Lorentz force | Equation, Properties, & Direction

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class="drawer d-flex flex-column open"> <div class="left-rail-section-content"> <div class="topic-left-rail-header text-truncate bg-gray-50 position-relative text-right d-flex align-items-center"> <div class="tlr-title px-20 py-15 text-left"> <em class="material-icons text-gray-400 d-lg-none" data-icon="toc"></em> <a class="font-serif font-weight-bold text-black link-blue" href="https://www.britannica.com/science/Lorentz-force">Lorentz force</a> </div> <button aria-label="Close" class="js-sections-close-button btn-link btn-sm btn d-lg-none position-absolute top-0 p-10 right-0" > <em class="material-icons font-26" data-icon="close"></em> </button> </div> <div class="section-content pl-10 pr-20 pl-sm-50 pr-sm-60 pl-lg-5 pr-lg-10 pt-10 pt-lg-0 bg-gray-50 clear-catfish-ad"> <div class="toc mb-20"> <div class="font-serif font-14 font-weight-bold mx-15 mb-15 mt-20"> Table of Contents </div> <span class="toc-extra-link selected link-gray-900 mt-15"> Introduction </span> <a 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in Earth's magnetic field" height="50" /> </a> <a href="https://cdn.britannica.com/78/478-004-A8BC3FA6/system-shells-magnetic-field-lines-ionosphere-magnetosphere.jpg" data-href="/media/1/348139/1168" class="media-overlay-link d-inline-block mr-5"> <img loading="lazy" src="https://cdn.britannica.com/78/478-004-A8BC3FA6/system-shells-magnetic-field-lines-ionosphere-magnetosphere.jpg" alt="field-aligned current system" height="50" /> </a> </div> <button disabled class="prev-button js-prev-button position-absolute btn btn-circle shadow btn-blue " aria-label="Previous"> <span class="material-icons md-24" data-icon="keyboard_arrow_left"></span> </button> <button disabled class="next-button js-next-button position-absolute btn btn-circle shadow btn-blue " aria-label="Next"> <span class="material-icons md-24" data-icon="keyboard_arrow_right"></span> </button> </div> </div> <div class="mb-30 tlr-related-questions"> <div class="text-gray-900 p-5 pb-0 font-serif font-14 font-weight-bold mx-10 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They write new content and verify and edit content received from contributors.</div> </a> <div data-popper-arrow></div> </div> <span class="btn btn-link editor-link p-0 qa-byline-link gtm-byline font-12 byline-contributor text-decoration-underline"> The Editors of Encyclopaedia Britannica</span></div> <div class="last-updated font-12 font-serif"> <span class="text-gray-700"> Last Updated: <time datetime="2025-01-28T00:00:00CST" >Jan 28, 2025</time> •</span> <a class="byline-edit-history" href="https://www.britannica.com/science/Lorentz-force/additional-info#history" rel="nofollow">Article History</a> </div></div> </div> <button class="d-flex d-lg-none btn btn-outline-blue border rounded-sm shadow-sm mobile-toc-button gtm-mobile-toc-inline-button d-none d-sm-block js-sections-inline-button module-spacing btn d-lg-none"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="toc"></em> Table of Contents </button> <div class="d-flex d-sm-none flex-row"> <button class="d-flex d-lg-none btn btn-outline-blue border rounded-sm shadow-sm mobile-toc-button gtm-mobile-toc-inline-button js-sections-inline-button module-spacing"> <em class="material-icons mr-5 ml-n10 my-n5 md-icon" data-icon="toc"></em> Table of Contents </button> <button class="ai-ask-button btn btn-sm border-2 btn-outline-red-400 border-red-400 module-spacing js-inline-ai-ask-button p-10 ml-5"> Ask the Chatbot </button> </div> <div class="js-qf-module qf-module px-40 px-sm-20 py-15 mx-auto module-spacing font-14 bg-gray-50 rounded"> <div class="facts-list mt-10"> <div class=""> <div class="js-fact mb-10 line-clamp clamp-3"> <dl> <dt>Related Topics: </dt> <dd><a href="/science/magnetic-field" topicid="357048">magnetic field</a></dd> </dl> <button class="js-more-btn d-none btn btn-unstyled font-12 bg-gray-50" aria-label="Toggle more/less