8C" title="بواری کارۆموگناتیسی – sentralkurdisk" lang="ckb" hreflang="ckb" data-title="بواری کارۆموگناتیسی" 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%BE%D0%BC%D0%B0%D0%B3%D0%BD%D0%B5%D1%82%D0%BD%D0%BE_%D0%BF%D0%BE%D1%99%D0%B5" title="Електромагнетно поље – serbisk" lang="sr" hreflang="sr" data-title="Електромагнетно 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/Elektromagnetno_polje" title="Elektromagnetno polje – serbokroatisk" lang="sh" hreflang="sh" data-title="Elektromagnetno 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-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/S%C3%A4hk%C3%B6magneettinen_kentt%C3%A4" title="Sähkömagneettinen kenttä – finsk" lang="fi" hreflang="fi" data-title="Sähkömagneettinen kenttä" data-language-autonym="Suomi" data-language-local-name="finsk" class="interlanguage-link-target"><span>Suomi</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%95%E0%AE%BE%E0%AE%A8%E0%AF%8D%E0%AE%A4%E0%AE%AA%E0%AF%8D_%E0%AE%AA%E0%AF%81%E0%AE%B2%E0%AE%AE%E0%AF%8D" title="மின்காந்தப் புலம் – tamil" lang="ta" hreflang="ta" data-title="மின்காந்தப் புலம்" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Elektromanyetik_alan" title="Elektromanyetik alan – tyrkisk" lang="tr" hreflang="tr" data-title="Elektromanyetik alan" 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%BE%D0%BC%D0%B0%D0%B3%D0%BD%D1%96%D1%82%D0%BD%D0%B5_%D0%BF%D0%BE%D0%BB%D0%B5" title="Електромагнітне поле – ukrainsk" lang="uk" hreflang="uk" data-title="Електромагнітне поле" data-language-autonym="Українська" data-language-local-name="ukrainsk" 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aria-label="Utseende"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Utseende</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">flytt til sidefeltet</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">skjul</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Fra Wikipedia, den frie encyklopedi</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="nb" dir="ltr"><figure typeof="mw:File/Thumb"><a href="/wiki/Fil:Felder_um_Dipol.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Felder_um_Dipol.svg/280px-Felder_um_Dipol.svg.png" decoding="async" width="280" height="260" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Felder_um_Dipol.svg/420px-Felder_um_Dipol.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c5/Felder_um_Dipol.svg/560px-Felder_um_Dipol.svg.png 2x" data-file-width="801" data-file-height="743" /></a><figcaption>Øyeblikksbilde av elektriske <b>E</b> og magnetiske <b>B</b> <a href="/wiki/Feltlinje" title="Feltlinje">feltlinjer</a> rundt en <a href="/wiki/Dipolantenne" title="Dipolantenne">dipolantenne</a>.</figcaption></figure> <p>Et <b>elektromagnetisk felt</b> oppstår i alle situasjoner hvor det finnes <a href="/wiki/Elektrisk_ladning" title="Elektrisk ladning">elektrisk ladning</a>. Er denne i ro, vil det bestå av et rent <a href="/wiki/Elektrisk_felt" title="Elektrisk felt">elektrisk felt</a>. Så snart den beveger seg, vil ladningen også omgi seg med et <a href="/wiki/Magnetisk_felt" class="mw-redirect" title="Magnetisk felt">magnetisk felt</a>. Feltet er beskrevet ved <a href="/wiki/Maxwells_ligninger" class="mw-redirect" title="Maxwells ligninger">Maxwells ligninger</a> som viser at det utbrer seg i <a href="/wiki/Vakuum" title="Vakuum">vakuum</a> med <a href="/wiki/Lyshastigheten" class="mw-redirect" title="Lyshastigheten">lyshastigheten</a>. </p><p>Egenskapene til det elektromagnetiske feltet lå til grunn for <a href="/wiki/Einstein" class="mw-redirect" title="Einstein">Einsteins</a> utvikling av den <a href="/wiki/Spesiell_relativitetsteori" class="mw-redirect" title="Spesiell relativitetsteori">spesielle relativitetsteorien</a> i 1905. Den viser at komponentene til feltet utgjør en antisymmetrisk <a href="/wiki/Tensor" title="Tensor">tensor</a> i det firedimensjonale <a href="/wiki/Romtid" class="mw-redirect" title="Romtid">tidrommet</a>. Om det skal ha hovedsakelig en elektrisk eller en magnetisk karakter, avhenger av hvordan det observeres. Feltet er invariant under <a href="/wiki/Gaugetransformasjon" title="Gaugetransformasjon">gaugetransformasjoner</a> slik at Maxwells ligninger definerer en <a href="/wiki/Gaugeteori" title="Gaugeteori">gaugeteori</a>. Siden 1983 har det vært kjent at det elektromagnetiske feltet inngår sammen med andre, tilsvarende felt for den <a href="/wiki/Svak_kjernekraft" title="Svak kjernekraft">svake kjernekraften</a> i en enhetlig, <a href="/wiki/Elektrosvak_vekselvirkning" title="Elektrosvak vekselvirkning">elektrosvak gaugeteori</a>. </p><p>Elektromagnetiske felt spiller en stadig større rolle i vårt moderne samfunn. All <a href="/wiki/Telekommunikasjon" title="Telekommunikasjon">telekommunikasjon</a>, <a href="/wiki/Radar" title="Radar">radar</a>, <a href="/wiki/Fiberoptikk" title="Fiberoptikk">fiberoptikk</a>, <a href="/wiki/Elektrisk_motor" title="Elektrisk motor">elektrisk motorer</a>, <a href="/wiki/Str%C3%B8mforsyning" title="Strømforsyning">strømforsyning</a> og mye annet er basert på de samme matematiske ligninger som ble funnet av <a href="/wiki/Maxwell" class="mw-disambig" title="Maxwell">Maxwell</a> for omtrent 150 år siden. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Maxwells_ligninger">Maxwells ligninger</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektromagnetisk_felt&veaction=edit&section=1" title="Rediger avsnitt: Maxwells ligninger" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektromagnetisk_felt&action=edit&section=1" title="Rediger kildekoden til seksjonen Maxwells ligninger"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Grunnlaget for den moderne forståelse av det elektromagnetiske feltet ble lagt av <a href="/wiki/Michael_Faraday" title="Michael Faraday">Michael Faraday</a> gjennom hans eksperimentelle undersøkelser av <a href="/wiki/Elektrisitet" title="Elektrisitet">elektriske</a> og <a href="/wiki/Magnetisme" title="Magnetisme">magnetiske</a> <a href="/wiki/Kraft" title="Kraft">krefter</a>. Disse forklarte han ved at de ble formidlet av et <a href="/wiki/Felt_(fysikk)" title="Felt (fysikk)">felt</a> som han kunne beskrive ved hjelp av <a href="/wiki/Feltlinje" title="Feltlinje">feltlinjer</a>. Den matematiske beskrivelsen av dette elektromagnetiske feltet ble funnet av <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a> i 1862 uttrykt ved tyve <a href="/wiki/Partielle_differensialligninger" title="Partielle differensialligninger">partielle differensialligninger</a> for de forskjellige komponentene til feltet. Et par tiår senere ble disse redusert til fire ligninger av <a href="/wiki/Oliver_Heaviside" title="Oliver Heaviside">Oliver Heaviside</a> ved bruk av <a href="/wiki/Vektoranalyse" title="Vektoranalyse">vektoranalyse</a>. </p><p>I denne moderne formuleringen er det elektromagnetiske feltet beskrevet ved to <a href="/wiki/Vektorfelt" title="Vektorfelt">vektorfelt</a> som kalles det <a href="/wiki/Elektrisk_felt" title="Elektrisk felt">elektriske feltet</a> <span class="nowrap"><b>E</b> = <b>E</b>(<b>r</b>,<i>t</i>)</span> og det <a href="/wiki/Magnetisk_felt" class="mw-redirect" title="Magnetisk felt">magnetiske fluksfeltet</a> <span class="nowrap"><b>B</b> = <b>B</b>(<b>r</b>,<i>t</i>)</span>. Hvis bare det elektriske feltet er tilstede og det ikke varierer med tiden, kalles det for et <a href="/wiki/Elektrostatikk" title="Elektrostatikk">elektrostatisk</a> felt. Likedan, hvis kun det magnetiske feltet er tilstede og det ikke varierer med tiden, kalles det for et <a href="/wiki/Magnetostatikk" title="Magnetostatikk">magnetostatisk</a> felt. Men hvis bare et av disse to vektorfeltene har en tidsavhengighet, vil også det andre få det gjennom Maxwells ligninger.<sup id="cite_ref-Griffiths_1-0" class="reference"><a href="#cite_note-Griffiths-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>For å beskrive feltet i materialer og forskjellige medier er det hensiktsmessig å innføre et <a href="/wiki/Elektrisk_felt#Elektrisk_polarisasjon" title="Elektrisk felt">elektrisk forskyvningsfelt</a> <span class="nowrap"><b>D</b> = <i>ε</i> <b>E</b></span> hvor <i>ε</i> er <a href="/wiki/Permittivitet" title="Permittivitet">permittiviteten</a> til materialet. På samme måte defineres et <a href="/wiki/Magnetisk_felt" class="mw-redirect" title="Magnetisk felt">magnetisk felt</a> <b>H</b> i et material med <a href="/wiki/Permeabilitet" class="mw-disambig" title="Permeabilitet">permeabilitet</a> <i>μ</i> ved sammenhengen <span class="nowrap"><b>B</b> = <i>μ</i> <b>H</b></span>. I <a href="/wiki/Vakuum" title="Vakuum">vakuum</a> er disse to materialkonstantene gitt ved henholdsvis den <a href="/wiki/Permittivitet" title="Permittivitet">elektriske konstanten</a> <i>ε</i><sub>0</sub>  og den <a href="/wiki/Permeabilitet_(fysikk)" title="Permeabilitet (fysikk)">magnetiske konstanten</a> <i>μ</i><sub>0</sub>. </p><p>Kilden til det elektromagnetiske feltet er <a href="/wiki/Elektrisk_ladning" title="Elektrisk ladning">elektrisk ladning</a> beskrevet ved en ladningstetthet <span class="nowrap"><i>ρ</i> = <i>ρ</i>(<b>r</b>,<i>t</i>)</span> og <a href="/wiki/Elektrisk_str%C3%B8m" title="Elektrisk strøm">elektriske strømmer</a> beskrevet ved strømtettheten <span class="nowrap"><b>J</b> = <b>J</b>(<b>r</b>,<i>t</i>) </span>. Disse inngår i de to første Maxwell-ligningene 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 {\boldsymbol {\nabla }}\cdot \mathbf {D} =\rho ,\;\;\;{\boldsymbol {\nabla }}\times \mathbf {H} =\mathbf {J} +{\frac {\partial \mathbf {D} }{\partial t}}}"> <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> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {D} =\rho ,\;\;\;{\boldsymbol {\nabla }}\times \mathbf {H} =\mathbf {J} +{\frac {\partial \mathbf {D} }{\partial t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/020ddcbb577dadb55948fbfe98060e8517d8ddb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:31.906ex; height:5.509ex;" alt="{\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {D} =\rho ,\;\;\;{\boldsymbol {\nabla }}\times \mathbf {H} =\mathbf {J} +{\frac {\partial \mathbf {D} }{\partial t}}}"></span></dd></dl> <p>Den første er <a href="/wiki/Gauss%27_lov" class="mw-redirect" title="Gauss' lov">Gauss' lov</a> for det elektriske feltet som har elektrisk ladning som kilde. Fra betydningen av <a href="/wiki/Divergens" title="Divergens">divergens</a>-operatoren følger at de tilsvarende <a href="/wiki/Feltlinje" title="Feltlinje">feltlinjene</a> går fra positive ladninger og ender opp på like store, negative ladninger. På samme vis viser den andre Maxwell-ligningen, hvor <a href="/wiki/Maxwells_forskyvningsstr%C3%B8m" title="Maxwells forskyvningsstrøm">forskyvningsstrømmen</a> <span class="nowrap">∂<b>D</b>/∂<i>t</i></span> inngår, at elektriske strømmer er kilden til det magnetiske feltet. Ligningen omtales vanligvis som <a href="/wiki/Amp%C3%A8res_sirkulasjonslov" title="Ampères sirkulasjonslov">Maxwell-Ampères sirkulasjonslov</a> da den er en generalisering av Ampères opprinnelige sirkulasjonslov for stasjonære strømmer til det generelle tilfellet hvor feltene varierer med tiden. Tar man divergensen av ligningen, fremkommer <a href="/wiki/Kontinuitetsligning" title="Kontinuitetsligning">kontinuitetsligningen</a> som et uttrykk for at elektrisk ladning er en bevart størrelse.<sup id="cite_ref-Darrigol_2-0" class="reference"><a href="#cite_note-Darrigol-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>De to siste Maxwell-ligningene kan 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 {B} =0,\;\;\;{\boldsymbol {\nabla }}\times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}"> <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">B</mi> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <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> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {B} =0,\;\;\;{\boldsymbol {\nabla }}\times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5444ad6c4d5f1edaed981dec4913af74997cfcd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.822ex; height:5.509ex;" alt="{\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {B} =0,\;\;\;{\boldsymbol {\nabla }}\times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}"></span></dd></dl> <p>og er uten kilder. Den første betyr at magnetiske flukslinjer alltid vil være lukkete <a href="/wiki/Kurve" title="Kurve">kurver</a> da det er ingen ladninger hvor de kan starte eller ende opp på. At det ikke finnes slike magnetiske ladninger, betyr også at i denne klassiske, elektromagnetiske teorien finnes ikke <a href="/wiki/Magnetisk_monopol" title="Magnetisk monopol">magnetiske monopoler</a>. Den siste ligningen her viser de hvordan elektriske og magnetiske felter er knyttet sammen i et elektromagnetisk felt slik at det ene kan gi opphav til det andre og representerer den matematiske formuleringen av <a href="/wiki/Faradays_induksjonslov" title="Faradays induksjonslov">Faradays induksjonslov</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Elektromagnetiske_potensial">Elektromagnetiske potensial</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektromagnetisk_felt&veaction=edit&section=2" title="Rediger avsnitt: Elektromagnetiske potensial" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektromagnetisk_felt&action=edit&section=2" title="Rediger kildekoden til seksjonen Elektromagnetiske potensial"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Den magnetiske Maxwell-ligningen <b>∇</b>⋅<b>B</b> = 0 er automatisk oppfylt ved å 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 {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>hvor <span class="nowrap"><b>A</b> = <b>A</b>(<b>r</b>,<i>t</i>)</span> er det <a href="/wiki/Magnetisk_felt#Vektorpotensialet" class="mw-redirect" title="Magnetisk felt">magnetiske vektorpotensialet</a>. Det ble først innført av <a href="/w/index.php?title=Franz_Ernst_Neumann&action=edit&redlink=1" class="new" title="Franz Ernst Neumann (ikke skrevet ennå)">Franz Neumann</a> og benyttet i stor grad av Maxwell i konstruksjon av sine ligninger.