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vector-toc-level-2"> <a class="vector-toc-link" href="#Kartesisk_beskrivelse"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Kartesisk beskrivelse</span> </div> </a> <ul id="toc-Kartesisk_beskrivelse-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Elektromagnetisk_induksjon" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Elektromagnetisk_induksjon"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Elektromagnetisk induksjon</span> </div> </a> <button aria-controls="toc-Elektromagnetisk_induksjon-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 induksjon</span> </button> <ul id="toc-Elektromagnetisk_induksjon-sublist" class="vector-toc-list"> <li id="toc-Relativitetsteori" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relativitetsteori"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Relativitetsteori</span> </div> </a> <ul id="toc-Relativitetsteori-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Kontinuerlig_ladningsfordeling" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kontinuerlig_ladningsfordeling"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Kontinuerlig ladningsfordeling</span> </div> </a> <ul id="toc-Kontinuerlig_ladningsfordeling-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Matematisk_utledning" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Matematisk_utledning"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Matematisk utledning</span> </div> </a> <ul id="toc-Matematisk_utledning-sublist" class="vector-toc-list"> </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">6</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">7</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">8</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">Lorentz-kraft</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 70 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-70" 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">70 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/Lorentzkraft" title="Lorentzkraft – norsk nynorsk" lang="nn" hreflang="nn" data-title="Lorentzkraft" 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/Lorentzkraft" title="Lorentzkraft – dansk" lang="da" hreflang="da" data-title="Lorentzkraft" 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/Lorentzkraft" title="Lorentzkraft – svensk" lang="sv" hreflang="sv" data-title="Lorentzkraft" 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/Lorentzkraftur" title="Lorentzkraftur – islandsk" lang="is" hreflang="is" data-title="Lorentzkraftur" data-language-autonym="Íslenska" data-language-local-name="islandsk" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Lorentzkrag" title="Lorentzkrag – afrikaans" lang="af" hreflang="af" data-title="Lorentzkrag" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%82%D9%88%D8%A9_%D9%84%D9%88%D8%B1%D9%86%D8%AA%D8%B3" title="قوة لورنتس – arabisk" lang="ar" hreflang="ar" data-title="قوة لورنتس" 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/Fuercia_de_Lorentz" title="Fuercia de Lorentz – asturisk" lang="ast" hreflang="ast" data-title="Fuercia de Lorentz" 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/Lorens_q%C3%BCvv%C9%99si" title="Lorens qüvvəsi – aserbajdsjansk" lang="az" hreflang="az" data-title="Lorens qüvvəsi" data-language-autonym="Azərbaycanca" data-language-local-name="aserbajdsjansk" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B2%E0%A7%8B%E0%A6%B0%E0%A7%87%E0%A6%A8%E0%A7%8D%E2%80%8C%E0%A7%8E%E0%A6%B8_%E0%A6%AC%E0%A6%B2" title="লোরেন্‌ৎস বল – bengali" lang="bn" hreflang="bn" data-title="লোরেন্‌ৎস বল" data-language-autonym="বাংলা" data-language-local-name="bengali" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A1%D1%96%D0%BB%D0%B0_%D0%9B%D0%BE%D1%80%D1%8D%D0%BD%D1%86%D0%B0" title="Сіла Лорэнца – belarusisk" lang="be" hreflang="be" data-title="Сіла Лорэнца" 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%A1%D1%96%D0%BB%D0%B0_%D0%9B%D1%91%D1%80%D1%8D%D0%BD%D1%86%D0%B0" title="Сіла Лёрэнца – belarusisk (klassisk ortografi)" lang="be-tarask" hreflang="be-tarask" data-title="Сіла Лёрэнца" 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%A1%D0%B8%D0%BB%D0%B0_%D0%BD%D0%B0_%D0%9B%D0%BE%D1%80%D0%B5%D0%BD%D1%86" title="Сила на Лоренц – bulgarsk" lang="bg" hreflang="bg" data-title="Сила на Лоренц" data-language-autonym="Български" data-language-local-name="bulgarsk" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/For%C3%A7a_de_Lorentz" title="Força de Lorentz – katalansk" lang="ca" hreflang="ca" data-title="Força de Lorentz" 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%9B%D0%BE%D1%80%D0%B5%D0%BD%D1%86_%D0%B2%C4%83%D0%B9%C4%95" title="Лоренц вăйĕ – tsjuvasjisk" lang="cv" hreflang="cv" data-title="Лоренц вăйĕ" 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/Lorentzova_s%C3%ADla" title="Lorentzova síla – tsjekkisk" lang="cs" hreflang="cs" data-title="Lorentzova síla" 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/Lorentzkraft" title="Lorentzkraft – tysk" lang="de" hreflang="de" data-title="Lorentzkraft" data-language-autonym="Deutsch" data-language-local-name="tysk" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Lorentzi_j%C3%B5ud" title="Lorentzi jõud – estisk" lang="et" hreflang="et" data-title="Lorentzi jõud" data-language-autonym="Eesti" data-language-local-name="estisk" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CF%8D%CE%BD%CE%B1%CE%BC%CE%B7_%CE%9B%CF%8C%CF%81%CE%B5%CE%BD%CF%84%CE%B6" title="Δύναμη Λόρεντζ – gresk" lang="el" hreflang="el" data-title="Δύναμη Λόρεντζ" 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/Lorentz_force" title="Lorentz force – engelsk" lang="en" hreflang="en" data-title="Lorentz force" 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/Fuerza_de_Lorentz" title="Fuerza de Lorentz – spansk" lang="es" hreflang="es" data-title="Fuerza de Lorentz" 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/Lorenca_forto" title="Lorenca forto – esperanto" lang="eo" hreflang="eo" data-title="Lorenca forto" 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/Lorentzen_indarra" title="Lorentzen indarra – baskisk" lang="eu" hreflang="eu" data-title="Lorentzen indarra" 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%86%DB%8C%D8%B1%D9%88%DB%8C_%D9%84%D9%88%D8%B1%D9%86%D8%AA%D8%B3" title="نیروی لورنتس – persisk" lang="fa" hreflang="fa" data-title="نیروی لورنتس" 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/Force_%C3%A9lectromagn%C3%A9tique" title="Force électromagnétique – fransk" lang="fr" hreflang="fr" data-title="Force électromagnétique" data-language-autonym="Français" data-language-local-name="fransk" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%A1%9C%EB%9F%B0%EC%B8%A0_%ED%9E%98" title="로런츠 힘 – koreansk" lang="ko" hreflang="ko" data-title="로런츠 힘" 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%BC%D5%B8%D6%80%D5%A5%D5%B6%D6%81%D5%AB_%D5%B8%D6%82%D5%AA" title="Լորենցի ուժ – armensk" lang="hy" hreflang="hy" data-title="Լորենցի ուժ" 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%B2%E0%A5%89%E0%A4%B0%E0%A5%87%E0%A4%82%E0%A4%9C_%E0%A4%AC%E0%A4%B2" title="लॉरेंज बल – hindi" lang="hi" hreflang="hi" data-title="लॉरेंज बल" 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/Lorentzova_sila" title="Lorentzova sila – kroatisk" lang="hr" hreflang="hr" data-title="Lorentzova sila" 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/Gaya_Lorentz" title="Gaya Lorentz – indonesisk" lang="id" hreflang="id" data-title="Gaya Lorentz" 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/Forza_di_Lorentz" title="Forza di Lorentz – italiensk" lang="it" hreflang="it" data-title="Forza di Lorentz" 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%9B%D7%95%D7%97_%D7%9C%D7%95%D7%A8%D7%A0%D7%A5" title="כוח לורנץ – hebraisk" lang="he" hreflang="he" data-title="כוח