fact data"> <em class="js-content link-blue">(Show more)</em> </button> </div> <div class="text-center"> <a class="btn btn-sm btn-link p-0" href="/facts/Lorentz-force"> See all related content </a> </div> </div> </div> </div><span class="marker before-article"></span><section data-level="1" id="ref1"><span class="marker PREMOD1 mod-inline"></span><p class="topic-paragraph"><strong><span id="ref1014975"></span>Lorentz force</strong>, the <a href="https://www.britannica.com/science/force-physics" class="md-crosslink autoxref " data-show-preview="true">force</a> <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="exerted" href="https://www.britannica.com/dictionary/exerted" data-type="EB">exerted</a> on a <a href="https://www.britannica.com/science/electric-charge" class="md-crosslink " data-show-preview="true">charged</a> particle <em>q</em> moving with velocity <em><strong>v</strong></em> through an <span id="ref1073051"></span><a href="https://www.britannica.com/science/electric-field" class="md-crosslink " data-show-preview="true">electric</a> <a href="https://www.britannica.com/science/field-physics" class="md-crosslink autoxref " data-show-preview="true">field</a> <em><strong>E</strong></em> and <span id="ref1073050"></span><a href="https://www.britannica.com/science/magnetic-field" class="md-crosslink " data-show-preview="true">magnetic field</a> <em><strong>B</strong></em>. The entire <a href="https://www.britannica.com/science/electromagnetism" class="md-crosslink " data-show-preview="true">electromagnetic</a> force <em><strong>F</strong></em> on the charged particle is called the Lorentz force (after the Dutch physicist <a href="https://www.britannica.com/biography/Hendrik-Antoon-Lorentz" class="md-crosslink " data-show-preview="true">Hendrik A. Lorentz</a>) and is given by <span class="md-formula"><em><strong>F</strong></em> = <em>q</em><em><strong>E</strong></em> + <em>q</em><em><strong>v</strong></em> × <em><strong>B</strong></em>.</span></p><span class="marker MOD1 mod-inline"></span><span class="marker PREMOD2 mod-inline"></span><p class="topic-paragraph">The first term is contributed by the <a href="https://www.britannica.com/science/electric-field" class="md-crosslink " data-show-preview="true">electric field</a>. The second term is the <a href="https://www.britannica.com/science/magnetism" class="md-crosslink " data-show-preview="true">magnetic</a> force and has a direction perpendicular to both the velocity and the magnetic field. The <a href="https://www.britannica.com/science/magnetic-force" class="md-crosslink autoxref " data-show-preview="true">magnetic force</a> is proportional to <em>q</em> and to the magnitude of the <a href="https://www.britannica.com/science/vector-mathematics" class="md-crosslink " data-show-preview="true">vector</a> <a href="https://www.britannica.com/science/cross-product" class="md-crosslink autoxref " data-show-preview="true">cross product</a> <em><strong>v</strong></em> × <em><strong>B</strong></em>. In terms of the angle ϕ between <em><strong>v</strong></em> and <em><strong>B</strong></em>, the magnitude of the force equals <em>q</em><em>v</em><em>B</em> sin ϕ. An interesting result of the Lorentz force is the motion of a charged particle in a uniform magnetic field. If <em><strong>v</strong></em> is perpendicular to <em><strong>B</strong></em> (i.e., with the angle ϕ between <em><strong>v</strong></em> and <em><strong>B</strong></em> of 90°), the particle will follow a circular <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="trajectory" href="https://www.britannica.com/dictionary/trajectory" data-type="EB">trajectory</a> with a radius of <em>r</em> = <em>m</em><em>v</em>/<em>q</em><em>B</em>. If the angle ϕ is less than 90°, the particle orbit will be a helix with an axis parallel to the field lines. If ϕ is zero, there will be no magnetic force on the particle, which will continue to move undeflected along the field lines. Charged <a