<sup id="cite_ref-Darrigol_2-1" class="reference"><a href="#cite_note-Darrigol-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Bruker man denne definisjonen i Faradays induksjonslov, ser man at <span class="nowrap"><b>E</b> + ∂<b>A</b>/∂<i>t</i></span>  må være <a href="/wiki/Gradient" title="Gradient">gradienten</a> til en <a href="/wiki/Skalar" title="Skalar">skalar</a> funksjon. Den er gitt ved det <a href="/wiki/Elektrisk_potensial" title="Elektrisk potensial">elektriske potensialet</a> <span class="nowrap">Φ = Φ(<b>r</b>,<i>t</i>) </span> slik at man skrive det elektriske feltet generelt 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 }}\Phi -{\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 mathvariant="normal">Φ</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 }}\Phi -{\partial \mathbf {A} \over \partial t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a93e18e7d9e284fa59e5ebfa8605a0fc29c3575" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.582ex; height:5.509ex;" alt="{\displaystyle \mathbf {E} =-{\boldsymbol {\nabla }}\Phi -{\partial \mathbf {A} \over \partial t}}"></span></dd></dl> <p>I det statiske tilfellet forenkles denne relasjonen til <b>E</b> = - <b>∇</b> φ. Det viser at dette potensialet da er direkte relatert til den <a href="/wiki/Potensiell_energi" title="Potensiell energi">potensielle energien</a> til en partikkel som befinner seg i et slikt elektrisk felt. </p><p>Derimot har det elektriske potensialet ingen slik direkte fysisk interpretasjon når feltet varierer med tiden da det alltid kan forandres ved en <a href="/wiki/Gaugetransformasjon" title="Gaugetransformasjon">gaugetransformasjon</a>. Det ser man ved å innføre en vilkårlig funkjon <span class="nowrap">χ = χ(<b>r</b>,<i>t</i>) </span>. Hvis man så definerer de transformerte potensialene, 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 {A} \rightarrow \mathbf {A} '=\mathbf {A} +{\boldsymbol {\nabla }}\chi ,\;\;\;\Phi \rightarrow \Phi '=\Phi -{\partial \chi \over \partial t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>′</mo> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mi>χ</mi> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">→</mo> <msup> <mi mathvariant="normal">Φ</mi> <mo>′</mo> </msup> <mo>=</mo> <mi mathvariant="normal">Φ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>χ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} \rightarrow \mathbf {A} '=\mathbf {A} +{\boldsymbol {\nabla }}\chi ,\;\;\;\Phi \rightarrow \Phi '=\Phi -{\partial \chi \over \partial t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/346264b01cdd37a828afcaa7024a5493fa8ba8bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:41.828ex; height:5.676ex;" alt="{\displaystyle \mathbf {A} \rightarrow \mathbf {A} '=\mathbf {A} +{\boldsymbol {\nabla }}\chi ,\;\;\;\Phi \rightarrow \Phi '=\Phi -{\partial \chi \over \partial t}}"></span></dd></dl> <p>vil de transformerte elektriske og magnetiske feltene forbli uforandret. Maxwell-teorien sies derfor å være <b>gaugeinvariant</b>, noe som på et vis definerer dens innhold. </p><p>Denne friheten kan også være av stor betydning ved bruk av teorien. Avhengig av hva man ønsker å oppnå kan man derfor anta at potensialene oppfyller en bestemt betingelse som medfører visse forenklinger. Så snart man tar en slik betingelse i bruk, sies man å arbeide innen en bestemt gauge. De mest vanlige gaugevalgene er <a href="/wiki/Gaugetransformasjon#Gaugefiksering" title="Gaugetransformasjon">Coulomb-gauge</a> og <a href="/wiki/Gaugetransformasjon#Gaugefiksering" title="Gaugetransformasjon">Lorenz-gauge</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Bølgeligninger"><span id="B.C3.B8lgeligninger"></span>Bølgeligninger</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektromagnetisk_felt&veaction=edit&section=3" title="Rediger avsnitt: Bølgeligninger" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektromagnetisk_felt&action=edit&section=3" title="Rediger kildekoden til seksjonen Bølgeligninger"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En viktig fordel ved å benytte potensial til å formulere lovene for det elektromagnetiske feltet, er det gir enklere <a href="/wiki/B%C3%B8lgeligning" title="Bølgeligning">bølgeligninger</a> som beskriver feltets utbredelse i tid og rom. For det magnetiske vektorpotensialet kan den utledes fra definisjonen <span class="nowrap"><b>B</b> = <b>∇</b> × <b>A</b></span>. Tar man <a href="/wiki/Curl" title="Curl">curl</a> av denne og bruker <a href="/wiki/Vektoranalyse" title="Vektoranalyse">identiteten</a> <span class="nowrap"><b>∇</b> × (<b>∇ </b>× <b>A</b>)</span> = <span class="nowrap"><b>∇</b> (<b>∇</b>⋅<b>A</b>) - <b>∇</b><sup> 2</sup><b>A</b></span> sammen med Maxwells to ligninger for <span class="nowrap"><b>∇</b> × <b>B</b></span> og <span class="nowrap"><b>∇</b> × <b>E</b></span>, får man </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 }}^{2}\mathbf {A} -{1 \over c^{2}}{\partial ^{2}\mathbf {A} \over \partial t^{2}}-{\boldsymbol {\nabla }}{\Big (}{\boldsymbol {\nabla }}\cdot \mathbf {A} +{1 \over c^{2}}{\partial \Phi \over \partial t}{\Big )}=-\mu \mathbf {J} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <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">∇</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Φ</mi> </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> <mo>−</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nabla }}^{2}\mathbf {A} -{1 \over c^{2}}{\partial ^{2}\mathbf {A} \over \partial t^{2}}-{\boldsymbol {\nabla }}{\Big (}{\boldsymbol {\nabla }}\cdot \mathbf {A} +{1 \over c^{2}}{\partial \Phi \over \partial t}{\Big )}=-\mu \mathbf {J} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ced922f332bb3cae1dbfb8a796407c6bbc1cfe0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:47.317ex; height:6.009ex;" alt="{\displaystyle {\boldsymbol {\nabla }}^{2}\mathbf {A} -{1 \over c^{2}}{\partial ^{2}\mathbf {A} \over \partial t^{2}}-{\boldsymbol {\nabla }}{\Big (}{\boldsymbol {\nabla }}\cdot \mathbf {A} +{1 \over c^{2}}{\partial \Phi \over \partial t}{\Big )}=-\mu \mathbf {J} }"></span></dd></dl> <p>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={\sqrt {1 \over \varepsilon \mu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mn>1</mn> <mrow> <mi>ε</mi> <mi>μ</mi> </mrow> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c={\sqrt {1 \over \varepsilon \mu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aea434809e32903b30009bcb201f2f215c9377d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:9.75ex; height:7.676ex;" alt="{\displaystyle c={\sqrt {1 \over \varepsilon \mu }}}"></span></dd></dl> <p>er <a href="/wiki/Lyshastigheten" class="mw-redirect" title="Lyshastigheten">lyshastigheten</a> i materialet der feltene befinner seg. Likedan, ved å ta <a href="/wiki/Divergens" title="Divergens">divergensen</a> av <a href="/wiki/Gauss%27_lov" class="mw-redirect" title="Gauss' lov">Gauss' lov</a> på differensiell form hvor man i alminnelighet har <span class="nowrap"><b>E</b> = - <b>∇</b>Φ - ∂<b>A</b>/∂<i>t</i></span>, kan resultatet 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 }}^{2}\Phi +{\frac {\partial }{\partial t}}\left(\mathbf {\nabla } \cdot \mathbf {A} \right)=-{\frac {\rho }{\varepsilon }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Φ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∇</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ρ</mi> <mi>ε</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nabla }}^{2}\Phi +{\frac {\partial }{\partial t}}\left(\mathbf {\nabla } \cdot \mathbf {A} \right)=-{\frac {\rho }{\varepsilon }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bbe60dc6062dc63adf5bda6f22bbde52fe943f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.215ex; height:5.509ex;" alt="{\displaystyle {\boldsymbol {\nabla }}^{2}\Phi +{\frac {\partial }{\partial t}}\left(\mathbf {\nabla } \cdot \mathbf {A} \right)=-{\frac {\rho }{\varepsilon }}.}"></span></dd></dl> <p>Dette er ingen vanlig bølgeligning for det elektriske potensialet. I tillegg involverer både dette og det forrige uttrykket begge potensialene Φ og <b>A</b> slik at an ikke uten videre kan holde disse to fra hverandre under en beregning. </p><p>Ligningene forenkles betraktelig ved bruk av <a href="/wiki/Gaugetransformasjon#Lorenz-gauge" title="Gaugetransformasjon">Lorenz-gaugen</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {A} +{1 \over c^{2}}{\partial \Phi \over \partial t}=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"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {A} +{1 \over c^{2}}{\partial \Phi \over \partial t}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5f9846b95be5979a9e166199e9a90ffe9719b39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.402ex; height:5.676ex;" alt="{\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {A} +{1 \over c^{2}}{\partial \Phi \over \partial t}=0.}"></span></dd></dl> <p>Det resulterer i den inhomogene <a href="/wiki/B%C3%B8lgeligning" title="Bølgeligning">bølgeligningen</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\nabla }}^{2}\mathbf {A} -{1 \over c^{2}}{\partial ^{2}\mathbf {A} \over \partial t^{2}}=-\mu \mathbf {J} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nabla }}^{2}\mathbf {A} -{1 \over c^{2}}{\partial ^{2}\mathbf {A} \over \partial t^{2}}=-\mu \mathbf {J} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/753294c66c2d584e3f3ee271791d6d195ca19f87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:23.98ex; height:6.009ex;" alt="{\displaystyle {\boldsymbol {\nabla }}^{2}\mathbf {A} -{1 \over c^{2}}{\partial ^{2}\mathbf {A} \over \partial t^{2}}=-\mu \mathbf {J} }"></span></dd></dl> <p>for det magnetiske vektorpotensialet. Samtidig vil det elektriske potensialet oppfylle den tilsvarende bølgeligningen </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 }}^{2}\Phi -{1 \over c^{2}}{\partial ^{2}\Phi \over \partial t^{2}}=-{\rho \over \varepsilon }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Φ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ρ</mi> <mi>ε</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nabla }}^{2}\Phi -{1 \over c^{2}}{\partial ^{2}\Phi \over \partial t^{2}}=-{\rho \over \varepsilon }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a6a27c85a44e138cf708cc15be29c0d41435a23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:23.199ex; height:6.009ex;" alt="{\displaystyle {\boldsymbol {\nabla }}^{2}\Phi -{1 \over c^{2}}{\partial ^{2}\Phi \over \partial t^{2}}=-{\rho \over \varepsilon }.}"></span></dd></dl> <p>De to potensialene er dermed blitt koblet fra hverandre og deres bevegelsesligninger kan løses hver for seg. De er kun indirekte avhengig av hverandre gjennom gaugebetingelsen. </p> <div class="mw-heading mw-heading3"><h3 id="Retarderte_løsninger"><span id="Retarderte_l.C3.B8sninger"></span>Retarderte løsninger</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektromagnetisk_felt&veaction=edit&section=4" title="Rediger avsnitt: Retarderte løsninger" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektromagnetisk_felt&action=edit&section=4" title="Rediger kildekoden til seksjonen Retarderte løsninger"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For gitt kilder <span class="nowrap"><i>ρ</i> = <i>ρ</i>(<b>r</b>,<i>t</i>)</span> og <span class="nowrap"><b>J</b> = <b>J</b>(<b>r</b>,<i>t</i>) </span> kan begge bølgeligningene løses ved bruk av <a href="/wiki/Greens_funksjon" title="Greens funksjon">Greens funksjon</a>. Siden et signal som er skapt i et kildepunkt <b>r'</b>, vil bruke en tid |<b>r</b> - <b>r'</b>|/<i>c</i>  for å nå frem til et feltpunkt <b>r</b>, vil feltet i dette punktet ved tiden <i>t</i> være avhengig av hva som forgikk i kilden ved det tidligere eller <b>retarderte tidspunktet</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 t'=t-{\frac {|\mathbf {r} -\mathbf {r} '|}{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>t</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <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> <mi>c</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t'=t-{\frac {|\mathbf {r} -\mathbf {r} '|}{c}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b82eff63595c595e4ab63d91b025ae20f2b34e63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.162ex; height:5.676ex;" alt="{\displaystyle t'=t-{\frac {|\mathbf {r} -\mathbf {r} '|}{c}}}"></span></dd></dl> <p>Løsningene av bølgeligningene i <a href="/wiki/Vakuum" title="Vakuum">vakuum</a> med <i>ε</i> = <i>ε</i><sub>0</sub> og <i>μ</i> = <i>μ</i><sub>0</sub> kan da skrives generelt 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 (\mathbf {r} ,t)={\frac {1}{4\pi \epsilon _{0}}}\int \!d^{3}x'{\frac {\rho (\mathbf {r'} ,t')}{|\mathbf {r} -\mathbf {r'} |}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>∫</mo> <mspace width="negativethinmathspace" /> <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> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>′</mo> </msup> </mrow> <mo>,</mo> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (\mathbf {r} ,t)={\frac {1}{4\pi \epsilon _{0}}}\int \!d^{3}x'{\frac {\rho (\mathbf {r'} ,t')}{|\mathbf {r} -\mathbf {r'} |}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2155c2e34ff3cf7aa93c7441ff521720b9fe115" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.95ex; height:6.509ex;" alt="{\displaystyle \Phi (\mathbf {r} ,t)={\frac {1}{4\pi \epsilon _{0}}}\int \!d^{3}x'{\frac {\rho (\mathbf {r'} ,t')}{|\mathbf {r} -\mathbf {r'} |}}}"></span></dd></dl> <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 {A} (\mathbf {r} ,t)={\frac {\mu _{0}}{4\pi }}\int \!d^{3}x'{\frac {\mathbf {J} (\mathbf {r'} ,t')}{|\mathbf {r} -\mathbf {r'} |}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>∫</mo> <mspace width="negativethinmathspace" /> <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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>′</mo> </msup> </mrow> <mo>,</mo> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} (\mathbf {r} ,t)={\frac {\mu _{0}}{4\pi }}\int \!d^{3}x'{\frac {\mathbf {J} (\mathbf {r'} ,t')}{|\mathbf {r} -\mathbf {r'} |}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5838dfe45d4cfec2a2e9cf3fb39960015cd28c8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.472ex; height:6.509ex;" alt="{\displaystyle \mathbf {A} (\mathbf {r} ,t)={\frac {\mu _{0}}{4\pi }}\int \!d^{3}x'{\frac {\mathbf {J} (\mathbf {r'} ,t')}{|\mathbf {r} -\mathbf {r'} |}}}"></span></dd></dl> <p>De ble for første gang utledet av den danske fysiker <a href="/wiki/Ludvig_Lorenz" title="Ludvig Lorenz">Ludvig Lorenz</a> på midten av 1800-tallet. All <a href="/wiki/Elektromagnetisk_str%C3%A5ling" title="Elektromagnetisk stråling">elektromagnetisk stråling</a> fra klassiske kilder kan beregnes fra disse to formlene. I enkleste approksimasjon beskrives kildene med sine <a href="/wiki/Dipol" title="Dipol">dipolmoment</a> som resultererer i elektrisk og <a href="/wiki/Magnetisk_dipol#Magnetisk_dipolstråling" title="Magnetisk dipol">magnetisk dipolstråling</a>.