לורנץ" data-language-autonym="עברית" data-language-local-name="hebraisk" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Gaya_Lorentz" title="Gaya Lorentz – javanesisk" lang="jv" hreflang="jv" data-title="Gaya Lorentz" data-language-autonym="Jawa" data-language-local-name="javanesisk" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9A%E1%83%9D%E1%83%A0%E1%83%94%E1%83%9C%E1%83%AA%E1%83%98%E1%83%A1_%E1%83%AB%E1%83%90%E1%83%9A%E1%83%90" title="ლორენცის ძალა – georgisk" lang="ka" hreflang="ka" data-title="ლორენცის ძალა" 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%9B%D0%BE%D1%80%D0%B5%D0%BD%D1%86_%D0%BA%D2%AF%D1%88%D1%96" title="Лоренц күші – kasakhisk" lang="kk" hreflang="kk" data-title="Лоренц күші" data-language-autonym="Қазақша" data-language-local-name="kasakhisk" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/F%C3%B2s_elektwomayetis" title="Fòs elektwomayetis – haitisk" lang="ht" hreflang="ht" data-title="Fòs elektwomayetis" data-language-autonym="Kreyòl ayisyen" data-language-local-name="haitisk" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Aequatio_Lorentziana" title="Aequatio Lorentziana – latin" lang="la" hreflang="la" data-title="Aequatio Lorentziana" data-language-autonym="Latina" data-language-local-name="latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Lorenca_sp%C4%93ks" title="Lorenca spēks – latvisk" lang="lv" hreflang="lv" data-title="Lorenca spēks" data-language-autonym="Latviešu" data-language-local-name="latvisk" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Lorentz-er%C5%91" title="Lorentz-erő – ungarsk" lang="hu" hreflang="hu" data-title="Lorentz-erő" 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%9B%D0%BE%D1%80%D0%B5%D0%BD%D1%86%D0%BE%D0%B2%D0%B0_%D1%81%D0%B8%D0%BB%D0%B0" title="Лоренцова сила – makedonsk" lang="mk" hreflang="mk" data-title="Лоренцова сила" data-language-autonym="Македонски" data-language-local-name="makedonsk" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B2%E0%A5%89%E0%A4%B0%E0%A5%87%E0%A4%82%E0%A4%9D_%E0%A4%AC%E0%A4%B2" title="लॉरेंझ बल – marathi" lang="mr" hreflang="mr" data-title="लॉरेंझ बल" data-language-autonym="मराठी" data-language-local-name="marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9B%D0%BE%D1%80%D0%B5%D0%BD%D1%86%D1%8B%D0%BD_%D1%85%D2%AF%D1%87" title="Лоренцын хүч – mongolsk" lang="mn" hreflang="mn" data-title="Лоренцын хүч" 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/Lorentzkracht" title="Lorentzkracht – nederlandsk" lang="nl" hreflang="nl" data-title="Lorentzkracht" 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/%E3%83%AD%E3%83%BC%E3%83%AC%E3%83%B3%E3%83%84%E5%8A%9B" title="ローレンツ力 – japansk" lang="ja" hreflang="ja" data-title="ローレンツ力" 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-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/F%C3%B2r%C3%A7a_de_Lorentz" title="Fòrça de Lorentz – oksitansk" lang="oc" hreflang="oc" data-title="Fòrça de Lorentz" 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/Lorents_kuchi" title="Lorents kuchi – usbekisk" lang="uz" hreflang="uz" data-title="Lorents kuchi" 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-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Equassion_%C3%ABd_Lorentz" title="Equassion ëd Lorentz – piemontesisk" lang="pms" hreflang="pms" data-title="Equassion ëd Lorentz" data-language-autonym="Piemontèis" data-language-local-name="piemontesisk" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Si%C5%82a_Lorentza" title="Siła Lorentza – polsk" lang="pl" hreflang="pl" data-title="Siła Lorentza" 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/For%C3%A7a_de_Lorentz" title="Força de Lorentz – portugisisk" lang="pt" hreflang="pt" data-title="Força de Lorentz" 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/For%C8%9B%C4%83_Lorentz" title="Forță Lorentz – rumensk" lang="ro" hreflang="ro" data-title="Forță Lorentz" 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%A1%D0%B8%D0%BB%D0%B0_%D0%9B%D0%BE%D1%80%D0%B5%D0%BD%D1%86%D0%B0" title="Сила Лоренца – russisk" lang="ru" hreflang="ru" data-title="Сила Лоренца" data-language-autonym="Русский" data-language-local-name="russisk" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Lorentz_force" title="Lorentz force – skotsk" lang="sco" hreflang="sco" data-title="Lorentz force" data-language-autonym="Scots" data-language-local-name="skotsk" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Forca_e_Lorencit" title="Forca e Lorencit – albansk" lang="sq" hreflang="sq" data-title="Forca e Lorencit" data-language-autonym="Shqip" data-language-local-name="albansk" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%BD%E0%B7%9C%E0%B6%BB%E0%B7%99%E0%B6%B1%E0%B7%8A%E0%B7%83%E0%B7%94_%E0%B7%80%E0%B7%90%E0%B6%BB" title="ලොරෙන්සු වැර – singalesisk" lang="si" hreflang="si" data-title="ලොරෙන්සු වැර" data-language-autonym="සිංහල" data-language-local-name="singalesisk" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Lorentz_force" title="Lorentz force – enkel engelsk" lang="en-simple" hreflang="en-simple" data-title="Lorentz force" data-language-autonym="Simple English" data-language-local-name="enkel engelsk" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Lorentzova_sila" title="Lorentzova sila – slovakisk" lang="sk" hreflang="sk" data-title="Lorentzova sila" 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/Lorentzeva_sila" title="Lorentzeva sila – slovensk" lang="sl" hreflang="sl" data-title="Lorentzeva sila" data-language-autonym="Slovenščina" data-language-local-name="slovensk" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9B%D0%BE%D1%80%D0%B5%D0%BD%D1%86%D0%BE%D0%B2%D0%B0_%D1%81%D0%B8%D0%BB%D0%B0" title="Лоренцова сила – serbisk" lang="sr" hreflang="sr" data-title="Лоренцова сила" 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/Lorentzova_sila" title="Lorentzova sila – serbokroatisk" lang="sh" hreflang="sh" data-title="Lorentzova sila" 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/Lorentzin_voima" title="Lorentzin voima – finsk" lang="fi" hreflang="fi" data-title="Lorentzin voima" data-language-autonym="Suomi" data-language-local-name="finsk" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Puwersang_Lorentz" title="Puwersang Lorentz – tagalog" lang="tl" hreflang="tl" data-title="Puwersang Lorentz" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%87%E0%AE%B2%E0%AE%BE%E0%AE%B0%E0%AE%A9%E0%AF%8D%E0%AE%9A%E0%AF%81_%E0%AE%B5%E0%AE%BF%E0%AE%9A%E0%AF%88" title="இலாரன்சு விசை – tamil" lang="ta" hreflang="ta" data-title="இலாரன்சு விசை" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9B%D0%BE%D1%80%D0%B5%D0%BD%D1%86_%D0%BA%D3%A9%D1%87%D0%B5" title="Лоренц көче – tatarisk" lang="tt" hreflang="tt" data-title="Лоренц көче" data-language-autonym="Татарча / tatarça" data-language-local-name="tatarisk" class="interlanguage-link-target"><span>Татарча / tatarça</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%81%E0%B8%A3%E0%B8%87%E0%B9%82%E0%B8%A5%E0%B9%80%E0%B8%A3%E0%B8%B4%E0%B8%99%E0%B8%95%E0%B8%AA%E0%B9%8C" title="แรงโลเรินตส์ – thai" lang="th" hreflang="th" data-title="แรงโลเรินตส์" data-language-autonym="ไทย" data-language-local-name="thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Lorentz_kuvveti" title="Lorentz kuvveti – tyrkisk" lang="tr" hreflang="tr" data-title="Lorentz kuvveti" 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%A1%D0%B8%D0%BB%D0%B0_%D0%9B%D0%BE%D1%80%D0%B5%D0%BD%D1%86%D0%B0" title="Сила Лоренца – ukrainsk" lang="uk" hreflang="uk" data-title="Сила Лоренца" data-language-autonym="Українська" data-language-local-name="ukrainsk" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%DB%8C%D9%88%D8%B1%DB%8C%D9%86%D9%B9%D8%B2_%D9%82%D9%88%D8%AA" title="یورینٹز قوت – urdu" lang="ur" hreflang="ur" data-title="یورینٹز قوت" data-language-autonym="اردو" data-language-local-name="urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/L%E1%BB%B1c_Lorentz" title="Lực