href="https://www.britannica.com/technology/particle-accelerator" class="md-crosslink " data-show-preview="true">particle accelerators</a> like <a href="https://www.britannica.com/technology/cyclotron" class="md-crosslink " data-show-preview="true">cyclotrons</a> make use of the fact that particles move in a circular orbit when <em><strong>v</strong></em> and <em><strong>B</strong></em> are at right angles. For each revolution, a carefully timed electric field gives the particles additional <a href="https://www.britannica.com/science/kinetic-energy" class="md-crosslink " data-show-preview="true">kinetic energy</a>, which makes them travel in increasingly larger orbits. When the particles have <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="acquired" href="https://www.britannica.com/dictionary/acquired" data-type="EB">acquired</a> the desired energy, they are extracted and used in a number of different ways, from studies of <a href="https://www.britannica.com/science/subatomic-particle" class="md-crosslink " data-show-preview="true">subatomic particles</a> to the medical treatment of <a href="https://www.britannica.com/science/cancer-disease" class="md-crosslink " data-show-preview="true">cancer</a>.</p><span class="marker MOD2 mod-inline"></span><span class="marker PREMOD3 mod-inline"></span><p class="topic-paragraph">The magnetic force on a moving charge reveals the sign of the <span id="ref1073055"></span><a href="https://www.britannica.com/science/charge-carrier" class="md-crosslink ">charge carriers</a> in a <span id="ref1073053"></span><a href="https://www.britannica.com/science/electrical-conductor" class="md-crosslink ">conductor</a>. A <a href="https://www.britannica.com/science/electric-current" class="md-crosslink " data-show-preview="true">current</a> flowing from right to left in a conductor can be the result of positive charge carriers moving from right to left or negative charges moving from left to right, or some combination of each. When a conductor is placed in a <em><strong>B</strong></em> field perpendicular to the current, the magnetic force on both types of charge carriers is in the same direction. This force gives rise to a small potential difference between the sides of the conductor. Known as the <span id="ref1073052"></span><a href="https://www.britannica.com/science/Hall-effect" class="md-crosslink " data-show-preview="true">Hall effect</a>, this phenomenon (discovered by the American physicist <a href="https://www.britannica.com/biography/Edward-Hall-English-historian" class="md-crosslink " data-show-preview="true">Edwin H. Hall</a>) results when an electric field is aligned with the direction of the magnetic force. The Hall effect shows that <a href="https://www.britannica.com/science/electron" class="md-crosslink " data-show-preview="true">electrons</a> dominate the <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="conduction" href="https://www.britannica.com/dictionary/conduction" data-type="EB">conduction</a> of <a href="https://www.britannica.com/science/electricity" class="md-crosslink autoxref " data-show-preview="true">electricity</a> in <a href="https://www.britannica.com/science/copper" class="md-crosslink " data-show-preview="true">copper</a>. In <a href="https://www.britannica.com/science/zinc" class="md-crosslink " data-show-preview="true">zinc</a>, however, conduction is dominated by the motion of positive charge carriers. Electrons in zinc that are excited from the <a href="https://www.britannica.com/science/valence-electron" class="md-crosslink " data-show-preview="true">valence</a> band leave holes, which are vacancies (i.e., unfilled levels) that behave like positive charge carriers. The motion of these holes accounts for most of the conduction of electricity in zinc.