<sup id="cite_ref-Brau_3-0" class="reference"><a href="#cite_note-Brau-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>For statiske kilder forenkles disse løsningene. Ligningen for Φ gir det <a href="/wiki/Elektrisk_potensial" title="Elektrisk potensial">elektriske potensialet</a> som skyldes en konstant ladningsfordeling, mens uttrykket for vektorpotensialet <b>A</b> er ekvivalent med <a href="/wiki/Biot-Savarts_lov" title="Biot-Savarts lov">Biot-Savarts lov</a> for det magnetiske feltet frembrakt av en stasjonær strømfordeling. </p> <div class="mw-heading mw-heading3"><h3 id="Coulomb-gauge">Coulomb-gauge</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektromagnetisk_felt&veaction=edit&section=5" title="Rediger avsnitt: Coulomb-gauge" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektromagnetisk_felt&action=edit&section=5" title="Rediger kildekoden til seksjonen Coulomb-gauge"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bølgeligningene kan også løses i Coulomb-gaugen <b>∇</b>⋅<b>A</b> = 0. Selv om løsningene da ser annerledes ut, vil de gi samme resultat ved beregning av fysiske størrelser. Med dette gaugevalget, vil den siste ligningen for det skalare potensialet Φ forenkles til <a href="/wiki/Poissons_ligning" class="mw-redirect" title="Poissons ligning">Poissons ligning</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\nabla }}^{2}\Phi (\mathbf {r} ,t)=-{1 \over \varepsilon _{0}}\rho (\mathbf {r} ,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <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">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nabla }}^{2}\Phi (\mathbf {r} ,t)=-{1 \over \varepsilon _{0}}\rho (\mathbf {r} ,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/360af016fe4e11127979e9cce04e6859ddb47847" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:23.611ex; height:5.509ex;" alt="{\displaystyle {\boldsymbol {\nabla }}^{2}\Phi (\mathbf {r} ,t)=-{1 \over \varepsilon _{0}}\rho (\mathbf {r} ,t)}"></span></dd></dl> <p>som ikke er en vanlig <a href="/wiki/B%C3%B8lgeligning" title="Bølgeligning">bølgeligning</a>. Den kan løses på samme måte som i det <a href="/wiki/Elektrostatikk" title="Elektrostatikk">elektrostatiske</a> tilfellet slik at man har </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 (\mathbf {r} ,t)={\frac {1}{4\pi \epsilon _{0}}}\int \!d^{3}x'{\frac {\rho (\mathbf {r'} ,t)}{|\mathbf {r} -\mathbf {r'} |}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>∫</mo> <mspace width="negativethinmathspace" /> <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> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>′</mo> </msup> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (\mathbf {r} ,t)={\frac {1}{4\pi \epsilon _{0}}}\int \!d^{3}x'{\frac {\rho (\mathbf {r'} ,t)}{|\mathbf {r} -\mathbf {r'} |}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44c2517b8373057964b45f95b5676b2f196a16a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.617ex; height:6.509ex;" alt="{\displaystyle \Phi (\mathbf {r} ,t)={\frac {1}{4\pi \epsilon _{0}}}\int \!d^{3}x'{\frac {\rho (\mathbf {r'} ,t)}{|\mathbf {r} -\mathbf {r'} |}}}"></span></dd></dl> <p>Her opptrer samme tidspunkt både i kildepunktet og feltpunktet. Selv om det ikke er noen retardasjon i denne løsningen, vil likevel et fysisk signal fra kilden ta en viss tid å nå frem siden det er sammensatt med den delen som blir overført gjennom vektorpotensialet. Og det forplanter seg med retardasjon som kommer frem ved å skrive den resulterende bølgeligningen i denne gaugen 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 }}^{2}\mathbf {A} -{1 \over c^{2}}{\partial ^{2}\mathbf {A} \over \partial t^{2}}=-\mu _{0}\mathbf {J} _{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nabla }}^{2}\mathbf {A} -{1 \over c^{2}}{\partial ^{2}\mathbf {A} \over \partial t^{2}}=-\mu _{0}\mathbf {J} _{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c83466bb34dee0d6e89e3cb6c3f1c56c07c5e87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:26.423ex; height:6.009ex;" alt="{\displaystyle {\boldsymbol {\nabla }}^{2}\mathbf {A} -{1 \over c^{2}}{\partial ^{2}\mathbf {A} \over \partial t^{2}}=-\mu _{0}\mathbf {J} _{T}}"></span></dd></dl> <p>etter å ha innført den <b>transverse</b> strømmen </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 {J} _{T}=\mathbf {J} -\varepsilon _{0}{\partial \over \partial t}{\boldsymbol {\nabla }}\Phi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>−</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mi mathvariant="normal">Φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J} _{T}=\mathbf {J} -\varepsilon _{0}{\partial \over \partial t}{\boldsymbol {\nabla }}\Phi .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb0f6cc6ba41431f4aafc176d93e1d789408ce07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.772ex; height:5.509ex;" alt="{\displaystyle \mathbf {J} _{T}=\mathbf {J} -\varepsilon _{0}{\partial \over \partial t}{\boldsymbol {\nabla }}\Phi .}"></span></dd></dl> <p>Løsningen for det magnetiske vektorpotensialet ved bruk av Coulomb-gaugen kan derfor skrives på den retarderte formen<sup id="cite_ref-Brau_3-1" class="reference"><a href="#cite_note-Brau-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} (\mathbf {r} ,t)={\frac {\mu _{0}}{4\pi }}\int \!d^{3}x'{\frac {\mathbf {J} _{T}(\mathbf {r'} ,t')}{|\mathbf {r} -\mathbf {r'} |}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo>∫</mo> <mspace width="negativethinmathspace" /> <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> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>′</mo> </msup> </mrow> <mo>,</mo> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} (\mathbf {r} ,t)={\frac {\mu _{0}}{4\pi }}\int \!d^{3}x'{\frac {\mathbf {J} _{T}(\mathbf {r'} ,t')}{|\mathbf {r} -\mathbf {r'} |}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1660b77264622dd2e611c115129d7d53ef9e111e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:30.508ex; height:6.509ex;" alt="{\displaystyle \mathbf {A} (\mathbf {r} ,t)={\frac {\mu _{0}}{4\pi }}\int \!d^{3}x'{\frac {\mathbf {J} _{T}(\mathbf {r'} ,t')}{|\mathbf {r} -\mathbf {r'} |}}.}"></span></dd></dl> <p>Den avhenger av den transverse delen av strømtettheten som har sitt navn fra egenskapen </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 {J} _{T}={\boldsymbol {\nabla }}\cdot \mathbf {J} -\varepsilon _{0}{\partial \over \partial t}{\boldsymbol {\nabla }}^{2}\Phi ={\boldsymbol {\nabla }}\cdot \mathbf {J} +{\partial \rho \over \partial t}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>−</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Φ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>ρ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {J} _{T}={\boldsymbol {\nabla }}\cdot \mathbf {J} -\varepsilon _{0}{\partial \over \partial t}{\boldsymbol {\nabla }}^{2}\Phi ={\boldsymbol {\nabla }}\cdot \mathbf {J} +{\partial \rho \over \partial t}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eecd37949e53d5b98cfed662853b82df0796c60b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:46.832ex; height:5.676ex;" alt="{\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {J} _{T}={\boldsymbol {\nabla }}\cdot \mathbf {J} -\varepsilon _{0}{\partial \over \partial t}{\boldsymbol {\nabla }}^{2}\Phi ={\boldsymbol {\nabla }}\cdot \mathbf {J} +{\partial \rho \over \partial t}=0}"></span></dd></dl> <p>når man benytter ligningen for det skalare potensialet sammen med <a href="/wiki/Kontinuitetsligning" title="Kontinuitetsligning">kontinuitetsligningen</a> for den elektriske strømmen. </p><p>For en partikkel med ladning <i>q</i> og en hastighet <b>v</b> som er mye mindre enn <a href="/wiki/Lyshastigheten" class="mw-redirect" title="Lyshastigheten">lyshastigheten</a>, kan man se bort retardasjonen i beregning av vektorpotensialet. Mens det er <span class="nowrap"><b>A</b> = <i>μ</i><sub>0</sub><i>q</i> <b>v</b>/4<i>π r</i> </span> i Lorenz-gaugen, blir det nå i Coulomb-gaugen </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 {A} ={\mu _{0}q \over 8\pi r}\left[\mathbf {v} +{\mathbf {r} (\mathbf {r} \cdot \mathbf {v} ) \over r^{2}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>q</mi> </mrow> <mrow> <mn>8</mn> <mi>π</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</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">v</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ={\mu _{0}q \over 8\pi r}\left[\mathbf {v} +{\mathbf {r} (\mathbf {r} \cdot \mathbf {v} ) \over r^{2}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/838d8bfb7991aa6d4094105317fea4082e1e9e13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.53ex; height:6.343ex;" alt="{\displaystyle \mathbf {A} ={\mu _{0}q \over 8\pi r}\left[\mathbf {v} +{\mathbf {r} (\mathbf {r} \cdot \mathbf {v} ) \over r^{2}}\right]}"></span></dd></dl> <p>hvor <b>r</b> nå angir separasjonen mellom partikkelen og punktet der potensialet virker. Dette oppfyller betingelsen <span class="nowrap"><b>∇</b>⋅<b>A</b> = 0</span> som definerer dette gaugevalget. Den magnetiske <a href="/wiki/Elektrodynamikk#Darwin-vekselvirkningen" title="Elektrodynamikk">Darwin-vekselvirkningen</a> mellom to ladninger i bevegelse følger nesten direkte fra dette potensialet.<sup id="cite_ref-Jackson_4-0" class="reference"><a href="#cite_note-Jackson-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Poyntings_teorem">Poyntings teorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektromagnetisk_felt&veaction=edit&section=6" title="Rediger avsnitt: Poyntings teorem" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektromagnetisk_felt&action=edit&section=6" title="Rediger kildekoden til seksjonen Poyntings teorem"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Når det elektromagnetiske feltet har både en elektrisk <b>E</b> og en magnetisk komponent <b>B</b>, vil kun det elektriske feltet utføre et arbeid på partiklene i feltet. Er disse beskrevet ved hastighetsfeltet <span class="nowrap"><b>v</b> = <b>v</b>(<b>x</b>,<i>t</i>)</span>, er <a href="/wiki/Lorentz-kraft" title="Lorentz-kraft">Lorentz-kraften</a> som virker på hvert volumelement av denne partikkelfordelingen med ladningstetthet <span class="nowrap"><i>ρ</i> = <i>ρ</i>(<b>x</b>,<i>t</i>)</span>, gitt ved ligningen <span class="nowrap"><b>f</b> = <i>ρ</i>(<b>E</b> + <b>v</b> × <b>B</b>).</span> Arbeidet som den utfører i hvert volumelement og per tidsenhet er 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 \mathbf {f} \cdot \mathbf {v} =\mathbf {J} \cdot \mathbf {E} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {f} \cdot \mathbf {v} =\mathbf {J} \cdot \mathbf {E} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d1b297ed4f0c950511f904bc0919fd4c488ca53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.059ex; height:2.176ex;" alt="{\displaystyle \mathbf {f} \cdot \mathbf {v} =\mathbf {J} \cdot \mathbf {E} }"></span></dd></dl> <p>der <b>J</b> = <i>ρ</i> <b>v</b>  er partiklenes strømtetthet. Denne kan nå uttrykkes ved feltene ved bruk av Maxwell-ligningen <span class="nowrap"><b>∇</b> × <b>H</b> = <b>J</b> + ∂<b>D</b>/∂<i>t</i></span>. Da vil produktet <span class="nowrap"><b>E</b>⋅(<b>∇</b> × <b>H</b>) </span> oppstå. Dette kan omskrives ved bruk av identiten </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} \times \mathbf {H} )=\mathbf {H} \cdot ({\boldsymbol {\nabla }}\times \mathbf {E} )-\mathbf {E} \cdot ({\boldsymbol {\nabla }}\times \mathbf {H} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <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> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nabla }}\cdot (\mathbf {E} \times \mathbf {H} )=\mathbf {H} \cdot ({\boldsymbol {\nabla }}\times \mathbf {E} )-\mathbf {E} \cdot ({\boldsymbol {\nabla }}\times \mathbf {H} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b60363003fe60b0f7845b3e4f17db45b7927d83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.149ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {\nabla }}\cdot (\mathbf {E} \times \mathbf {H} )=\mathbf {H} \cdot ({\boldsymbol {\nabla }}\times \mathbf {E} )-\mathbf {E} \cdot ({\boldsymbol {\nabla }}\times \mathbf {H} )}"></span></dd></dl> <p>fra <a href="/wiki/Vektoranalyse" title="Vektoranalyse">vektoranalysen</a>. På samme måte kan man her benytte at <span class="nowrap"><b>∇</b> × <b>E</b> = - ∂<b>B</b>/∂<i>t</i></span>. De leddene som på den måten oppstår, kan ordnes og skrives på 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 -\mathbf {J} \cdot \mathbf {E} ={\partial u \over \partial t}+{\boldsymbol {\nabla }}\cdot (\mathbf {E} \times \mathbf {H} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>u</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\mathbf {J} \cdot \mathbf {E} ={\partial u \over \partial t}+{\boldsymbol {\nabla }}\cdot (\mathbf {E} \times \mathbf {H} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3b3cf37d860ae9c2997de59daddac6a2bd0a766" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.451ex; height:5.509ex;" alt="{\displaystyle -\mathbf {J} \cdot \mathbf {E} ={\partial u \over \partial t}+{\boldsymbol {\nabla }}\cdot (\mathbf {E} \times \mathbf {H} )}"></span></dd></dl> <p>og uttrykker energibalansen i dette systemet. Dette matematiske resultatet omtales som <b>Poyntings teorem</b>. I det første leddet oppter den skalare 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 u={1 \over 2}{\Big (}\mathbf {E} \cdot \mathbf {D} +\mathbf {B} \cdot \mathbf {H} {\Big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <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"> <mi mathvariant="bold">D</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</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 u={1 \over 2}{\Big (}\mathbf {E} \cdot \mathbf {D} +\mathbf {B} \cdot \mathbf {H} {\Big )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d96ffb47b9f83ac1f4bc5bc739981f2036b29cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.2ex; height:5.176ex;" alt="{\displaystyle u={1 \over 2}{\Big (}\mathbf {E} \cdot \mathbf {D} +\mathbf {B} \cdot \mathbf {H} {\Big )}}"></span></dd></dl> <p>som er den elektromagnetiske feltenergitettheten. I <a href="/wiki/Vakuum" title="Vakuum">vakuum</a> består den av <span class="nowrap"><i>u<sub>E</sub></i> = (1/2)<b>E⋅D</b></span> = <span class="nowrap"><i>ε</i><sub>0</sub><i>E</i><sup> 2</sup>/2</span> som er bidraget fra det <a href="/wiki/Elektrisk_felt#Elektrisk_feltenergi" title="Elektrisk felt">elektriske feltet</a> <b>E</b>, mens <span class="nowrap"><i>u<sub>B</sub></i> = (1/2)<b>B⋅H</b></span> = <span class="nowrap"><i>B</i><sup> 2</sup>/2<i>μ</i><sub>0</sub></span> er den delen som skyldes det <a href="/wiki/Magnetisk_felt#Magnetisk_energi" class="mw-redirect" title="Magnetisk felt">magnetiske feltet</a> <b>B</b>. Da denne utledningen av teoremet følger direkte fra Maxwells ligninger, er det også gyldig i et <a href="/wiki/Dielektrisk_materiale" title="Dielektrisk materiale">dielektrisk materiale</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Poyntings_vektor">Poyntings vektor</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektromagnetisk_felt&veaction=edit&section=7" title="Rediger avsnitt: Poyntings vektor" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektromagnetisk_felt&action=edit&section=7" title="Rediger kildekoden til seksjonen Poyntings vektor"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ved å sammenligne det matematiske resultatet for energibalansen med <a href="/wiki/Kontinuitetsligning" title="Kontinuitetsligning">kontinuitetsligningen</a> for elektrisk ladning, ser man samme struktur som denne på høyre siden av ligning. Mens venstre side beskriver hvor mye energi som overføres til partiklene i et lite volumelemnt, vil leddet ∂<i>u</i>/∂<i>t</i>  si hvor raskt denne energien tas fra selve feltet. Det resterende leddet <b>∇</b>⋅<b>S</b> hvor vektoren </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 {S} =\mathbf {E} \times \mathbf {H} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {S} =\mathbf {E} \times \mathbf {H} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12a535a7bdc520ab5cfa9e856b5d61dade695ff4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.92ex; height:2.509ex;" alt="{\displaystyle \mathbf {S} =\mathbf {E} \times \mathbf {H} ,}"></span></dd></dl> <p>sier hvor mye energi som strømmer ut av volumelementet. Den blir omtalt som <a href="/wiki/Poyntings_vektor" title="Poyntings vektor">Poyntings vektor</a>. Mens <i>u</i>  angir tettheten av elektromagnetiske feltenergi i hvert punkt, beskriver <b>S</b>  strømmen av denne energien. </p><p>Poyntings teorem sammenfatter bevarelse av elektromagnetisk energi på en kompakt form. Siden arbeidet <b>f</b>⋅<b>v</b> som feltet utfører, går med til å øke den indre energitettheten <i>w</i> til partiklene, kan man skrive teoremet på den alternative 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 {\partial \over \partial t}{\big (}u+w{\big )}+{\boldsymbol {\nabla }}\cdot \mathbf {S} =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mi>u</mi> <mo>+</mo> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\partial \over \partial t}{\big (}u+w{\big )}+{\boldsymbol {\nabla }}\cdot \mathbf {S} =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eaafb2e5fb099f597dc7d2eb86a8135978973b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.45ex; height:5.509ex;" alt="{\displaystyle {\partial \over \partial t}{\big (}u+w{\big )}+{\boldsymbol {\nabla }}\cdot \mathbf {S} =0}"></span></dd></dl> <p>som gjør mer tydelig at den totale energien til både feltet og partiklene holder seg konstant. Øker den ene, vil den andre komponenten avta like mye - og omvendt. </p> <div class="mw-heading mw-heading3"><h3 id="Elektromagnetisk_impulstetthet">Elektromagnetisk impulstetthet</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektromagnetisk_felt&veaction=edit&section=8" title="Rediger avsnitt: Elektromagnetisk impulstetthet" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektromagnetisk_felt&action=edit&section=8" title="Rediger kildekoden til seksjonen Elektromagnetisk impulstetthet"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mens <b>f</b>⋅<b>v</b>  angir hvor raskt energien til partiklene forandres gjennom deres vekselvirkning med feltet, vil selve volumkraften <span class="nowrap"><b>f</b> = <i>ρ</i> <b>E</b> + <b>J</b> × <b>B</b> </span> si hvor raskt deres impuls forandres med tiden. Det følger fra <a href="/wiki/Newtons_lover" class="mw-redirect" title="Newtons lover">Newtons andre lov</a> <span class="nowrap"><b>f</b> = ∂<b>P</b>/∂<i>t</i> </span> hvor <b>P</b> angir impulstettheten til partiklene. Igjen kan man uttrykke ladningstettheten <i>ρ</i>  og strømtettheten <b>J</b>  ved feltene gjennom bruk av Maxwells ligninger. Etter noen <a href="/wiki/Vektoranalyse" title="Vektoranalyse">vekoranalytiske</a> omforminger, finner man da resultatet kan skrives på formen<sup id="cite_ref-Griffiths_1-1" class="reference"><a href="#cite_note-Griffiths-1"><span class="cite-bracket">[</span>1<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 {\partial \over \partial t}{\big (}\mathbf {P} +\mathbf {G} {\big )}={\boldsymbol {\nabla }}\cdot {\boldsymbol {\sigma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">G</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\partial \over \partial t}{\big (}\mathbf {P} +\mathbf {G} {\big )}={\boldsymbol {\nabla }}\cdot {\boldsymbol {\sigma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f1844396b9feeaa214934c5cc2c0957bfb93027" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:20.49ex; height:5.509ex;" alt="{\displaystyle {\partial \over \partial t}{\big (}\mathbf {P} +\mathbf {G} {\big )}={\boldsymbol {\nabla }}\cdot {\boldsymbol {\sigma }}}"></span></dd></dl> <p>hvor vektoren <b>G</b> = <b>D</b> × <b>B</b> = <b>S</b>/<i>c</i><sup>2</sup>  må forstås som impulstettheten til det elektromagnetiske feltet. På høyre side av ligningen står <a href="/wiki/Divergens" title="Divergens">divergensen</a> av <a href="/w/index.php?title=Maxwells_spenningstensor&action=edit&redlink=1" class="new" title="Maxwells spenningstensor (ikke skrevet ennå)">Maxwells spenningstensor</a>. Den kan 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 \sigma _{ij}=E_{i}D_{j}+B_{i}H_{j}-{1 \over 2}\delta _{ij}{\big (}\mathbf {E} \cdot \mathbf {D} +\mathbf {B} \cdot \mathbf {H} {\big )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{ij}=E_{i}D_{j}+B_{i}H_{j}-{1 \over 2}\delta _{ij}{\big (}\mathbf {E} \cdot \mathbf {D} +\mathbf {B} \cdot \mathbf {H} {\big )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad06a97465786c9d293c04216d8d0e88136710b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.973ex; height:5.176ex;" alt="{\displaystyle \sigma _{ij}=E_{i}D_{j}+B_{i}H_{j}-{1 \over 2}\delta _{ij}{\big (}\mathbf {E} \cdot \mathbf {D} +\mathbf {B} \cdot \mathbf {H} {\big )}}"></span></dd></dl> <p>når man gjør bruk av <a href="/wiki/Kronecker-delta" title="Kronecker-delta">Kronecker-deltaet</a> <i>δ<sub>ij</sub></i>. I denne ligningen for bevarelse av den totale impulsen til systemet, kan da <a href="/wiki/Tensor" title="Tensor">tensoren</a> <span class="nowrap"><i>σ<sub>ij</sub></i> </span> forstås som <a href="/wiki/Fluks" title="Fluks">fluks</a> av elektromagnetisk impuls på samme måte som Poynting-vektoren uttrykker fluks av elektromagnetisk energi. </p> <div class="mw-heading mw-heading2"><h2 id="Kovariant_formulering">Kovariant formulering</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektromagnetisk_felt&veaction=edit&section=9" title="Rediger avsnitt: Kovariant formulering" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektromagnetisk_felt&action=edit&section=9" title="Rediger kildekoden til seksjonen Kovariant formulering"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Maxwellls teori for det elektromagnetiske feltet lå til grunn for Einsteins etablering av <a href="/wiki/Spesiell_relativitetsteori" class="mw-redirect" title="Spesiell relativitetsteori">spesiell relativitetsteori</a>. Det betyr at den også kan formuleres på <a href="/wiki/Kovariant_relativitetsteori" title="Kovariant relativitetsteori">kovariant</a> måte slik at den er den samme i alle <a href="/wiki/Inertialsystem" class="mw-redirect" title="Inertialsystem">inertialsystem</a>. Det elektriske potensialet Φ  og det magnetiske vektorpotensialet <b>A</b> utgjør da en firevektor <span class="nowrap"><i>A<sup>μ</sup></i> = (Φ/<i>c</i>, <b>A</b>)</span>. Komponentene til to feltene <b>E</b> og <b>B</b> inngår da i den antisymmetriske <a href="/wiki/Tensor" title="Tensor">tensoren</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <mo>−</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77fbf0c39a13f9706357f83b041774749598ded1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.137ex; height:2.843ex;" alt="{\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }}"></span></dd></dl> <p>hvor de kovariante komponentene av vektorpotensialet er <i>A<sub>μ</sub></i> = <i>η<sub>μν</sub>A<sup>ν</sup></i> når man benytter <a href="/wiki/Einsteins_summekonvensjon" title="Einsteins summekonvensjon">Einsteins summekonvensjon</a> og <a href="/wiki/Kovariant_relativitetsteori#Minkowski-metrikken" title="Kovariant relativitetsteori">Minkowski-metrikken</a> <i>η<sub>μν</sub></i> med diagonale komponenter (1, -1,-1,-1). </p><p>I mange sammenhenger er det hensiktsmessig å la de kovariante komponentene <i>A<sub>μ</sub></i> være komponenter av en <a href="/wiki/Differensialform#Eksempel:_Kovariant_elektromagnetisme" title="Differensialform">1-form</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{A}}=A_{\mu }{\text{d}}x^{\mu }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{A}}=A_{\mu }{\text{d}}x^{\mu }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8de134eac9ad5c9668a920e584cac32ac21864b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.301ex; height:3.009ex;" alt="{\displaystyle {\text{A}}=A_{\mu }{\text{d}}x^{\mu }.}"></span> Dens <a href="/wiki/Differensialform#Ytre_derivasjon" title="Differensialform">ytre deriverte</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{F}}={\text{d}}{\text{A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>F</mtext> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>A</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{F}}={\text{d}}{\text{A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/365cd8fbc600cc3951af3fd0c124ff780f0b6bc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.652ex; height:2.176ex;" alt="{\displaystyle {\text{F}}={\text{d}}{\text{A}}}"></span> er da en 2-form hvis komponenter er <i>F<sub> μν</sub></i>. Fra <a href="/wiki/Differensialform#Ytre_derivasjon" title="Differensialform">Poincairés lemma</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{d}}^{2}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{d}}^{2}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3251932769a900c97258e6d528bf82cd0e970479" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.608ex; height:2.676ex;" alt="{\displaystyle {\text{d}}^{2}=0}"></span> følger nå at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{d}}{\text{F}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>d</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>F</mtext> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{d}}{\text{F}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b21b715c950cab671cfa8b6e01c54d8504d5d85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.071ex; height:2.176ex;" alt="{\displaystyle {\text{d}}{\text{F}}=0}"></span> som gir to av Maxwellls to ligninger på formen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{\lambda }F_{\mu \nu }+\partial _{\nu }F_{\lambda \mu }+\partial _{\mu }F_{\nu \lambda }=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mi>μ</mi> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> <mi>λ</mi> </mrow> </msub> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{\lambda }F_{\mu \nu }+\partial _{\nu }F_{\lambda \mu }+\partial _{\mu }F_{\nu \lambda }=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95254b838545776a63e29972dc8710bc0c30dae6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.631ex; height:2.843ex;" alt="{\displaystyle \partial _{\lambda }F_{\mu \nu }+\partial _{\nu }F_{\lambda \mu }+\partial _{\mu }F_{\nu \lambda }=0.}"></span> </p><p>De kovariante komponentene til denne «Faraday-tensoren» 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 F_{\mu \nu }={\begin{pmatrix}0&E_{x}/c&E_{y}/c&E_{z}/c\\-E_{x}/c&0&-B_{z}&B_{y}\\-E_{y}/c&B_{z}&0&-B_{x}\\-E_{z}/c&-B_{y}&B_{x}&0\end{pmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mtd> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mtd> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mtd> <mtd> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> </mtd> <mtd> <mo>−</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\mu \nu }={\begin{pmatrix}0&E_{x}/c&E_{y}/c&E_{z}/c\\-E_{x}/c&0&-B_{z}&B_{y}\\-E_{y}/c&B_{z}&0&-B_{x}\\-E_{z}/c&-B_{y}&B_{x}&0\end{pmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1078b5d4d00e62b07a4ac38fd5000f48ff19f177" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:40.863ex; height:13.509ex;" alt="{\displaystyle F_{\mu \nu }={\begin{pmatrix}0&E_{x}/c&E_{y}/c&E_{z}/c\\-E_{x}/c&0&-B_{z}&B_{y}\\-E_{y}/c&B_{z}&0&-B_{x}\\-E_{z}/c&-B_{y}&B_{x}&0\end{pmatrix}}.