Lorentz – vietnamesisk" lang="vi" hreflang="vi" data-title="Lực Lorentz" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamesisk" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%B4%9B%E4%BC%A6%E5%85%B9%E5%8A%9B" title="洛伦兹力 – wu" lang="wuu" hreflang="wuu" data-title="洛伦兹力" data-language-autonym="吴语" data-language-local-name="wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%B4%9B%E5%80%AB%E8%8C%B2%E5%8A%9B" title="洛倫茲力 – kantonesisk" lang="yue" hreflang="yue" data-title="洛倫茲力" data-language-autonym="粵語" data-language-local-name="kantonesisk" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%B4%9B%E4%BC%A6%E5%85%B9%E5%8A%9B" title="洛伦兹力 – kinesisk" lang="zh" hreflang="zh" data-title="洛伦兹力" data-language-autonym="中文" data-language-local-name="kinesisk" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q172137#sitelinks-wikipedia" title="Rediger lenker til artikkelen på andre språk" class="wbc-editpage">Rediger lenker</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Navnerom"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Lorentz-kraft" title="Vis innholdssiden [c]" accesskey="c"><span>Artikkel</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Diskusjon:Lorentz-kraft" rel="discussion" 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srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Lorentzkraft.png/360px-Lorentzkraft.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/30/Lorentzkraft.png/480px-Lorentzkraft.png 2x" data-file-width="852" data-file-height="918" /></a><figcaption>Lorentz-kraften <b>F</b> virker på en ladning <i>q</i> som har hastighet <b>v</b> i en kombinasjon av elektrisk <b>E</b> og magnetisk <b>B</b> felt.</figcaption></figure> <p><b>Lorentz-kraften</b> er den <a href="/wiki/Kraft" title="Kraft">kraften</a> som virker på en <a href="/wiki/Elektrisk_ladning" title="Elektrisk ladning">elektrisk ladning</a> som befinner seg i et <a href="/wiki/Elektromagnetisk_felt" title="Elektromagnetisk felt">elektromagnetisk felt</a>. Mens den elektriske delen av kraften er uavhengig av ladningens bevegelse, er størrelsen til den magnetiske delen proporsjonal med ladningens hastighet og avhengig av retningen til bevegelsen. Kraften er oppkalt etter den nederlandske <a href="/wiki/Fysikk" title="Fysikk">fysiker</a> <a href="/wiki/Hendrik_Antoon_Lorentz" title="Hendrik Antoon Lorentz">Hendrik Antoon Lorentz</a>. </p><p>Betegnes det elektriske feltet med <b>E</b> og det magnetiske feltet med <b>B</b>, er Lorentz-kraften <b>F</b> som virker på ladningen <i>q</i> gitt ved det matematiske uttrykket </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =q(\mathbf {E} +\mathbf {v} \times \mathbf {B} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mi>q</mi> <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">v</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =q(\mathbf {E} +\mathbf {v} \times \mathbf {B} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3dd99e3bd55cbeff1cd2506d944405f3efa9e0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.41ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} =q(\mathbf {E} +\mathbf {v} \times \mathbf {B} )}"></span></dd></dl> <p>når ladningen har hastighet <b>v</b>. Da den elektriske kraften <i>q</i> <b>E</b> definerer det <a href="/wiki/Elektrisk_felt" title="Elektrisk felt">elektriske feltet</a>, blir ofte Lorentz-kraften forbundet med kun den magnetiske delen. Dens størrelse involverer et vektorielt <a href="/wiki/Kryssprodukt" class="mw-redirect" title="Kryssprodukt">kryssprodukt</a> som betyr at kraften er maksimal i størrelse når ladningen beveger seg vinkelrett på magnetfeltet. Den er av samme grunn null når ladningen beveger langs dette feltet. Kraftens retning finnes ved bruk av <a href="/wiki/H%C3%B8yreh%C3%A5ndsregelen" title="Høyrehåndsregelen">høyrehåndsregelen</a>. Formelen for kraften er også gyldig når feltene varierer med tiden. </p><p>Lorentz-kraften er i overensstemmelse med <a href="/wiki/Einstein" class="mw-redirect" title="Einstein">Einsteins</a> <a href="/wiki/Spesiell_relativitetsteori" class="mw-redirect" title="Spesiell relativitetsteori">spesielle relativitetsteori</a>. Den kan der forklares som et resultat av hvordan <a href="/wiki/Coulombs_lov" title="Coulombs lov">Coulomb-kraften</a> opptrer i et <a href="/wiki/Inertialsystem" class="mw-redirect" title="Inertialsystem">inertialsystem</a> som beveger seg. Formelt beskrives denne situasjonen ved en <a href="/wiki/Kovariant_relativitetsteori#Kontravariante_komponenter" title="Kovariant relativitetsteori">Lorentz-transformasjon</a>. </p><p>I alle system hvor elektriske ladninger befinner seg i et magnetfelt, spiller Lorentz-kraften en avgjørende rolle. Den benyttes i gammeldagse TV-apparat til å fokusere elektronstrålen i <a href="/wiki/Bilder%C3%B8r" title="Bilderør">bilderøret</a> og avbøyer de ladete partiklene som akselereres i <a href="/wiki/Syklotron" title="Syklotron">syklotroner</a> og andre <a href="/wiki/Partikkelakselerator" title="Partikkelakselerator">partikkelakseleratorer</a>. På samme måte blir <a href="/wiki/Elektron" title="Elektron">elektroner</a> fra <a href="/wiki/Solen" title="Solen">Solen</a> styrt inn mot nordlige breddegrader av det <a href="/wiki/Jordens_magnetfelt" title="Jordens magnetfelt">jordmagnetiske feltet</a> slik at det dannes <a href="/wiki/Nordlys" class="mw-redirect" title="Nordlys">nordlys</a> i de områdene. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Historie">Historie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz-kraft&veaction=edit&section=1" title="Rediger avsnitt: Historie" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lorentz-kraft&action=edit&section=1" title="Rediger kildekoden til seksjonen Historie"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sammenhengen mellom <a href="/wiki/Elektrisk_str%C3%B8m" title="Elektrisk strøm">elektriske strømmer</a> og <a href="/wiki/Magnetfelt" title="Magnetfelt">magnetfelt</a> ble først systematisk undersøkt av <a href="/wiki/Hans_Christian_%C3%98rsted" title="Hans Christian Ørsted">Ørsted</a> og <a href="/wiki/Andr%C3%A9-Marie_Amp%C3%A8re" title="André-Marie Ampère">Ampère</a> på begynnelsen av 1800-tallet. Et <a href="/wiki/Magnetfelt" title="Magnetfelt">magnetfelt</a> <b>B</b> ble vist å utøve en kraft på en strømførende ledning. Hvis den fører strømmen <i>I</i> gjennom et lite stykke med lengde <i>d</i> <b>s</b> som befinner seg i feltet, kan kraften på dette strømelementet 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 d\mathbf {F} =Id\mathbf {s} \times \mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mi>I</mi> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">s</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\mathbf {F} =Id\mathbf {s} \times \mathbf {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02da38e3d6fb3bfccd1f090aa5902ac51d552c75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.182ex; height:2.176ex;" alt="{\displaystyle d\mathbf {F} =Id\mathbf {s} \times \mathbf {B} }"></span></dd></dl> <p>Dette resultatet gir en kompakt sammenfatning av <a href="/wiki/Amp%C3%A8res_kraftlov" title="Ampères kraftlov">Ampères kraftlov</a> som har en lang og omstendelig historie.<sup id="cite_ref-Tricker_1-0" class="reference"><a href="#cite_note-Tricker-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Det lyktes å finne denne loven selv om man på den tiden ikke hadde noen detaljert kunnskap om hva en elektrisk strøm besto av. </p><p>Men etter mange år med arbeider, spesielt innen kjemisk <a href="/wiki/Elektrolyse" title="Elektrolyse">elektrolyse</a>, ble det på slutten av århundret klart at strømmen besto av ladete <a href="/wiki/Elektron" title="Elektron">elektroner</a>. Den magnetiske kraften på en elektrisk strøm kunne da forstås som feltets virkning på hver av disse partiklene. Resultatet ble presentert av <a href="/wiki/Hendrik_Antoon_Lorentz" title="Hendrik Antoon Lorentz">Lorentz</a> i 1892 og kan utledes direkte fra Ampères kraftlov. Hvis en liten ladning <i>dq</i> i løpet av den korte tiden <i>dt</i> går gjennom et ledningselement <i>d</i> <b>s</b> med hastigheten <b>v</b>, vil man ha at <span class="nowrap"><i>d</i> <b>s</b> = <b>v</b> <i>dt</i>.