}"></span></dd></dl> <p>Herav finnes også de kontravariante komponentene <i>F<sup> μν</sup> = η<sup>μρ</sup>η<sup>νσ</sup>F<sub>ρσ</sub></i> ved å skifte fortegn på de elektriske komponentene i denne matrisen.<sup id="cite_ref-Jackson_4-1" class="reference"><a href="#cite_note-Jackson-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Feltligninger">Feltligninger</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektromagnetisk_felt&veaction=edit&section=10" title="Rediger avsnitt: Feltligninger" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektromagnetisk_felt&action=edit&section=10" title="Rediger kildekoden til seksjonen Feltligninger"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>De to andre av Maxwells ligninger kan kalles bevegelsesligningene til det elektromagnetiske feltet. Mest direkte kan de utledes fra <a href="/wiki/Hamiltons_virkningsprinsipp" title="Hamiltons virkningsprinsipp">Hamiltons virkningsprinsipp</a> basert på en fundamental <a href="/wiki/Hamiltons_virkningsprinsipp#Elektromagnetiske_felt" title="Hamiltons virkningsprinsipp">Lagrange-funksjon</a> for feltene koblet til elektriske strømmer <span class="nowrap"><b>J</b> = <b>J</b>(<b>r</b>,<i>t</i>)</span> og ladninger <span class="nowrap"><i>ρ</i> = <i>ρ</i>(<b>r</b>,<i>t</i>).</span> Den 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 {\mathcal {L}}={\varepsilon _{0} \over 2}\mathbf {E} ^{2}-{1 \over 2\mu _{0}}\mathbf {B} ^{2}-\rho \Phi +\mathbf {J} \cdot \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mn>2</mn> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>ρ</mi> <mi mathvariant="normal">Φ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\varepsilon _{0} \over 2}\mathbf {E} ^{2}-{1 \over 2\mu _{0}}\mathbf {B} ^{2}-\rho \Phi +\mathbf {J} \cdot \mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d703e9ec228bb735e0ba3c20c429869dacc60f10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.378ex; height:5.676ex;" alt="{\displaystyle {\mathcal {L}}={\varepsilon _{0} \over 2}\mathbf {E} ^{2}-{1 \over 2\mu _{0}}\mathbf {B} ^{2}-\rho \Phi +\mathbf {J} \cdot \mathbf {A} }"></span></dd></dl> <p>som nå kan skrives på den mer kompakte, kovariante 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 {\mathcal {L}}=-{1 \over 4\mu _{0}}F_{\mu \nu }F^{\mu \nu }-J^{\mu }A_{\mu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">L</mi> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}=-{1 \over 4\mu _{0}}F_{\mu \nu }F^{\mu \nu }-J^{\mu }A_{\mu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2a45ead49fcfa8735232842686f193c63743bf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.02ex; height:5.676ex;" alt="{\displaystyle {\mathcal {L}}=-{1 \over 4\mu _{0}}F_{\mu \nu }F^{\mu \nu }-J^{\mu }A_{\mu }}"></span></dd></dl> <p>etter å ha definert den elektriske firestrømmen som <span class="nowrap"><i>J<sup> μ</sup></i> = (<i>ρc</i>, <b>J</b>).</span> </p><p>Under en variasjon <i>A<sub>μ</sub></i> → <i>A<sub>μ</sub></i> + <i>δA<sub>μ</sub></i> av vektorpotensialet sier Hamiltons virkningsprinsipp at den totale virkningen skal være stasjonær. Det vil si at den resulterende variasjonen </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 \delta S=\delta \!\int \!d^{4}x\left[-{1 \over 4\mu _{0}}F_{\mu \nu }F^{\mu \nu }-J^{\mu }A_{\mu }\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>S</mi> <mo>=</mo> <mi>δ</mi> <mspace width="negativethinmathspace" /> <mo>∫</mo> <mspace width="negativethinmathspace" /> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>x</mi> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>4</mn> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msup> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta S=\delta \!\int \!d^{4}x\left[-{1 \over 4\mu _{0}}F_{\mu \nu }F^{\mu \nu }-J^{\mu }A_{\mu }\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9bc8a1424ef74554653f92bbadeb61cc531c0a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.65ex; height:6.176ex;" alt="{\displaystyle \delta S=\delta \!\int \!d^{4}x\left[-{1 \over 4\mu _{0}}F_{\mu \nu }F^{\mu \nu }-J^{\mu }A_{\mu }\right]}"></span></dd></dl> <p>skal være null. Her 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 \delta (F_{\mu \nu }F^{\mu \nu })=2F^{\mu \nu }\delta F_{\mu \nu }=4F^{\mu \nu }\partial _{\mu }\delta A_{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <mi>δ</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msub> <mo>=</mo> <mn>4</mn> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <mi>δ</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (F_{\mu \nu }F^{\mu \nu })=2F^{\mu \nu }\delta F_{\mu \nu }=4F^{\mu \nu }\partial _{\mu }\delta A_{\nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49111bae4fc8d240324d7cf55f54103f19349b75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.688ex; height:3.009ex;" alt="{\displaystyle \delta (F_{\mu \nu }F^{\mu \nu })=2F^{\mu \nu }\delta F_{\mu \nu }=4F^{\mu \nu }\partial _{\mu }\delta A_{\nu }}"></span></dd></dl> <p>slik 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 \delta S=\int \!d^{4}x\left[{1 \over \mu _{0}}\partial _{\mu }F^{\mu \nu }-J^{\nu }\right]\delta A_{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>S</mi> <mo>=</mo> <mo>∫</mo> <mspace width="negativethinmathspace" /> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>x</mi> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mi>δ</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta S=\int \!d^{4}x\left[{1 \over \mu _{0}}\partial _{\mu }F^{\mu \nu }-J^{\nu }\right]\delta A_{\nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c4375fcad85cf567960faca14963c8f0eb22f43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.696ex; height:6.176ex;" alt="{\displaystyle \delta S=\int \!d^{4}x\left[{1 \over \mu _{0}}\partial _{\mu }F^{\mu \nu }-J^{\nu }\right]\delta A_{\nu }}"></span></dd></dl> <p>etter en partielll integrasjon i første delen hvor overflateleddet kan sees bort fra. Dette gir de resterende to andre av Maxwell ligninger på den kovariante 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 \partial _{\mu }F^{\mu \nu }=\mu _{0}J^{\nu }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> </mrow> </msub> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mi>ν</mi> </mrow> </msup> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{\mu }F^{\mu \nu }=\mu _{0}J^{\nu }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64280a1794049909a4b2013fc9757fcac6ecf923" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.551ex; height:3.009ex;" alt="{\displaystyle \partial _{\mu }F^{\mu \nu }=\mu _{0}J^{\nu }}"></span></dd></dl> <p>For komponenten med <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 \nu =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7229c47b5bc20ef0a1371a4f3c09459ccb6909ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.493ex; height:2.176ex;" alt="{\displaystyle \nu =0}"></span> gir dette <a href="/wiki/Gauss%27_lov" class="mw-redirect" title="Gauss' lov">Gauss' lov</a> for det elektriske feltet når man benytter definisjonen <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^{2}=1/\mu _{0}\varepsilon _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{2}=1/\mu _{0}\varepsilon _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2610851db35f77ba272eabfcb674595e377ff6e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.078ex; height:3.176ex;" alt="{\displaystyle c^{2}=1/\mu _{0}\varepsilon _{0}}"></span> av <a href="/wiki/Lyshastighet" class="mw-redirect" title="Lyshastighet">lyshastigheten</a>, mens de andre komponentene gir <a href="/wiki/Amp%C3%A8res_sirkulasjonslov" title="Ampères sirkulasjonslov">Ampères sirkulasjonslov</a> for det magnetiske feltet. </p> <div class="mw-heading mw-heading2"><h2 id="Plane_bølger"><span id="Plane_b.C3.B8lger"></span>Plane bølger</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektromagnetisk_felt&veaction=edit&section=11" title="Rediger avsnitt: Plane bølger" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektromagnetisk_felt&action=edit&section=11" title="Rediger kildekoden til seksjonen Plane bølger"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>I områder av rommet hvor det er hverken elektriske ladninger eller strømmer, vil det elektromagnetiske feltet bre seg utover som en fri <a href="/wiki/B%C3%B8lge" title="Bølge">bølge</a> eller en <a href="/wiki/Superposisjon" class="mw-redirect" title="Superposisjon">superposisjon</a> av slike. Den elektriske komponenten vil oppfylle <a href="/wiki/B%C3%B8lgeligning#Elektromagnetiske_bølger" title="Bølgeligning">bølgeligningen</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\nabla }}^{2}\mathbf {E} -{1 \over c^{2}}{\partial ^{2}\mathbf {E} \over \partial t^{2}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nabla }}^{2}\mathbf {E} -{1 \over c^{2}}{\partial ^{2}\mathbf {E} \over \partial t^{2}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/517774ffd016cdd9f9d5586dc1012349a2a41d4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.027ex; height:6.009ex;" alt="{\displaystyle {\boldsymbol {\nabla }}^{2}\mathbf {E} -{1 \over c^{2}}{\partial ^{2}\mathbf {E} \over \partial t^{2}}=0}"></span></dd></dl> <p>og tilsvarende for det magnetiske feltet som opptrer samtidig via Maxwell-ligningen <b>∇</b> × <b>E</b> = - ∂<b>B</b>/∂<i>t</i>. Har bølgen <a href="/wiki/Vinkelfrekvens" title="Vinkelfrekvens">vinkelfrekvensen</a> <i>ω</i> = <i>c</i> |<b>k</b>|  hvor <b>k</b> er <a href="/wiki/B%C3%B8lge" title="Bølge">bølgevektoren</a>, vil den enkleste, <a href="/wiki/B%C3%B8lge#Plane_bølger" title="Bølge">plane bølge</a> kunne 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 \mathbf {E} (\mathbf {r} ,t)=\mathbf {E} _{0}\cos(\mathbf {k} \cdot \mathbf {r} -\omega t)}"> <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>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} (\mathbf {r} ,t)=\mathbf {E} _{0}\cos(\mathbf {k} \cdot \mathbf {r} -\omega t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf95e50ac9b91c1e08b3fab33946e0bfed0f4838" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.077ex; height:2.843ex;" alt="{\displaystyle \mathbf {E} (\mathbf {r} ,t)=\mathbf {E} _{0}\cos(\mathbf {k} \cdot \mathbf {r} -\omega t)}"></span></dd></dl> <p>hvor den konstante vektoren <b>E</b><sub>0</sub>  angir dens <a href="/wiki/Amplitude" title="Amplitude">amplitude</a>. Bølgen brer seg i retning <b>k</b> og fra <span class="nowrap"><b>∇</b>⋅<b>E</b> = 0</span> følger at <span class="nowrap"><b>k</b>⋅<b>E</b> = 0</span>. Den elektriske feltvektoren står derfor <a href="/wiki/Vinkelrett" title="Vinkelrett">vinkelrett</a> på utbredelsesretningen, noe som karakteriserer en <b>transvers</b> bølge. Denne transvere egenskapen har også det magnetiske feltet som er gitt ved <span class="nowrap"><b>B</b> = <b>k</b> × <b>E</b>/<i>ω</i></span>. I vakum er <b>H</b> = <b>B</b>/<i>μ</i><sub>0</sub> slik at forholdet mellom det elektriske og det magnetiske feltet til bølgen er <span class="nowrap"><i>E</i>/<i>H</i> = <i>μ</i><sub>0</sub><i>c</i></span>. Denne størrelsen kalles vanligvis for «rommets bølgemotstand» eller <b>vakumimpedansen</b> <i>Z</i><sub>0</sub>. Settes inn størrelsene for den <a href="/wiki/Permittivitet" title="Permittivitet">elektriske</a> <i>ε</i><sub>0</sub>  og den <a href="/wiki/Permeabilitet" class="mw-disambig" title="Permeabilitet">magnetiske konstanten</a> <i>μ</i><sub>0</sub>, finner man at den har en verdi </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 Z_{0}=\mu _{0}c={\sqrt {\mu _{0} \over \varepsilon _{0}}}=377\,{\text{ohm}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>c</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </msqrt> </mrow> <mo>=</mo> <mn>377</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>ohm</mtext> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z_{0}=\mu _{0}c={\sqrt {\mu _{0} \over \varepsilon _{0}}}=377\,{\text{ohm}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22819522274276986b51a11f5ffc6cc8fdb808ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.928ex; height:6.176ex;" alt="{\displaystyle Z_{0}=\mu _{0}c={\sqrt {\mu _{0} \over \varepsilon _{0}}}=377\,{\text{ohm}}.}"></span></dd></dl> <p>Det er den som bestemmer <a href="/wiki/Str%C3%A5lingsmotstand" title="Strålingsmotstand">strålingsmotstanden</a> i <a href="/wiki/Antenne" title="Antenne">antenner</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Energi_og_impuls">Energi og impuls</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektromagnetisk_felt&veaction=edit&section=12" title="Rediger avsnitt: Energi og impuls" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektromagnetisk_felt&action=edit&section=12" title="Rediger kildekoden til seksjonen Energi og impuls"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Den midlere energitetthet i en slik plan bølge med elektrisk amplitude <b>E</b><sub>0</sub> 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 \langle u\rangle ={1 \over 2}\langle \mathbf {E} \cdot \mathbf {D} +\mathbf {B} \cdot \mathbf {H} \rangle ={1 \over 2}\varepsilon _{0}E_{0}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>u</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <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> <msubsup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle u\rangle ={1 \over 2}\langle \mathbf {E} \cdot \mathbf {D} +\mathbf {B} \cdot \mathbf {H} \rangle ={1 \over 2}\varepsilon _{0}E_{0}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dddb39b06244b2d2a4df8d7b2b26ab8b1cedd24b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.126ex; height:5.176ex;" alt="{\displaystyle \langle u\rangle ={1 \over 2}\langle \mathbf {E} \cdot \mathbf {D} +\mathbf {B} \cdot \mathbf {H} \rangle ={1 \over 2}\varepsilon _{0}E_{0}^{2}}"></span></dd></dl> <p>da faktoren cos<sup>2</sup> i middel gir 1/2. På samme måte blir den midlere verdien av Poynting-vektoren </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 \langle \mathbf {S} \rangle =c\langle u\rangle {\widehat {\mathbf {k} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>c</mi> <mo fence="false" stretchy="false">⟨</mo> <mi>u</mi> <mo fence="false" stretchy="false">⟩</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \mathbf {S} \rangle =c\langle u\rangle {\widehat {\mathbf {k} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7516658aa9f0ab83ef94873c806a5ba1aded784" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.95ex; height:3.343ex;" alt="{\displaystyle \langle \mathbf {S} \rangle =c\langle u\rangle {\widehat {\mathbf {k} }}}"></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 {\widehat {\mathbf {k} }}}"> <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">k</mi> </mrow> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\mathbf {k} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1016ce512a1faca3fa0b39d0293c9430c39390f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:2.843ex;" alt="{\displaystyle {\widehat {\mathbf {k} }}}"></span> er i samme retning som bølgevektoren <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 {k} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {k} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ea699cbc1f843f2e855577d57529430ec33a1ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:2.176ex;" alt="{\displaystyle \mathbf {k} }"></span>. Dette er i overensstemmelse med at <b>S</b> beskriver en strøm av energi med tetthet <<i>u</i>> som transporteres med lyshastigheten <i>c</i>. Dette gir nå direkte impulstettheten i bølgen 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 \langle \mathbf {G} \rangle ={\langle u\rangle \over c}{\widehat {\mathbf {k} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">G</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>u</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mi>c</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \mathbf {G} \rangle ={\langle u\rangle \over c}{\widehat {\mathbf {k} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1821f9fa22789b6eefb5e5fc135fc2925ee8bd51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.395ex; height:5.676ex;" alt="{\displaystyle \langle \mathbf {G} \rangle ={\langle u\rangle \over c}{\widehat {\mathbf {k} }}}"></span></dd></dl> <p>Når det elektromagnetiske feltet <a href="/wiki/Kvantemekanikk" title="Kvantemekanikk">kvantiseres</a> i <a href="/wiki/Kvanteelektrodynamikk" title="Kvanteelektrodynamikk">kvanteelektrodynamikken</a>, vil denne plane bølgen beskrives ved <a href="/wiki/Foton" title="Foton">fotoner</a> som hver har energien <span class="nowrap"><i>ħω</i></span> uttrykt ved den reduserte <a href="/wiki/Plancks_konstant" title="Plancks konstant">Planck-konstanten</a> <i>ħ</i> = <i>h</i>/2<i>π</i>. Hvis den romlige tettheten av fotoner er <i>n</i>, vil man da ha sammenhengene <span class="nowrap"><<i>u</i>> = <i>nħω</i></span> og <span class="nowrap"><<i>G</i>> = <i>nħk</i></span>  da <i>ω</i>/<i>c</i> = <i>k</i>. Det tilsvarer at hvert foton har impulsen <b>p</b> = <i>ħ</i> <b>k</b> som ble først utledet av <a href="/wiki/Einstein" class="mw-redirect" title="Einstein">Einstein</a> i hans forklaring av <a href="/wiki/Varmestr%C3%A5ling" title="Varmestråling">varmestrålingen</a>.<sup id="cite_ref-SMM_5-0" class="reference"><a href="#cite_note-SMM-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Polarisasjon">Polarisasjon</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektromagnetisk_felt&veaction=edit&section=13" title="Rediger avsnitt: Polarisasjon" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektromagnetisk_felt&action=edit&section=13" title="Rediger kildekoden til seksjonen Polarisasjon"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Denne plane bølgen har en elektrisk feltvektor som oscillerer langs en konstant retning gitt ved amplituden <b>E</b><sub>0</sub>. Av denne grunn sies bølgen å være <a href="/wiki/Polarisering_(elektromagnetisme)" title="Polarisering (elektromagnetisme)">planpolarisert</a>. Mer generelt vil retningen til denne feltvektoren ikke være konstant og bølgen sies å være <b>elliptisk polarisert</b>. Det kan enklest anskueliggjøres ved å betrakte en bølge som beveger seg langs <i>z</i>-aksen. Feltvektoren vil derfor ha en komponent <i>E<sub>x</sub></i> langs <i>x</i>-aksen og en komponent <i>E<sub>y</sub></i> langs <i>y</i>-aksen.<sup id="cite_ref-HLL_6-0" class="reference"><a href="#cite_note-HLL-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Hver komponent oppfyller den samme bølgeligningen, men løsningen av denne vil i alminnelighet inneholde en faseforskjell <i>φ</i> mellom disse to komponentene. Dermed vil det elektriske feltet for denne plane bølgen generelt 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 \mathbf {E} (z,t)=E_{0x}\cos(kz-\omega t)\mathbf {e} _{x}+E_{0y}\cos(kz-\omega t+\phi )\mathbf {e} _{y}}"> <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> <mi>z</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mi>x</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>z</mi> <mo>−</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mi>y</mi> </mrow> </msub> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>z</mi> <mo>−</mo> <mi>ω</mi> <mi>t</mi> <mo>+</mo> <mi>ϕ</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} (z,t)=E_{0x}\cos(kz-\omega t)\mathbf {e} _{x}+E_{0y}\cos(kz-\omega t+\phi )\mathbf {e} _{y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67acf477f9f84fbb24404eb3e21aa669b9e4774c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:54.126ex; height:3.009ex;" alt="{\displaystyle \mathbf {E} (z,t)=E_{0x}\cos(kz-\omega t)\mathbf {e} _{x}+E_{0y}\cos(kz-\omega t+\phi )\mathbf {e} _{y}}"></span></dd></dl> <p>hvor <b>e</b><sub><i>x</i></sub>  og <b>e</b><sub><i>y</i></sub>  er basisvektorer i de to transverse retningene. Komponentene <i>E</i><sub><i>x</i></sub>  og <i>E</i><sub><i>y</i></sub>  til feltvektoren vil nå alltid oppfylle ligningen </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 {\Big (}{E_{x} \over E_{0x}}{\Big )}^{2}+{\Big (}{E_{y} \over E_{0y}}{\Big )}^{2}-2{\Big (}{E_{x} \over E_{0x}}{\Big )}{\Big (}{E_{y} \over E_{0y}}{\Big )}\cos \phi =\sin ^{2}\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mi>x</mi> </mrow> </msub> </mfrac> </mrow> <msup> <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"> <mn>2</mn> </mrow> </msup> <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"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mi>y</mi> </mrow> </msub> </mfrac> </mrow> <msup> <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"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <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"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mi>x</mi> </mrow> </msub> </mfrac> </mrow> <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"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mi>y</mi> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>ϕ</mi> <mo>=</mo> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\Big (}{E_{x} \over E_{0x}}{\Big )}^{2}+{\Big (}{E_{y} \over E_{0y}}{\Big )}^{2}-2{\Big (}{E_{x} \over E_{0x}}{\Big )}{\Big (}{E_{y} \over E_{0y}}{\Big )}\cos \phi =\sin ^{2}\phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c7944c9b98da0145d6539f697781b1a9009211f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:52.042ex; height:6.176ex;" alt="{\displaystyle {\Big (}{E_{x} \over E_{0x}}{\Big )}^{2}+{\Big (}{E_{y} \over E_{0y}}{\Big )}^{2}-2{\Big (}{E_{x} \over E_{0x}}{\Big )}{\Big (}{E_{y} \over E_{0y}}{\Big )}\cos \phi =\sin ^{2}\phi }"></span></dd></dl> <p>som viser at de beveger seg på en <a href="/wiki/Ellipse" title="Ellipse">ellipse</a> med hovedakser som danner en viss vinkel med koordinataksene. I det generelle tilfellet sies derfor bølgen å være elliptisk polarisert. Ellipsen degenerer til en rett linje for det spesielle tilfellet <span class="nowrap"><i>φ</i> = 0</span> som gir en planpolarisert bølge. Når <span class="nowrap"><i>φ</i> = ± <i>π</i> /2</span> faller ellipseaksene sammen med koordinataksene. For denne faseforskyvningen går ellipsen over til å bli en <a href="/wiki/Sirkel" title="Sirkel">sirkel</a> når <span class="nowrap"><i>E</i><sub>0<i>x</i></sub> = <i>E</i><sub>0<i>y</i></sub> </span> og bølgen sies å være <a href="/wiki/Polarisering_(elektromagnetisme)" title="Polarisering (elektromagnetisme)">sirkulært polarisert</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Elektromagnetisk_stråling"><span id="Elektromagnetisk_str.C3.A5ling"></span>Elektromagnetisk stråling</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektromagnetisk_felt&veaction=edit&section=14" title="Rediger avsnitt: Elektromagnetisk stråling" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektromagnetisk_felt&action=edit&section=14" title="Rediger kildekoden til seksjonen Elektromagnetisk stråling"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mens det elektriske feltet fra en ladning i ro avtar med avstanden som 1/<i>r</i><sup> 2</sup>, vil det kunne avta som 1/<i>r</i> i stor avstand fra en strømfordeling som varierer med tiden. Dette kalles et <b>strålingsfelt</b> og kan beregnes fra den inhomogene bølgeligningen. Det gjøres enklest i <a href="/wiki/Gaugetransformasjon#Lorenz-gauge" title="Gaugetransformasjon">Lorenz-gaugen</a> hvor vektorpotensialet 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 \mathbf {A} (\mathbf {r} ,t)={\mu _{0} \over 4\pi }\int d^{3}x'{\frac {\mathbf {J} (\mathbf {r'} ,t')}{|\mathbf {r} -\mathbf {r'} |}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>′</mo> </msup> </mrow> <mo>,</mo> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>′</mo> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} (\mathbf {r} ,t)={\mu _{0} \over 4\pi }\int d^{3}x'{\frac {\mathbf {J} (\mathbf {r'} ,t')}{|\mathbf {r} -\mathbf {r'} |}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a202bc61d31f3d56e0239235cf68335ba11c8e94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.859ex; height:6.509ex;" alt="{\displaystyle \mathbf {A} (\mathbf {r} ,t)={\mu _{0} \over 4\pi }\int d^{3}x'{\frac {\mathbf {J} (\mathbf {r'} ,t')}{|\mathbf {r} -\mathbf {r'} |}}}"></span></dd></dl> <p>hvor <i>t' </i> = <i>t</i> - |<b>r</b> - <b>r' </b>|/<i>c </i> er den <a href="/wiki/Helmholtz-ligning#Eksempel" class="mw-redirect" title="Helmholtz-ligning">retarderte tiden</a>. Når feltpunktet <b>r</b> ligger langt fra borte fra kildepunktet <b>r'</b>, er det nærliggende å tro at integralet kan forenkles 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 \mathbf {A} (\mathbf {r} ,t)={\mu _{0} \over 4\pi r}\int \!d^{3}x'\mathbf {J} (\mathbf {r'} ,t-r/c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mn>4</mn> <mi>π</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo>∫</mo> <mspace width="negativethinmathspace" /> <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"> <mi mathvariant="bold">J</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">r</mi> <mo>′</mo> </msup> </mrow> <mo>,</mo> <mi>t</mi> <mo>−</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} (\mathbf {r} ,t)={\mu _{0} \over 4\pi r}\int \!d^{3}x'\mathbf {J} (\mathbf {r'} ,t-r/c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13fab514a8ea762f0623ed08a7bc2a6db8cf8f04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.058ex; height:5.676ex;" alt="{\displaystyle \mathbf {A} (\mathbf {r} ,t)={\mu _{0} \over 4\pi r}\int \!d^{3}x'\mathbf {J} (\mathbf {r'} ,t-r/c)}"></span></dd></dl> <p>Men for at dette skal være riktig, må <b>r'</b> ikke forandre seg mye under integrasjonen. Hvis strømkilden derfor består av ladete partikler, må disse bevege seg med ikke-relativistisk hastigheter <i>v</i> << <i>c</i>. Under denne forutsetning kan denne formelen for vektorpotensialet benyttes i i mange sammenhenger for å beregne elektromagnetisk stråling.