</span> Men da strømmen i dette tilfellet er <span class="nowrap"><i>I</i> = <i>dq</i>/<i>dt</i></span>, vil uttrykket for kraften på strømelementet kunne omformes til <span class="nowrap"><i>d</i> <b>F</b> = <i>dq</i> <b>v</b> × <b>B</b></span>. For en endelig stor ladning blir da formelen for den magnetiske kraften </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =q\mathbf {v} \times \mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =q\mathbf {v} \times \mathbf {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8208e984160e97a12a3837a497561fd379bb048e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.003ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} =q\mathbf {v} \times \mathbf {B} }"></span></dd></dl> <p>Selv om samme resultat var funnet av <a href="/wiki/Maxwell" class="mw-disambig" title="Maxwell">Maxwell</a> og <a href="/wiki/Oliver_Heaviside" title="Oliver Heaviside">Heaviside</a> flere år tidligere,<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> har dette uttrykket blitt oppkalt etter Lorentz som gjennom sine publikasjoner gjorde det kjent.<sup id="cite_ref-Lorentz_3-0" class="reference"><a href="#cite_note-Lorentz-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>En mer generell utledning av Lorentz-kraften ble presentert av <a href="/wiki/Karl_Schwarzschild" title="Karl Schwarzschild">Schwarzschild</a> i 1903.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Han baserte sin beregning på den <a href="/wiki/Potensiell_energi" title="Potensiell energi">potensielle energien</a> for et elektron som befinner seg i en kombinasjon av elektriske og magnetiske felt. Ved bruk av <a href="/wiki/Lagrange-mekanikk" title="Lagrange-mekanikk">Lagrange-mekanikk</a> kom han dermed frem til den generelle formen til kraftloven. Siden den elektromagnetiske vekselvirkningen han benyttet, viste seg å være invariant under <a href="/wiki/Kovariant_relativitetsteori#Kontravariante_komponenter" title="Kovariant relativitetsteori">Lorentz-transformasjoner</a>, er derfor også Lorentz-kraften i overensstemmelse med <a href="/wiki/Einstein" class="mw-redirect" title="Einstein">Einsteins</a> <a href="/wiki/Spesiell_relativitetsteori" class="mw-redirect" title="Spesiell relativitetsteori">spesielle relativitetsteori</a>.<sup id="cite_ref-Griffiths_5-0" class="reference"><a href="#cite_note-Griffiths-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Partikkeldynamikk">Partikkeldynamikk</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz-kraft&veaction=edit&section=2" title="Rediger avsnitt: Partikkeldynamikk" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lorentz-kraft&action=edit&section=2" title="Rediger kildekoden til seksjonen Partikkeldynamikk"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Fil:Lorentz_force.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Lorentz_force.svg/280px-Lorentz_force.svg.png" decoding="async" width="280" height="228" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Lorentz_force.svg/420px-Lorentz_force.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8c/Lorentz_force.svg/560px-Lorentz_force.svg.png 2x" data-file-width="675" data-file-height="550" /></a><figcaption>En ladning <i>q</i> med hastighet <b>v</b> vil avbøyes oppover når den er negativ og nedover når den er positiv i et magnetfelt <b>B</b> som kommer ut av papirplanet.</figcaption></figure> <p>Når Lorentz-kraften virker på en partikkel med hastighet <b>v</b>, vil den utføre et <a href="/wiki/Arbeid" class="mw-disambig" title="Arbeid">arbeid</a> som er proporsjonal med <b>v</b>⋅<b>F</b>. Da den magnetiske delen <b>v</b> × <b>B</b> av kraften står <a href="/wiki/Vinkelrett" title="Vinkelrett">vinkelrett</a> på hastigheten, vil denne kraften derfor ikke bevirke noe arbeid slik at partikkelens energi forblir konstant. </p><p>Mer formelt følger det fra <a href="/wiki/Newtons_lover" class="mw-redirect" title="Newtons lover">Newtons andre lov</a>. Så lenge partikkelen med masse <i>m</i> beveger seg ikke-relativistisk, sier den at <span class="nowrap"><i>md</i> <b>v</b>/<i>dt</i> = <b>F</b></span>. Multipliseres denne ligningen med <b>v</b>, 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 m\mathbf {v} \cdot {d\mathbf {v} \over dt}={d \over dt}{\Big (}{1 \over 2}mv^{2}{\Big )}=q\mathbf {v} \cdot \mathbf {E} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <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> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>=</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\mathbf {v} \cdot {d\mathbf {v} \over dt}={d \over dt}{\Big (}{1 \over 2}mv^{2}{\Big )}=q\mathbf {v} \cdot \mathbf {E} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c30a64c9a95ee790465dc862ba956ae651b7a568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:32.595ex; height:5.509ex;" alt="{\displaystyle m\mathbf {v} \cdot {d\mathbf {v} \over dt}={d \over dt}{\Big (}{1 \over 2}mv^{2}{\Big )}=q\mathbf {v} \cdot \mathbf {E} }"></span></dd></dl> <p>Hvis det elektriske feltet <b>E</b> = 0, er derfor den <a href="/wiki/Kinetisk_energi" title="Kinetisk energi">kinetiske energien</a> <i>K</i> = <i>mv</i><sup>2</sup>/2  til partikkelen konstant. Men det forutsetter at det magnetiske feltet <b>B</b> ikke varierer med tiden. Hvis ikke, vil det skape et elektriske felt som en konsekvens av <a href="/wiki/Faradays_induksjonslov" title="Faradays induksjonslov">Faradays induksjonslov</a> <span class="nowrap"><b>∇</b> × <b>E</b> = - ∂<b>B</b>/∂<i>t</i>.</span> </p><p>Et konstant, elektrisk felt kan uttrykkes ved et <a href="/wiki/Elektrisk_potensial" title="Elektrisk potensial">elektrisk potensial</a> som <b>E</b> = - <b>∇</b> Φ. Da man har den matematiske sammenhengen <span class="nowrap"><b>v</b>⋅<b>∇</b> Φ = <i>d</i> Φ/<i>dt</i></span>, betyr det 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 {d \over dt}{\Big (}{1 \over 2}mv^{2}+q\Phi {\Big )}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>m</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>q</mi> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d \over dt}{\Big (}{1 \over 2}mv^{2}+q\Phi {\Big )}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/307126e72aae83c353efb86597d9ce097ac810f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:21.737ex; height:5.509ex;" alt="{\displaystyle {d \over dt}{\Big (}{1 \over 2}mv^{2}+q\Phi {\Big )}=0}"></span></dd></dl> <p>Den totale energien til partikkelen er derfor bevart i dette mer generelle tilfellet med elektriske og magnetiske felt som ikke forandrer seg med tiden. </p> <div class="mw-heading mw-heading3"><h3 id="Syklotronbevegelse">Syklotronbevegelse</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz-kraft&veaction=edit&section=3" title="Rediger avsnitt: Syklotronbevegelse" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lorentz-kraft&action=edit&section=3" title="Rediger kildekoden til seksjonen Syklotronbevegelse"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Fil:Lorentz_force_-_mural_Leiden_1,_2016.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Lorentz_force_-_mural_Leiden_1%2C_2016.jpg/300px-Lorentz_force_-_mural_Leiden_1%2C_2016.jpg" decoding="async" width="300" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/96/Lorentz_force_-_mural_Leiden_1%2C_2016.jpg/450px-Lorentz_force_-_mural_Leiden_1%2C_2016.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/96/Lorentz_force_-_mural_Leiden_1%2C_2016.jpg/600px-Lorentz_force_-_mural_Leiden_1%2C_2016.jpg 2x" data-file-width="5472" data-file-height="3648" /></a><figcaption>Illustrasjon på en husvegg i <a href="/wiki/Leiden" title="Leiden">Leiden</a> av partikkelbevegelse forårsaket av Lorentz-kraften.