<sup id="cite_ref-Jackson_4-2" class="reference"><a href="#cite_note-Jackson-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Stråling_fra_punktpartikkel"><span id="Str.C3.A5ling_fra_punktpartikkel"></span>Stråling fra punktpartikkel</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektromagnetisk_felt&veaction=edit&section=15" title="Rediger avsnitt: Stråling fra punktpartikkel" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektromagnetisk_felt&action=edit&section=15" title="Rediger kildekoden til seksjonen Stråling fra punktpartikkel"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>En liten ladning <i>q</i> som beveger seg med hastighet <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 {v} ={\dot {\mathbf {r} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</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>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} ={\dot {\mathbf {r} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b196d3e4d97a8fbf318dbc860c923d3e4971d1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.672ex; height:2.176ex;" alt="{\displaystyle \mathbf {v} ={\dot {\mathbf {r} }}}"></span> kan beskrives som en <a href="/wiki/Kontinuitetsligning#Punktpartikler" title="Kontinuitetsligning">punktpartikkel</a>. Den gir opphav til strømtettheten </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 {J} (\mathbf {r} ,t)=q\mathbf {v} (t)\delta (\mathbf {r} -\mathbf {r} (t))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <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"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J} (\mathbf {r} ,t)=q\mathbf {v} (t)\delta (\mathbf {r} -\mathbf {r} (t))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0b65d5e545a2d6fd5ff46e16b920b2c5a5ade3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.945ex; height:2.843ex;" alt="{\displaystyle \mathbf {J} (\mathbf {r} ,t)=q\mathbf {v} (t)\delta (\mathbf {r} -\mathbf {r} (t))}"></span></dd></dl> <p>hvor <a href="/wiki/Diracs_deltafunksjon" title="Diracs deltafunksjon">Diracs <i>δ</i>-funksjon</a> uttrykker at partikkelens utstrekning er forsvinnende liten. Vektorpotensialet finnes nå direkte 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 {A} (\mathbf {r} ,t)={q\mu _{0} \over 4\pi r}\mathbf {v} (t-r/c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} (\mathbf {r} ,t)={q\mu _{0} \over 4\pi r}\mathbf {v} (t-r/c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/089c3e8dc61b6c2f552fe79cf1960d8a7adc8282" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.4ex; height:4.843ex;" alt="{\displaystyle \mathbf {A} (\mathbf {r} ,t)={q\mu _{0} \over 4\pi r}\mathbf {v} (t-r/c)}"></span></dd></dl> <p>Den retarderte tiden viser her at en forandring i hastigheten tar en tid <i>r</i>/<i>c</i> for å nå frem til feltpunktet som ligger i en avstand <i>r </i> fra partikkelen. </p><p>Magnetfeltet finnes fra definisjonen <span class="nowrap"><b>B</b>(<b>r</b>,<i>t</i>) = <b>∇</b> × <b>A</b>(<b>r</b>,<i>t</i>)</span> som får to bidrag. Det første kommer fra derivasjon av faktoren <span class="nowrap">1/<i>r</i></span>. Det resulterer i et ledd som går som <span class="nowrap">1/<i>r</i><sup> 2</sup> </span> og kan derfor neglisjeres ved store avstander. Men det er også en <i>r</i> -avhengighet i den retarderte tiden slik 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 }}t'=-{1 \over c}{\boldsymbol {\nabla }}r=-{\mathbf {n} \over c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>c</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mi>r</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mi>c</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nabla }}t'=-{1 \over c}{\boldsymbol {\nabla }}r=-{\mathbf {n} \over c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3283abee102f54eb01c113549b355a4ccae1d993" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.159ex; height:5.176ex;" alt="{\displaystyle {\boldsymbol {\nabla }}t'=-{1 \over c}{\boldsymbol {\nabla }}r=-{\mathbf {n} \over c}}"></span></dd></dl> <p>hvor enhetsvektoren <b>n</b> = <b>r</b>/<i>r</i> peker mot feltpunktet i retning <b>r</b>. Ved å bruke <a href="/wiki/Kjerneregelen" class="mw-redirect" title="Kjerneregelen">kjerneregelen</a> for derivasjon får man dermed for magnetfeltet </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} (\mathbf {r} ,t)={q\mu _{0} \over 4\pi r}{\boldsymbol {\nabla }}t'\times {d\mathbf {v} \over dt'}(t')=-{q\mu _{0} \over 4\pi rc}{\mathbf {n} \times {\dot {\mathbf {v} }}}(t-r/c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mo>′</mo> </msup> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <mi>r</mi> <mi>c</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} (\mathbf {r} ,t)={q\mu _{0} \over 4\pi r}{\boldsymbol {\nabla }}t'\times {d\mathbf {v} \over dt'}(t')=-{q\mu _{0} \over 4\pi rc}{\mathbf {n} \times {\dot {\mathbf {v} }}}(t-r/c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2d687e6973f7a83b806106e24a19b81a9e3c0e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:52.401ex; height:5.509ex;" alt="{\displaystyle \mathbf {B} (\mathbf {r} ,t)={q\mu _{0} \over 4\pi r}{\boldsymbol {\nabla }}t'\times {d\mathbf {v} \over dt'}(t')=-{q\mu _{0} \over 4\pi rc}{\mathbf {n} \times {\dot {\mathbf {v} }}}(t-r/c)}"></span></dd></dl> <p>hvor <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 {\dot {\mathbf {v} }}=\mathbf {a} }"> <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">v</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\mathbf {v} }}=\mathbf {a} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34530cddb7e6f90a48e2f3acd23278d5fdedce0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.809ex; height:2.176ex;" alt="{\displaystyle {\dot {\mathbf {v} }}=\mathbf {a} }"></span> er <a href="/wiki/Akselerasjon" title="Akselerasjon">akselerasjonen</a> til partikkelen. Dette bidraget varierer som <span class="nowrap">1/<i>r</i></span> og er det magnetiske strålingsfeltet.<sup id="cite_ref-Griffiths_1-2" class="reference"><a href="#cite_note-Griffiths-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Elektrisk_strålingsfelt"><span id="Elektrisk_str.C3.A5lingsfelt"></span>Elektrisk strålingsfelt</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektromagnetisk_felt&veaction=edit&section=16" title="Rediger avsnitt: Elektrisk strålingsfelt" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektromagnetisk_felt&action=edit&section=16" title="Rediger kildekoden til seksjonen Elektrisk strålingsfelt"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Det elektriske strålingsfeltet kan finnes fra betingelsen </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 {A} +{1 \over c^{2}}{\partial \Phi \over \partial t}=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"> <mi mathvariant="bold">A</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {A} +{1 \over c^{2}}{\partial \Phi \over \partial t}=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5f9846b95be5979a9e166199e9a90ffe9719b39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.402ex; height:5.676ex;" alt="{\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {A} +{1 \over c^{2}}{\partial \Phi \over \partial t}=0.}"></span></dd></dl> <p>som definerer <a href="/wiki/Gaugetransformasjon#Lorenz-gauge" title="Gaugetransformasjon">Lorenz-gaugen</a>. Her er nå på samme måte </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^{2}{\boldsymbol {\nabla }}\cdot \mathbf {A} =-{q\mu _{0}c \over 4\pi r}\mathbf {n} \cdot {\dot {\mathbf {v} }}=-{\partial \Phi \over \partial t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>c</mi> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c^{2}{\boldsymbol {\nabla }}\cdot \mathbf {A} =-{q\mu _{0}c \over 4\pi r}\mathbf {n} \cdot {\dot {\mathbf {v} }}=-{\partial \Phi \over \partial t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/885597efafe98394b1e94384699385ef4a9c4477" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:31.575ex; height:5.509ex;" alt="{\displaystyle c^{2}{\boldsymbol {\nabla }}\cdot \mathbf {A} =-{q\mu _{0}c \over 4\pi r}\mathbf {n} \cdot {\dot {\mathbf {v} }}=-{\partial \Phi \over \partial t}}"></span></dd></dl> <p>som ved integrasjon viser hvordan det elektriske potensialet varierer i tid og rom, </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 (\mathbf {r} ,t)={q\mu _{0}c \over 4\pi r}\mathbf {n} \cdot \mathbf {v} (t-r/c)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>c</mi> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (\mathbf {r} ,t)={q\mu _{0}c \over 4\pi r}\mathbf {n} \cdot \mathbf {v} (t-r/c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b722f115c98718190a4c4e511f62fbbd9fa18fef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.213ex; height:4.843ex;" alt="{\displaystyle \Phi (\mathbf {r} ,t)={q\mu _{0}c \over 4\pi r}\mathbf {n} \cdot \mathbf {v} (t-r/c)}"></span></dd></dl> <p>Det elektriske feltet følger fra definisjonen </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 }}\Phi -{\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 mathvariant="normal">Φ</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 }}\Phi -{\partial \mathbf {A} \over \partial t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a93e18e7d9e284fa59e5ebfa8605a0fc29c3575" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.582ex; height:5.509ex;" alt="{\displaystyle \mathbf {E} =-{\boldsymbol {\nabla }}\Phi -{\partial \mathbf {A} \over \partial t}}"></span></dd></dl> <p>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 {\partial \mathbf {A} \over \partial t}={q\mu _{0} \over 4\pi r}{\dot {\mathbf {v} }}={q\mu _{0} \over 4\pi r}(\mathbf {n} \cdot \mathbf {n} ){\dot {\mathbf {v} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\partial \mathbf {A} \over \partial t}={q\mu _{0} \over 4\pi r}{\dot {\mathbf {v} }}={q\mu _{0} \over 4\pi r}(\mathbf {n} \cdot \mathbf {n} ){\dot {\mathbf {v} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9ef60515a65c7d36117877e93599a61ab5e49a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.41ex; height:5.509ex;" alt="{\displaystyle {\partial \mathbf {A} \over \partial t}={q\mu _{0} \over 4\pi r}{\dot {\mathbf {v} }}={q\mu _{0} \over 4\pi r}(\mathbf {n} \cdot \mathbf {n} ){\dot {\mathbf {v} }}}"></span></dd></dl> <p>Dermed 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 {E} ={q\mu _{0} \over 4\pi r}{\Big [}\mathbf {n} (\mathbf {n} \cdot {\dot {\mathbf {v} }})-(\mathbf {n} \cdot \mathbf {n} ){\dot {\mathbf {v} }}{\Big ]}={q\mu _{0} \over 4\pi r}\mathbf {n} \times (\mathbf {n} \times {\dot {\mathbf {v} }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <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">n</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo>×</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} ={q\mu _{0} \over 4\pi r}{\Big [}\mathbf {n} (\mathbf {n} \cdot {\dot {\mathbf {v} }})-(\mathbf {n} \cdot \mathbf {n} ){\dot {\mathbf {v} }}{\Big ]}={q\mu _{0} \over 4\pi r}\mathbf {n} \times (\mathbf {n} \times {\dot {\mathbf {v} }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/221d5bd5381a3c776d666df3d575df2a8472afca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:49.359ex; height:4.843ex;" alt="{\displaystyle \mathbf {E} ={q\mu _{0} \over 4\pi r}{\Big [}\mathbf {n} (\mathbf {n} \cdot {\dot {\mathbf {v} }})-(\mathbf {n} \cdot \mathbf {n} ){\dot {\mathbf {v} }}{\Big ]}={q\mu _{0} \over 4\pi r}\mathbf {n} \times (\mathbf {n} \times {\dot {\mathbf {v} }})}"></span></dd></dl> <p>Feltet står som forventet <a href="/wiki/Vinkelrett" title="Vinkelrett">vinkelrett</a> både på strålingsretningen og det magnetiske feltet da <b>E</b> = <i>c</i> <b>B</b> × <b>n</b>. Men dette er ikke <a href="/wiki/B%C3%B8lge#Harmoniske_bølger" title="Bølge">harmoniske bølger</a> med et visst bølgetall og frekvens, men med en bølgeform som er gitt ved funksjonen <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 {\dot {\mathbf {v} }}(t-r/c)}"> <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">v</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>c</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {\mathbf {v} }}(t-r/c)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0b27147e0b6211b5cc3ae79173de05510cd04f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.118ex; height:2.843ex;" alt="{\displaystyle {\dot {\mathbf {v} }}(t-r/c)}"></span> som beskriver partikkelens akselerasjonen. </p> <div class="mw-heading mw-heading3"><h3 id="Larmors_formel">Larmors formel</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektromagnetisk_felt&veaction=edit&section=17" title="Rediger avsnitt: Larmors formel" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektromagnetisk_felt&action=edit&section=17" title="Rediger kildekoden til seksjonen Larmors formel"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Den utstrålte energien kan beregnes fra <a href="/wiki/Poyntings_vektor" title="Poyntings vektor">Poyntings vektor</a> <b>S</b> = <b>E</b> × <b>H</b> hvor <span class="nowrap"><b>H</b> = <b>B</b>/<i>μ</i><sub>0</sub></span> i vakum. Med de funne strålingsfeltene 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 {S} ={c \over \mu _{0}}\mathbf {B} ^{2}(\mathbf {r} ,t)={q^{2}\mu _{0} \over 16\pi ^{2}r^{2}c}(\mathbf {n} \times {\dot {\mathbf {v} }})^{2}\mathbf {n} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <mn>16</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>c</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {S} ={c \over \mu _{0}}\mathbf {B} ^{2}(\mathbf {r} ,t)={q^{2}\mu _{0} \over 16\pi ^{2}r^{2}c}(\mathbf {n} \times {\dot {\mathbf {v} }})^{2}\mathbf {n} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d691e4fc67a33c53871fc00db1e1d5225ba5be4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:37.459ex; height:6.176ex;" alt="{\displaystyle \mathbf {S} ={c \over \mu _{0}}\mathbf {B} ^{2}(\mathbf {r} ,t)={q^{2}\mu _{0} \over 16\pi ^{2}r^{2}c}(\mathbf {n} \times {\dot {\mathbf {v} }})^{2}\mathbf {n} }"></span></dd></dl> <p>Betrakter man en liten <a href="/wiki/Romvinkel" title="Romvinkel">romvinkel</a> <i>dΩ</i> i avstand <i>r</i> fra partikkelen, blir intensiteten i denne retningen </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 {dP \over d\Omega }=r^{2}\mathbf {n} \cdot \mathbf {S} ={q^{2} \over 16\pi ^{2}\varepsilon _{0}c^{3}}(\mathbf {n} \times {\dot {\mathbf {v} }})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>P</mi> </mrow> <mrow> <mi>d</mi> <mi mathvariant="normal">Ω</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>16</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {dP \over d\Omega }=r^{2}\mathbf {n} \cdot \mathbf {S} ={q^{2} \over 16\pi ^{2}\varepsilon _{0}c^{3}}(\mathbf {n} \times {\dot {\mathbf {v} }})^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15c608990a32831130246ad3f3d4d1cd1b5d0e30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.095ex; height:6.343ex;" alt="{\displaystyle {dP \over d\Omega }=r^{2}\mathbf {n} \cdot \mathbf {S} ={q^{2} \over 16\pi ^{2}\varepsilon _{0}c^{3}}(\mathbf {n} \times {\dot {\mathbf {v} }})^{2}}"></span></dd></dl> <p>når man benytter at <i>cμ</i><sub>0</sub> = 1/<i>cε</i><sub>0</sub> fra definisjonen av <a href="/wiki/Lyshastigheten" class="mw-redirect" title="Lyshastigheten">lyshastigheten</a>. Dette er Larmors formel som spiller en viktig rolle i mange sammenhenger.