</figcaption></figure> <p>I et konstant, magnetisk felt er bevegelsen til partikkelen gitt ved 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 m{d\mathbf {v} \over dt}=q\mathbf {v} \times \mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <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> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{d\mathbf {v} \over dt}=q\mathbf {v} \times \mathbf {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2d676b19af84a00d747cec47270ec9f0a07eaa9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.824ex; height:5.509ex;" alt="{\displaystyle m{d\mathbf {v} \over dt}=q\mathbf {v} \times \mathbf {B} }"></span></dd></dl> <p>Da Lorentz-kraften virker normalt på retningen til <b>B</b>-feltet som kan tas å være langs <i>z</i>-aksen, vil komponenten av hastigheten <b>v</b> langs denne retningen forbli uforandret. Derimot vil de to transverse komponentene <span class="nowrap"><b>v</b><sub><i>T</i></sub> = (<i>v<sub>x</sub></i>,<i>v<sub>y</sub></i>)</span> forandre retning på en måte som følger fra </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 {d\mathbf {v} _{T} \over dt}={\boldsymbol {\omega }}\times \mathbf {v} _{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d\mathbf {v} _{T} \over dt}={\boldsymbol {\omega }}\times \mathbf {v} _{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca5fbf371cdebfda580324a03f9faadd0ad6aaab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.26ex; height:5.509ex;" alt="{\displaystyle {d\mathbf {v} _{T} \over dt}={\boldsymbol {\omega }}\times \mathbf {v} _{T}}"></span></dd></dl> <p>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 {\boldsymbol {\omega }}=-{q \over m}\mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <mi>m</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\omega }}=-{q \over m}\mathbf {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d10d8c170736673ba36802ce1e848b8d2bb7cde2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.353ex; height:4.843ex;" alt="{\displaystyle {\boldsymbol {\omega }}=-{q \over m}\mathbf {B} }"></span></dd></dl> <p>er konstant. Denne sammenhengen viser at hastigheten <b>v</b><sub><i>T</i></sub>  roterer om magnetfeltet <b>B</b> med <b>syklotronfrekvensen</b> <i>ω</i> = |<i><b>ω</b></i>| = <i>qB</i>/<i>m</i>. Dette er prinsippet som benyttes i <a href="/wiki/Syklotron" title="Syklotron">syklotronen</a> og i andre <a href="/wiki/Partikkelakselerator" title="Partikkelakselerator">partikkelakseleratorer</a>. </p><p>Under denne rotasjonen forblir størrelsen av hastigheten den samme, bare dens retning forandres. Det kan sees mer direkte ved å innføre partikkelens posisjon <b>r</b>. Da kan dens hastighet skrives som <b>v</b><sub><i>T</i></sub> = <i>d</i> <b>r</b>/<i>dt</i>. Bevegelsesligningen lar seg dermed direkte integereres og 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 \mathbf {v} _{T}={\boldsymbol {\omega }}\times \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ω</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{T}={\boldsymbol {\omega }}\times \mathbf {r} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53b1f01f13069d6d65cfccb89a1ef46a1f54f350" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.51ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{T}={\boldsymbol {\omega }}\times \mathbf {r} }"></span></dd></dl> <p>Hastigheten står derfor alltid <a href="/wiki/Vinkelrett" title="Vinkelrett">normalt</a> på posisjonsvektoren, noe som viser at partikkelen beveger seg i en sirkel med <a href="/wiki/Vinkelfrekvens" title="Vinkelfrekvens">vinkelfrekvens</a> lik med <i>ω</i>. Radius <i>r</i> til denne sirkelbevegelsen blir ofte omtalt som <b>gyroradius</b>. I et magnetfelt med en gitt størrelse bestemmer den energien til partikkelen da denne er proporsjonal med kvadratet av hastigheten. </p> <div class="mw-heading mw-heading3"><h3 id="Kartesisk_beskrivelse">Kartesisk beskrivelse</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz-kraft&veaction=edit&section=4" title="Rediger avsnitt: Kartesisk beskrivelse" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lorentz-kraft&action=edit&section=4" title="Rediger kildekoden til seksjonen Kartesisk beskrivelse"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ved å skrive ut bevegelsesligningene for de to kartesiske hastighetskomponentene <i>v<sub>x</sub></i>  og <i>v<sub>y</sub></i>, fremkommer de to første ordens <a href="/wiki/Differensialligning" title="Differensialligning">differensialligningene</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 {dv_{x} \over dt}=\omega v_{y},\;\;\;{dv_{y} \over dt}=-\omega v_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>ω</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>ω</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {dv_{x} \over dt}=\omega v_{y},\;\;\;{dv_{y} \over dt}=-\omega v_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67667d881dd8b3e1d02eb4f35ffa5bd1beb4d2af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.924ex; height:5.843ex;" alt="{\displaystyle {dv_{x} \over dt}=\omega v_{y},\;\;\;{dv_{y} \over dt}=-\omega v_{x}}"></span></dd></dl> <p>Etter å ha tatt den tidsderiverte av den første ligningen hvor så den siste blir satt inn, står man igjen med </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 {d^{2}v_{x} \over dt^{2}}+\omega ^{2}v_{x}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d^{2}v_{x} \over dt^{2}}+\omega ^{2}v_{x}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6631ea181d51648091ed4e49c2488391043dc02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.31ex; height:6.009ex;" alt="{\displaystyle {d^{2}v_{x} \over dt^{2}}+\omega ^{2}v_{x}=0}"></span>.</dd></dl> <p>som er <i>svingeligningen</i> for en <a href="/wiki/Harmonisk_oscillator" title="Harmonisk oscillator">harmonisk oscillator</a>. Da den er av andre orden, vil dens løsning inneholde to integrasjonskonstanter. De kan man for eksempel fastsette ved å anta at partikkelen krysser <i>x</i>-aksen ved tiden <i>t</i> = 0 med hastigheten <i>v<sub>T</sub></i>. Den betingelsen gir at <span class="nowrap"><i>v<sub>x</sub></i> = <i>v<sub>T</sub> </i> sin<i>ωt</i> </span> som igjen medfører at <span class="nowrap"><i>v<sub>y</sub></i> = <i>v<sub>T</sub> </i> cos<i>ωt</i></span>. Dermed er også kravet <span class="nowrap"><i>v<sub>x</sub></i><sup>2</sup> + <i>v<sub>y</sub></i><sup>2</sup> = <i>v<sub>T</sub></i><sup>2</sup> </span> oppfylt. Begge hastighetskomponentene varierer derfor harmonisk med tiden med en periode gitt ved syklotronfrekvensen som <span class="nowrap"><i>T</i> = 2<i>π</i> /<i>ω</i></span>. Tilsammen beskriver de bevegelsen til partikkelen i en sirkulær bane normalt på magnetfeltet. </p> <div class="mw-heading mw-heading2"><h2 id="Elektromagnetisk_induksjon">Elektromagnetisk induksjon</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz-kraft&veaction=edit&section=5" title="Rediger avsnitt: Elektromagnetisk induksjon" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lorentz-kraft&action=edit&section=5" title="Rediger kildekoden til seksjonen Elektromagnetisk induksjon"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Fil:Lorentzkraft_und_Induktion.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Lorentzkraft_und_Induktion.svg/300px-Lorentzkraft_und_Induktion.svg.png" decoding="async" width="300" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/Lorentzkraft_und_Induktion.svg/450px-Lorentzkraft_und_Induktion.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/16/Lorentzkraft_und_Induktion.svg/600px-Lorentzkraft_und_Induktion.svg.png 2x" data-file-width="720" data-file-height="560" /></a><figcaption>Beveger lederen seg i magnetfeltet, flyttes ladningene slik at det induseres et elektrisk felt.</figcaption></figure> <p>Lorentz-kraften kan gi seg utslag på mange forskjellige måter. Et eksempel er <a href="/wiki/Elektromagnetisk_induksjon" title="Elektromagnetisk induksjon">elektromagnetisk induksjon</a> hvor en ledning beveger seg med hastigheten <b>v</b> i et magnetfelt <b>B</b> slik at det oppstår en <a href="/wiki/Elektromotorisk_spenning" title="Elektromotorisk spenning">elektromotorisk spenning</a> i den. Kraften vil påvirke ladningene i lederen slik at de forflyttes. Dermed oppstår det et elektrisk felt <b>E</b> i den. Når dette blir sterkt nok, vil forflytningen av ladning opphøre. Da har man likevekt i ledningen definert av at totalkraften <span class="nowrap"><b>F</b> = <i>q</i>(<b>E</b> + <b>v</b> × <b>B</b>)</span> = 0. Det betyr at det induserte, elektriske feltet er gitt som <span class="nowrap"><b>E</b> = - <b>v</b> × <b>B</b>.</span> Står hastigheten og feltet vinkelrett på hverandre, er derfor den induserte spenningen <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 {E}}=vBL}"> <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">E</mi> </mrow> </mrow> <mo>=</mo> <mi>v</mi> <mi>B</mi> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}=vBL}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50b709fa8d4f2664407f1b4d74ba7f6d7cb4c084" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.884ex; height:2.176ex;" alt="{\displaystyle {\mathcal {E}}=vBL}"></span> over en lengde <i>L</i> av ledningen. Dette er samme resultat som kommer frem ved bruk av <a href="/wiki/Faradays_induksjonslov" title="Faradays induksjonslov">Faradays induksjonslov</a> basert på forandringen av den magnetiske fluksen som lederen omslutter.<sup id="cite_ref-Griffiths_5-1" class="reference"><a href="#cite_note-Griffiths-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/Fil:Lorentzkraft_und_Lenzsche_Regel.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Lorentzkraft_und_Lenzsche_Regel.svg/240px-Lorentzkraft_und_Lenzsche_Regel.svg.png" decoding="async" width="240" height="206" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/83/Lorentzkraft_und_Lenzsche_Regel.svg/360px-Lorentzkraft_und_Lenzsche_Regel.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/83/Lorentzkraft_und_Lenzsche_Regel.svg/480px-Lorentzkraft_und_Lenzsche_Regel.svg.png 2x" data-file-width="700" data-file-height="600" /></a><figcaption>Lorentz-kraften forklarer <a href="/wiki/Lenz%27_lov" class="mw-redirect" title="Lenz' lov">Lenz' lov</a> for en leder som beveges i et magnetfelt.</figcaption></figure> <p>Når ladningene i ledningen kan bevege seg fritt med hastighet <b>v</b><sub>1</sub>, vil de påvirkes av Lorentz-kraften <b>F</b><sub>1</sub> = <i>q</i> <b>v</b><sub>1</sub>× <b>B</b>  som gir dem en indusert hastighet <b>v</b><sub>2</sub> vinkelrett på <b>v</b><sub>1</sub>. Den igjen gir opphav til en ny Lorentz-kraft <b>F</b><sub>2</sub> = <i>q</i> <b>v</b><sub>2</sub>× <b>B</b>  som har motsatt retning av den opprinnelige bevegelsen av lederen og prøver å motvirke denne. Dette er forklaringen på <a href="/wiki/Lenz%27_lov" class="mw-redirect" title="Lenz' lov">Lenz' lov</a> i dette tilfellet. </p><p>Dette er også samme mekanismen som virker i <a href="/wiki/Hall-effekt" title="Hall-effekt">Hall-effekten</a>. Her ledes strøm gjennom et materiale vinkelrett på et magnetisk felt. Igjen vil det bygges opp i elektrisk felt i materialet med en retning som er avhengig av fortegnet til ladningsbærerne i materialet. Disse kan være elektroner med negativ ladning eller <a href="/wiki/Ladningsb%C3%A6rere" title="Ladningsbærere">hull</a> som oppfører seg som partikler med positiv ladning. I det ene tilfellet blir den induserte spenningen positiv og i det andre tilfellet negativ. På den måten har man påvist at <a href="/wiki/Ladningsb%C3%A6rere" title="Ladningsbærere">ladningsbærne</a> i <a href="/wiki/Kobber" title="Kobber">kobber</a> er elektroner, mens de i <a href="/wiki/Zink" class="mw-redirect" title="Zink">zink</a> er hovedsakelig positive hull.<sup id="cite_ref-AM_6-0" class="reference"><a href="#cite_note-AM-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Relativitetsteori">Relativitetsteori</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz-kraft&veaction=edit&section=6" title="Rediger avsnitt: Relativitetsteori" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lorentz-kraft&action=edit&section=6" title="Rediger kildekoden til seksjonen Relativitetsteori"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Hendrik_Antoon_Lorentz" title="Hendrik Antoon Lorentz">Lorentz</a> uviklet sin <i>Elektronteori</i> basert på eksistensen av <a href="/wiki/Eter_(fysikk)" title="Eter (fysikk)">eter</a> som definerer et bestemt <a href="/wiki/Inertialsystem" class="mw-redirect" title="Inertialsystem">inertialsystem</a> hvor <a href="/wiki/Maxwells_ligninger" class="mw-redirect" title="Maxwells ligninger">Maxwells ligninger</a> er gyldige.<sup id="cite_ref-elektronteori_7-0" class="reference"><a href="#cite_note-elektronteori-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Noen år senere viste <a href="/wiki/Einstein" class="mw-redirect" title="Einstein">Einstein</a> at man ikke behøver noen eter og at ligningene er gyldige i alle inertialsystem Det betyr også at uttrykket for Lorentz-kraften er gyldig i slike system når den uttrykkes ved hastigheter og felt som måles i det samme systemet. </p><p>En konsekvens av hans <a href="/wiki/Spesiell_relativitetsteori" class="mw-redirect" title="Spesiell relativitetsteori">spesielle relativitetsteori</a> er at et magnetfelt <b>B</b> som finnes i et inertialsystem, vil også observeres i et annet inertialsystem som beveger seg med hastigheten <b>v</b>, men sammen med et nytt elektrisk felt <b>E</b> = <b>v</b> × <b>B</b> så lenge hastigheten <i>v</i> er mye mindre enn <a href="/wiki/Lyshastigheten" class="mw-redirect" title="Lyshastigheten">lyshastigheten</a> <i>c</i>. Dette induserte feltet vil flytte ladningen i en leder som befinner seg i ro i det bevegelige inertialsystemet og dermed bygge opp en elektromotorisk spenning i denne. </p><p>På samme måte vil et elektrisk felt <b>E</b> i det første inertialsystemet gi opphav til et magnetisk felt <b>B</b> = - <b>v</b> × <b>E</b>/<i>c</i><sup> 2</sup>  i det andre inertialsystemet. Det var ikke uten grunn at Einstein kalte den opprinnelige artikkelen om den spesielle relativitetsteorien for <i>Zur Elektrodynamik bewegter Körper</i> (<i>Elektrodynamikk til bevegte legemer</i>). Han viste her at det <a href="/wiki/Elektromagnetisk_felt" title="Elektromagnetisk felt">elektromagnetiske feltet</a> vil være hovedsakelig elektrisk eller magnetisk avhengig av hvem som observerer det. </p> <div class="mw-heading mw-heading2"><h2 id="Kontinuerlig_ladningsfordeling">Kontinuerlig ladningsfordeling</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz-kraft&veaction=edit&section=7" title="Rediger avsnitt: Kontinuerlig ladningsfordeling" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lorentz-kraft&action=edit&section=7" title="Rediger kildekoden til seksjonen Kontinuerlig ladningsfordeling"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/Fil:Lorentz_force_continuum.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Lorentz_force_continuum.svg/200px-Lorentz_force_continuum.svg.png" decoding="async" width="200" height="251" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Lorentz_force_continuum.svg/300px-Lorentz_force_continuum.