<sup id="cite_ref-Longair_7-0" class="reference"><a href="#cite_note-Longair-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>Intensiteten retningen <b>r</b> av strålingen er naturlig å beskrive i <a href="/wiki/Kulekoordinater" title="Kulekoordinater">kulekoordinater</a> (<i>r, θ, φ</i>) med den polare aksen langs akselerasjonen <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} ={\dot {\mathbf {v} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} ={\dot {\mathbf {v} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87386c0e800ec0fd5a124170c6e72c8277e8e25d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.809ex; height:2.176ex;" alt="{\displaystyle \mathbf {a} ={\dot {\mathbf {v} }}}"></span>. Det 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 {dP \over d\Omega }={q^{2}a^{2} \over 16\pi ^{2}\varepsilon _{0}c^{3}}\sin ^{2}\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>P</mi> </mrow> <mrow> <mi>d</mi> <mi mathvariant="normal">Ω</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>16</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {dP \over d\Omega }={q^{2}a^{2} \over 16\pi ^{2}\varepsilon _{0}c^{3}}\sin ^{2}\theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e89f3f1c2f651f8cc2e28829bb41092c4a34c188" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.419ex; height:6.343ex;" alt="{\displaystyle {dP \over d\Omega }={q^{2}a^{2} \over 16\pi ^{2}\varepsilon _{0}c^{3}}\sin ^{2}\theta }"></span></dd></dl> <p>hvor romvinkelelementet er <i>dΩ</i> = 2<i>π</i> sin<i>θdθ</i>. Dette resultatet er <a href="/wiki/Larmors_formel" title="Larmors formel">Larmors formel</a> og er den samme som for en oscillerende, <a href="/wiki/Dipol#Elektrisk_dipolstråling" title="Dipol">elektrisk dipol</a>. Intensiteten er maksimal normalt på akselerasjonen og er uavhengig av partikkelens hastighet. </p><p>Utstrålt energi per tidsenhet alle retninger 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 P={q^{2}a^{2} \over 16\pi ^{2}\varepsilon _{0}c^{3}}2\pi \!\int _{0}^{\pi }\!d\theta \sin ^{3}\theta ={q^{2}a^{2} \over 6\pi \varepsilon _{0}c^{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>16</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mn>2</mn> <mi>π</mi> <mspace width="negativethinmathspace" /> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msubsup> <mspace width="negativethinmathspace" /> <mi>d</mi> <mi>θ</mi> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>6</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P={q^{2}a^{2} \over 16\pi ^{2}\varepsilon _{0}c^{3}}2\pi \!\int _{0}^{\pi }\!d\theta \sin ^{3}\theta ={q^{2}a^{2} \over 6\pi \varepsilon _{0}c^{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b092bfce5faef5191718da6a478f2d09b63665d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.434ex; height:6.343ex;" alt="{\displaystyle P={q^{2}a^{2} \over 16\pi ^{2}\varepsilon _{0}c^{3}}2\pi \!\int _{0}^{\pi }\!d\theta \sin ^{3}\theta ={q^{2}a^{2} \over 6\pi \varepsilon _{0}c^{3}}}"></span></dd></dl> <p>Dette energitapet betyr at partikkelen må miste <a href="/wiki/Kinetisk_energi" title="Kinetisk energi">kinetisk energi</a> slik at dens hastighet reduseres. I <a href="/wiki/Klassisk_mekanikk" title="Klassisk mekanikk">klassisk mekanikk</a> betyr det at strålingen må utøve en tilbakevirkende kraft på partikkelen. En slik effekt kalles for «strålingsreaksjon» og er vanskelig å gi en god beskrivelse.<sup id="cite_ref-Jackson_4-3" class="reference"><a href="#cite_note-Jackson-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Sirkulær_bevegelse"><span id="Sirkul.C3.A6r_bevegelse"></span>Sirkulær bevegelse</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Elektromagnetisk_felt&veaction=edit&section=18" title="Rediger avsnitt: Sirkulær bevegelse" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektromagnetisk_felt&action=edit&section=18" title="Rediger kildekoden til seksjonen Sirkulær bevegelse"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Den enkleste bruk av Larmors formel er når akselerasjon har en konstant en størrelse. I så fall har den da også en konstant retning eller den roterer med en konstant <a href="/wiki/Vinkelhastighet" title="Vinkelhastighet">vinkelhastighet</a> <i>ω</i> i en sirkulær bane. Har denne radius <i>R</i> og hastigheten til partikkelen er <i>v</i>, er <span class="nowrap"><i>ω</i> = <i>v</i>/<i>R</i></span> og <a href="/wiki/Sentripetalakselerasjon" title="Sentripetalakselerasjon">sentripetalakselerasjonen</a> <span class="nowrap"><i>a</i> = <i>v</i><sup>2</sup>/<i>R</i></span> = <i>ωv</i>. Utstrålt energi per tidsenhet blir da </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 {dP \over d\Omega }={q^{4}v^{4} \over 16\pi ^{2}\varepsilon _{0}R^{2}c^{3}}\sin ^{2}\theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>P</mi> </mrow> <mrow> <mi>d</mi> <mi mathvariant="normal">Ω</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mrow> <mn>16</mn> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <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> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {dP \over d\Omega }={q^{4}v^{4} \over 16\pi ^{2}\varepsilon _{0}R^{2}c^{3}}\sin ^{2}\theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaa2c34915055e81683040d8d8386b68b9833ec5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.237ex; height:6.343ex;" alt="{\displaystyle {dP \over d\Omega }={q^{4}v^{4} \over 16\pi ^{2}\varepsilon _{0}R^{2}c^{3}}\sin ^{2}\theta }"></span></dd></dl> <p>hvor vinkelen <i>θ</i> er relativ til akselerasjonen <b>a</b> som roterer med konstant hastighet. Strålingen er derfor konsentrert i retninger langs banen. Hastigheten <i>v</i> kan ikke bli større enn lyshastigheten. Men da må formelen generaliseres til å gjelde ved store hastigheter som i <a href="/wiki/Elektrodynamikk" title="Elektrodynamikk">relativistisk elektrodynamikk</a>. Samtidig øker denne strålingsenergien når baneradius <i>R</i> blir mindre. </p><p>I en <a href="/wiki/Syklotron" title="Syklotron">syklotron</a> beveger en ladet partikkel seg i en sirkelbane ved at den blir holdt på plass av et konstant magnetfelt <i>B</i>. Den får da en konstant vinkelhastighet gitt ved <a href="/wiki/Lorentz-kraft#Syklotronbevegelse" title="Lorentz-kraft">syklotronfrekvensen</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{c}={qB \over m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <mi>B</mi> </mrow> <mi>m</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{c}={qB \over m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e2faa18426b06a1badd3dbbfd8051d44cd3376f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.158ex; height:5.343ex;" alt="{\displaystyle \omega _{c}={qB \over m}}"></span></dd></dl> <p>Ved økende hastighet må den bevege seg i sirkler med stadig større radius da <i>v = ω<sub>c</sub>R</i>. Derimot i en <a href="/w/index.php?title=Synkrotron&action=edit&redlink=1" class="new" title="Synkrotron (ikke skrevet ennå)">synkrotron</a> som kan akselerere partikler til enda høyere energier, beveger de seg i en sirkulær bane med konstant radius. Da må magnetfeltet øke i synkront i takt med økende hastigheter. Den energien som partikkelen stråler ut, kalles for <a href="/wiki/Synkrotronstr%C3%A5ling" title="Synkrotronstråling">synkrotronstråling</a>.<sup id="cite_ref-HLL_6-1" class="reference"><a href="#cite_note-HLL-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>Utstrålt energi er spesielt stort for lette partikler som for <a href="/wiki/Elektron" title="Elektron">elektroner</a>. Hvis det beveger seg om en positiv ladning som i et <a href="/wiki/Hydrogenatom" title="Hydrogenatom">hydrogenatom</a>, betyr strålingstapet at etter omtrent <span class="nowrap">10<sup> -16</sup></span> sekund vil det ha falt inn mot sentrum. Atomer ville derfor ikke kunne eksistere som stabile byggestener i naturen.<sup id="cite_ref-Princeton_8-0" class="reference"><a href="#cite_note-Princeton-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Allerede før etableringen av <a href="/wiki/Bohrs_atommodell" title="Bohrs atommodell">Bohrs atommodell</a> hadde <a href="/wiki/Hendrik_Antoon_Lorentz" title="Hendrik Antoon Lorentz">Lorentz</a> foreslått at elektronene kunne være bundet i et <a href="/wiki/Atom" title="Atom">atom</a> hvor de beveget seg i lukkete baner. Han kunne dermed forklare det som i dag kalles den normale <a href="/wiki/Zeeman-effekt" title="Zeeman-effekt">Zeeman-effekten</a>. Det var i denne forbindelsen at <a href="/w/index.php?title=Joseph_Larmor&action=edit&redlink=1" class="new" title="Joseph Larmor (ikke skrevet ennå)">Larmor</a> fant sin formel for tapet av energi for et slikt bundet elektron. Strålingstapet ville gjøre atomet ustabilt. </p><p>I <a href="/wiki/Thomsons_rosinbollemodell" title="Thomsons rosinbollemodell">Thomsons rosinbollemodell</a> ble dette problemet unngått ved å plassere elektronene sammen med en kompenserende, positiv ladning jevnt utover hele atomet. Men etter at <a href="/wiki/Rutherford" class="mw-redirect" title="Rutherford">Rutherford</a> i sitt <a href="/wiki/Gullfolieeksperimentet" class="mw-redirect" title="Gullfolieeksperimentet">gullfolieeksperimentet</a> viste at den positive ladningen er konsentrert i en liten <a href="/wiki/Atomkjerne" title="Atomkjerne">kjerne</a>, unngikk Bohr dette problemet ved å ganske enkelt postulere at det finnes stabile baner hvor elektronene ikke vil miste energi. Dette var et brudd på klassisk, <a href="/wiki/Elektromagnetisme" title="Elektromagnetisme">elektromagnetisk teori</a> og danner grunnlaget for moderne <a href="/wiki/Kvanteteori" class="mw-redirect" title="Kvanteteori">kvanteteori</a>.<sup id="cite_ref-Longair_7-1" class="reference"><a href="#cite_note-Longair-7"><span class="cite-bracket">[</span>7<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=Elektromagnetisk_felt&veaction=edit&section=19" 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=Elektromagnetisk_felt&action=edit&section=19" 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/Elektrisk_felt" title="Elektrisk felt">Elektrisk felt</a></li> <li><a href="/wiki/Magnetisk_felt" class="mw-redirect" title="Magnetisk felt">Magnetisk felt</a></li> <li><a href="/wiki/Elektromagnetisme" title="Elektromagnetisme">Elektromagnetisme</a></li> <li><a href="/wiki/Elektromagnetisk_str%C3%A5ling" title="Elektromagnetisk stråling">Elektromagnetisk stråling</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=Elektromagnetisk_felt&veaction=edit&section=20" title="Rediger avsnitt: Referanser" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Elektromagnetisk_felt&action=edit&section=20" 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-Griffiths-1"><b>^</b> <a href="#cite_ref-Griffiths_1-0"><sup>a</sup></a> <a href="#cite_ref-Griffiths_1-1"><sup>b</sup></a> <a href="#cite_ref-Griffiths_1-2"><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-Darrigol-2"><b>^</b> <a href="#cite_ref-Darrigol_2-0"><sup>a</sup></a> <a href="#cite_ref-Darrigol_2-1"><sup>b</sup></a> <span class="reference-text"> O. Darrigol, <i>Electrodynamics from Ampère to Einstein</i>, Oxford University Press, Oxford (2003). <a href="/wiki/Spesial:Bokkilder/0198505930" class="internal mw-magiclink-isbn">ISBN 0-19-850593-0</a>.</span> </li> <li id="cite_note-Brau-3"><b>^</b> <a href="#cite_ref-Brau_3-0"><sup>a</sup></a> <a href="#cite_ref-Brau_3-1"><sup>b</sup></a> <span class="reference-text"> C.A. Brau, <i>Modern Problems in Classical Electrodynamics</i>, Oxford University Press, Oxford (2004). <a href="/wiki/Spesial:Bokkilder/0195146654" class="internal mw-magiclink-isbn">ISBN 0-19-514665-4</a>.</span> </li> <li id="cite_note-Jackson-4"><b>^</b> <a href="#cite_ref-Jackson_4-0"><sup>a</sup></a> <a href="#cite_ref-Jackson_4-1"><sup>b</sup></a> <a href="#cite_ref-Jackson_4-2"><sup>c</sup></a> <a href="#cite_ref-Jackson_4-3"><sup>d</sup></a> <span class="reference-text"> J. D. Jackson, <i>Classical Electrodynamics</i>, John Wiley & Sons, New York (1998). <a href="/wiki/Spesial:Bokkilder/047130932X" class="internal mw-magiclink-isbn">ISBN 0-4713-0932-X</a>.</span> </li> <li id="cite_note-SMM-5"><b><a href="#cite_ref-SMM_5-0">^</a></b> <span class="reference-text"> R.A. Serway, C.J. Moser and C.A. Moyer, <i>Modern Physics</i>, Saunders College Publishing, Philadelphia (1989). <a href="/wiki/Spesial:Bokkilder/0030297974" class="internal mw-magiclink-isbn">ISBN 0-03-029797-4</a>.</span> </li> <li id="cite_note-HLL-6"><b>^</b> <a href="#cite_ref-HLL_6-0"><sup>a</sup></a> <a href="#cite_ref-HLL_6-1"><sup>b</sup></a> <span class="reference-text">O. Hunderi, J.R. Lien og G. Løvhøiden, <i>Generell fysikk for universiteter og høgskoler, Bind 2</i>, 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-Longair-7"><b>^</b> <a href="#cite_ref-Longair_7-0"><sup>a</sup></a> <a href="#cite_ref-Longair_7-1"><sup>b</sup></a> <span class="reference-text"> M.S. Longair, <i>Theoretical Concepts in Physics</i>, Cambridge University Press, Cambridge (2003). <a href="/wiki/Spesial:Bokkilder/9780521528788" class="internal mw-magiclink-isbn">ISBN 978-0-521-52878-8</a>.</span> </li> <li id="cite_note-Princeton-8"><b><a href="#cite_ref-Princeton_8-0">^</a></b> <span class="reference-text">J.D. Olsen and K.T. McDonald, <a rel="nofollow" class="external text" href="http://www.physics.princeton.edu/~mcdonald/examples/orbitdecay.pdf"><i>Classical Lifetime of a Bohr Atom</i></a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190909221112/http://www.physics.princeton.edu/~mcdonald/examples/orbitdecay.pdf">Arkivert</a> 9. september 2019 hos <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>., Joseph Henry Laboratories, Princeton University, (2017).</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=Elektromagnetisk_felt&veaction=edit&section=21" 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=Elektromagnetisk_felt&action=edit&section=21" title="Rediger kildekoden til seksjonen Eksterne lenker"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Johannes Skaar, <a rel="nofollow" class="external text" href="https://www.uio.no/studier/emner/matnat/fys/FYS1120/h17/pensumliste/elektromagnetisme.pdf"><i>Elektromagnetisme</i></a>, forelesninger ved UiO, 2017.</li> <li>ETHW, <a rel="nofollow" class="external text" href="http://ethw.org/Maxwell's_Equations">Short history of Maxwell's equations.</a></li> <li>J.D. Jackson, <a rel="nofollow" class="external text" href="https://archive.org/details/ClassicalElectrodynamics"><i>Classical Electrodynamics</i></a>, John Wiley & Sons, New York (1982). <a href="/wiki/Spesial:Bokkilder/0471431311" class="internal mw-magiclink-isbn">ISBN 0-471-43131-1</a>. Digital utgave.</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 ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist 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Elektromagnetisk felt – Wikipedia