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Lorentz_force_continuum.svg/400px-Lorentz_force_continuum.svg.png 2x" data-file-width="348" data-file-height="437" /></a><figcaption>Lorentz-kraften som virker på et lite volumelement <i>dV</i> i en ladningsfordeling med tetthet <i>ρ</i> og strømtetthet <b>J</b> = <i>ρ</i> <b>v</b>, kan uttrykkes ved volumkraften <b>f</b> = <i>ρ</i> <b>E</b> + <b>J</b> × <b>B</b></figcaption></figure> <p>Hvis man betrakter et system bestående av en kontinuerling fordeling av partikler med ladningstetthet <span class="nowrap"><i>ρ</i> = <i>ρ</i>(<b>x</b>,<i>t</i>)</span>, vil ladningen i et lite volumelement <i>dV</i> være <i>dq</i> = <i>ρdV</i>. Kraften som virker på partiklene i dette volumelementet er da <span class="nowrap"><i>d</i> <b>F</b> = <i>ρ</i>(<b>E</b> + <b>v</b> × <b>B</b>)<i>dV</i></span>  hvor <span class="nowrap"><b>v</b> = <b>v</b>(<b>x</b>,<i>t</i>)</span> er <a href="/wiki/Kontinuitetsligning#Hastighetsfelt" title="Kontinuitetsligning">hastighetsfeltet</a> som sier hvordan partiklene beveger seg. Her vil nå <b>J</b> = <i>ρ</i> <b>v</b>  være <a href="/wiki/Elektrisk_str%C3%B8m" title="Elektrisk strøm">strømtettheten</a> i ladningsfordelingen. </p><p>Denne formen for den differensielle Lorentz-kraften gjør det naturlig å uttrykke den ved en <b>volumkraft</b> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} =\rho \mathbf {E} +\mathbf {J} \times \mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">f</mi> </mrow> <mo>=</mo> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</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">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {f} =\rho \mathbf {E} +\mathbf {J} \times \mathbf {B} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2b874c8e0f6ad41637c0c165c369b07a7a8cebc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.073ex; height:2.676ex;" alt="{\displaystyle \mathbf {f} =\rho \mathbf {E} +\mathbf {J} \times \mathbf {B} }"></span></dd></dl> <p>som virker på hvert differensielt volumelement <i>dV</i> av ladningsfordelingen. Man kan her bruke <a href="/wiki/Maxwells_ligninger" class="mw-redirect" title="Maxwells ligninger">Maxwells ligninger</a> til å uttrykke ladningstettheten <i>ρ</i> og strømtettheten <b>J</b> ved de <a href="/wiki/Elektromagnetisk_felt" title="Elektromagnetisk felt">elektromagnetiske feltene</a> slik at volumkraften tar 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 {f} +{\dfrac {1}{c^{2}}}{\dfrac {\partial \mathbf {S} }{\partial t}}={\boldsymbol {\nabla }}\cdot {\boldsymbol {\sigma }}}"> <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"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mn>1</mn> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> </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 \mathbf {f} +{\dfrac {1}{c^{2}}}{\dfrac {\partial \mathbf {S} }{\partial t}}={\boldsymbol {\nabla }}\cdot {\boldsymbol {\sigma }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2faf432d36d03e9d7dca7cdcf2fa06bcf4cbfe64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.029ex; height:5.676ex;" alt="{\displaystyle \mathbf {f} +{\dfrac {1}{c^{2}}}{\dfrac {\partial \mathbf {S} }{\partial t}}={\boldsymbol {\nabla }}\cdot {\boldsymbol {\sigma }}}"></span></dd></dl> <p>hvor <b>S</b> = <b>E</b> × <b>H</b>  er <a href="/wiki/Poyntings_vektor" title="Poyntings vektor">Poyntings vektor</a> som angir strømmen av energi fra ladningsfordelingen når feltene varierer med tiden. I tillegg er <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 {\sigma }}=(\sigma _{ij})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\sigma }}=(\sigma _{ij})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcefd32ad3d1f91dbf95dbd9eb1f89f0cc3249db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.307ex; height:3.009ex;" alt="{\displaystyle {\boldsymbol {\sigma }}=(\sigma _{ij})}"></span> <a href="/w/index.php?title=Maxwells_spenningstensor&action=edit&redlink=1" class="new" title="Maxwells spenningstensor (ikke skrevet ennå)">Maxwells spenningstensor</a> som gjør det mulig å beregne de elektromagnetiske kreftene som virker både inni og utenpå ladningsfordelingen.<sup id="cite_ref-Brau_8-0" class="reference"><a href="#cite_note-Brau-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Den totale Lorentz-kraften er gitt ved det tredimensjonale volumintegralet </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} =\int \!dV(\rho \mathbf {E} +\mathbf {J} \times \mathbf {B} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mo>∫</mo> <mspace width="negativethinmathspace" /> <mi>d</mi> <mi>V</mi> <mo stretchy="false">(</mo> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</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">B</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =\int \!dV(\rho \mathbf {E} +\mathbf {J} \times \mathbf {B} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/102b7814ac8a1dc863c443a3434498f5982a905e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.709ex; height:5.676ex;" alt="{\displaystyle \mathbf {F} =\int \!dV(\rho \mathbf {E} +\mathbf {J} \times \mathbf {B} )}"></span></dd></dl> <p>som går over hele rommet hvor ladningene befinner seg. Det spiller en viktig rolle innen <a href="/wiki/Plasma_(fysikk)" title="Plasma (fysikk)">plasmafysikken</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Matematisk_utledning">Matematisk utledning</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz-kraft&veaction=edit&section=8" title="Rediger avsnitt: Matematisk utledning" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lorentz-kraft&action=edit&section=8" title="Rediger kildekoden til seksjonen Matematisk utledning"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Grunnen til at Lorentz-kraften er konsistent med Einsteins relativitetsteori, er at den kan utledes fra en <a href="/wiki/Kovariant_relativitetsteori#Relativistisk_bevegelsesligning" title="Kovariant relativitetsteori">Lagrange-funksjon</a> som er invariant under <a href="/wiki/Spesiell_relativitetsteori" class="mw-redirect" title="Spesiell relativitetsteori">Lorentz-transformasjoner</a>. For en relativistisk partikkel med hastighet <b>v</b> kan den 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 L=-mc^{2}{\sqrt {1-v^{2}/c^{2}}}-q(\Phi -\mathbf {v} \cdot \mathbf {A} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mo>−</mo> <mi>m</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>−</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Φ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=-mc^{2}{\sqrt {1-v^{2}/c^{2}}}-q(\Phi -\mathbf {v} \cdot \mathbf {A} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71a82d7165289db0ee583b9777fe9cbdcefb53b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:37.67ex; height:4.843ex;" alt="{\displaystyle L=-mc^{2}{\sqrt {1-v^{2}/c^{2}}}-q(\Phi -\mathbf {v} \cdot \mathbf {A} )}"></span></dd></dl> <p>hvor Φ er det <a href="/wiki/Elektrisk_potensial" title="Elektrisk potensial">elektriske potensialet</a> og <b>A</b> det <a href="/wiki/Magnetfelt" title="Magnetfelt">magnetiske vektorpotensialet</a>. De inngår i den elektromagnetiske vekselvirkningen på en slik måte at den er invariant under gaugetransformasjoner. </p><p>I <a href="/wiki/Lagrange-mekanikk" title="Lagrange-mekanikk">Euler-Lagrange-ligningen</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 {\partial L \over \partial \mathbf {x} }-{d \over dt}{\partial L \over \partial \mathbf {v} }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\partial L \over \partial \mathbf {x} }-{d \over dt}{\partial L \over \partial \mathbf {v} }=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a2160c10049d64a206e59411bbb6e6229de387e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.467ex; height:5.509ex;" alt="{\displaystyle {\partial L \over \partial \mathbf {x} }-{d \over dt}{\partial L \over \partial \mathbf {v} }=0}"></span></dd></dl> <p>hvor <b>v</b> = <i>d</i> <b>x</b>/<i>dt</i>, inngår den kanoniske impulsen </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {p} ={\partial L \over \partial \mathbf {v} }={m\mathbf {v} \over {\sqrt {1-v^{2}/c^{2}}}}+q\mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">p</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {p} ={\partial L \over \partial \mathbf {v} }={m\mathbf {v} \over {\sqrt {1-v^{2}/c^{2}}}}+q\mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25eae29bbfbf01a5611999bde574d94e259dada6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.917ex; height:6.676ex;" alt="{\displaystyle \mathbf {p} ={\partial L \over \partial \mathbf {v} }={m\mathbf {v} \over {\sqrt {1-v^{2}/c^{2}}}}+q\mathbf {A} }"></span></dd></dl> <p>sammen med </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 L \over \partial \mathbf {x} }=-q{\boldsymbol {\nabla }}\Phi +q{\boldsymbol {\nabla }}(\mathbf {v} \cdot \mathbf {A} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mi mathvariant="normal">Φ</mi> <mo>+</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\partial L \over \partial \mathbf {x} }=-q{\boldsymbol {\nabla }}\Phi +q{\boldsymbol {\nabla }}(\mathbf {v} \cdot \mathbf {A} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a17534c175e2071095a0bc6d18bd63edf90bede5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:26.672ex; height:5.509ex;" alt="{\displaystyle {\partial L \over \partial \mathbf {x} }=-q{\boldsymbol {\nabla }}\Phi +q{\boldsymbol {\nabla }}(\mathbf {v} \cdot \mathbf {A} )}"></span></dd></dl> <p>Ved å benytte 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 {d\mathbf {A} \over dt}={\partial \mathbf {A} \over \partial t}+(\mathbf {v} \cdot {\boldsymbol {\nabla }})\mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d\mathbf {A} \over dt}={\partial \mathbf {A} \over \partial t}+(\mathbf {v} \cdot {\boldsymbol {\nabla }})\mathbf {A} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0574fbd10f0c5d26f17efc807461b461c1e60dec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.329ex; height:5.509ex;" alt="{\displaystyle {d\mathbf {A} \over dt}={\partial \mathbf {A} \over \partial t}+(\mathbf {v} \cdot {\boldsymbol {\nabla }})\mathbf {A} }"></span></dd></dl> <p>sammen med <a href="/wiki/Vektoranalyse" title="Vektoranalyse">vektoridentiteten</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} \times ({\boldsymbol {\nabla }}\times \mathbf {A} )={\boldsymbol {\nabla }}(\mathbf {v} \cdot \mathbf {A} )-(\mathbf {v} \cdot {\boldsymbol {\nabla }})\mathbf {A} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</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">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} \times ({\boldsymbol {\nabla }}\times \mathbf {A} )={\boldsymbol {\nabla }}(\mathbf {v} \cdot \mathbf {A} )-(\mathbf {v} \cdot {\boldsymbol {\nabla }})\mathbf {A} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee407bb3c7ea179af8bf55e217c05ab1f6277151" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.023ex; height:2.843ex;" alt="{\displaystyle \mathbf {v} \times ({\boldsymbol {\nabla }}\times \mathbf {A} )={\boldsymbol {\nabla }}(\mathbf {v} \cdot \mathbf {A} )-(\mathbf {v} \cdot {\boldsymbol {\nabla }})\mathbf {A} ,}"></span></dd></dl> <p>kommer man dermed fram til bevegelsesligningen </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 {d \over dt}{m\mathbf {v} \over {\sqrt {1-v^{2}/c^{2}}}}=q(\mathbf {E} +\mathbf {v} \times \mathbf {B} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>−</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>q</mi> <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">v</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {d \over dt}{m\mathbf {v} \over {\sqrt {1-v^{2}/c^{2}}}}=q(\mathbf {E} +\mathbf {v} \times \mathbf {B} ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0888334add4583a7936732d194ba9be4ff5e8a44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.834ex; height:6.676ex;" alt="{\displaystyle {d \over dt}{m\mathbf {v} \over {\sqrt {1-v^{2}/c^{2}}}}=q(\mathbf {E} +\mathbf {v} \times \mathbf {B} ).}"></span></dd></dl> <p>På venstre side inngår den tidsderiverte av den relativistiske impulsen til partikkelen som er <i>m</i> <b>v</b> ved lave hastigheter. Lorentz-kraften opptrer på høyre side av resultatet med det elektriske feltet <span class="nowrap"><b>E</b> = -<b>∇</b> Φ - <i>∂</i> <b>A</b>/<i>∂ t</i></span> og det magnetiske feltet <span class="nowrap"><b>B</b> = <b>∇</b> × <b>A</b>.</span> </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=Lorentz-kraft&veaction=edit&section=9" 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=Lorentz-kraft&action=edit&section=9" 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_potensial" title="Elektrisk potensial">Elektrisk potensial</a></li> <li><a href="/wiki/Magnetisk_felt" class="mw-redirect" title="Magnetisk felt">Magnetisk felt</a></li> <li><a href="/wiki/Maxwells_likninger" title="Maxwells likninger">Maxwells likninger</a></li> <li><a href="/wiki/Elektromagnetiske_felt" class="mw-redirect" title="Elektromagnetiske felt">Elektromagnetiske felt</a></li> <li><a href="/wiki/Elektrodynamikk" title="Elektrodynamikk">Elektrodynamikk</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Referanser">Referanser</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Lorentz-kraft&veaction=edit&section=10" title="Rediger avsnitt: Referanser" class="mw-editsection-visualeditor"><span>rediger</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Lorentz-kraft&action=edit&section=10" 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-Tricker-1"><b><a href="#cite_ref-Tricker_1-0">^</a></b> <span class="reference-text"> R.A.R. Tricker, <i>Early Electrodynamics</i>, Pergamon Press, London (1965).</span> </li> <li id="cite_note-Darrigol-2"><b><a href="#cite_ref-Darrigol_2-0">^</a></b> <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-Lorentz-3"><b><a href="#cite_ref-Lorentz_3-0">^</a></b> <span class="reference-text"> H.A. Lorentz, <a rel="nofollow" class="external text" href="https://archive.org/stream/electronstheory00lorerich#page/n3/mode/2up"><i>The Theory of Electrons</i></a>, B.G. Teubner, Leipzig (1916). </span> </li> <li id="cite_note-4"><b><a href="#cite_ref-4">^</a></b> <span class="reference-text"> K. Schwarzschild, <i>Zur Elektrodynamik</i>. I: <i>Zwei Formen des Princips der kleinsten Action in der Elektronentheorie</i>, Gött. Nach., Math.-Phys. Kl. 126-131 (1903).</span> </li> <li id="cite_note-Griffiths-5"><b>^</b> <a href="#cite_ref-Griffiths_5-0"><sup>a</sup></a> <a href="#cite_ref-Griffiths_5-1"><sup>b</sup></a> <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-AM-6"><b><a href="#cite_ref-AM_6-0">^</a></b> <span class="reference-text"> N.W. Ashcroft and N.D. Mermin, <i>Solid State Physics</i>, Holt-Saunders International Editions, Tokyo (1981). <a href="/wiki/Spesial:Bokkilder/0030493463" class="internal mw-magiclink-isbn">ISBN 0-03-049346-3</a>.</span> </li> <li id="cite_note-elektronteori-7"><b><a href="#cite_ref-elektronteori_7-0">^</a></b> <span class="reference-text"> Salmonsens Konversationsleksikon (1915-1930), <a rel="nofollow" class="external text" href="https://runeberg.org/salmonsen/2/7/0078.html"><i>Elektronteorien</i>, Bind VII</a>, Projekt Runeberg elektronisk utgave. </span> </li> <li id="cite_note-Brau-8"><b><a href="#cite_ref-Brau_8-0">^</a></b> <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> </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=Lorentz-kraft&veaction=edit&section=11" 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=Lorentz-kraft&action=edit&section=11" title="Rediger kildekoden til seksjonen Eksterne lenker"><span>rediger kilde</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>R.P. Feynman, <a rel="nofollow" class="external text" href="http://www.feynmanlectures.caltech.edu/II_26.html"><i>Lorentz transformations of the fields</i></a>, Lectures on Physics, Volume II, Caltech, Pasadena (2013).</li> <li>NCERT, <a rel="nofollow" class="external text" href="http://ncert.nic.in/ncerts/l/leph104.pdf"><i>Moving charges and magnetism</i></a>, del av indisk forelesningsserie om elektromagnetisme.</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 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