</p><div class="module-spacing"> </div><span class="marker MOD3 mod-inline"></span><span class="marker PREMOD4 mod-inline"></span><p class="topic-paragraph">If a wire with a current <em>i</em> is placed in an external magnetic field <em><strong>B</strong></em>, how will the force on the wire depend on the orientation of the wire? Since a current represents a movement of charges in the wire, the Lorentz force acts on the moving charges. Because these charges are bound to the conductor, the magnetic forces on the moving charges are transferred to the wire. The force on a small length <em>d</em><em><strong>l</strong></em> of the wire depends on the orientation of the wire with respect to the field. The magnitude of the force is given by <em>i</em><em>d</em><em><strong>lB</strong></em> sin ϕ, where ϕ is the angle between <em><strong>B</strong></em> and <em>d</em><em><strong>l</strong></em>. There is no force when ϕ = 0 or 180°, both of which correspond to a current along a direction parallel to the field. The force is at a maximum when the current and field are perpendicular to each other. The force is given by <span class="md-formula"><em>d</em><em><strong>F</strong></em>= <em>i</em><em>d</em><em><strong>l</strong></em> × <em><strong>B</strong></em>.</span></p><div class="one-good-fact-module"> </div><span class="marker MOD4 mod-inline"></span><span class="marker PREMOD5 mod-inline"></span><p class="topic-paragraph">Again, the <a class="md-dictionary-link md-dictionary-tt-off eb" data-term="vector" href="https://www.britannica.com/dictionary/vector" data-type="EB">vector</a> cross product denotes a direction perpendicular to both <em>d</em><em><strong>l</strong></em> and <em><strong>B</strong></em>.</p><span class="marker MOD5 mod-inline"></span></section> <span class="md-signature font-12"><a href="/editor/The-Editors-of-Encyclopaedia-Britannica/4419">The Editors of Encyclopaedia Britannica</a></span> <span class="md-signature font-12">This article was most recently revised and updated by <a href="/editor/Erik-Gregersen/6723">Erik Gregersen</a>.</span><span class="marker end-of-content"></span><span class="marker after-article"></span></div> <div id="chatbot-simplify-root"></div> <div id="chatbot-root"></div> </div> </div> </div> <div class="ai-dialog-placeholder"></div> </div> </div> <aside class="col-md-da-320"></aside> </div> </div> </div> </div> </article> </div> </div> </div> </div> </main> <div id="md-footer"></div> <noscript><iframe src="//www.googletagmanager.com/ns.html?id=GTM-5W6NC8" height="0" width="0" style="display:none;visibility:hidden"></iframe></noscript>  <script> window.ap3c = window.ap3c || {}; var ap3c = window.ap3c; ap3c.cmd = ap3c.cmd || []; ap3c.cmd.push(function() { ap3c.init('ZO4siT4cLwnykPnzZWJtd3Byb2Q', 'https://engage.email.britannica.com/'); ap3c.track({v: 0}); }); ap3c.activity = function(act) { ap3c.act = (ap3c.act || []); ap3c.act.push(act); }; var s, t; s = document.createElement('script'); s.type = 'text/javascript'; s.src = "https://engage.email.britannica.com/app.js"; t = document.getElementsByTagName('script')[0]; t.parentNode.insertBefore(s, t); </script> <script class="marketing-page-info" type="application/json"> {"pageType":"Topic","templateName":"DESKTOP","pageNumber":1,"pagesTotal":1,"pageId":348139,"pageLength":657,"initialLoad":true,"lastPageOfScroll":false} </script> <script class="marketing-content-info" type="application/json"> [] </script> <script src="https://cdn.britannica.com/mendel-resources/3-133/js/libs/jquery-3.5.0.min.js?v=3.133.36"></script> <script type="text/javascript" data-type="Init Mendel Code Splitting"> (function() { $.ajax({ dataType: 'script', cache: true, url: 'https://cdn.britannica.com/mendel-resources/3-133/dist/topic-page.js?v=3.133.36' }); })(); </script> <script class="analytics-metadata" type="application/json"> {"leg":"C","adLeg":"C","userType":"ANONYMOUS","pageType":"Topic","pageSubtype":null,"articleTemplateType":"MEDIUM","gisted":false,"pageNumber":1,"hasSummarizeButton":false,"hasAskButton":true} </script> <script type="text/javascript"> EBStat={accountId:-1,hostnameOverride:'webstats.eb.com',domain:'www.britannica.com', json:''}; </script> <script type="text/javascript"> ( function() { $.ajax( { dataType: 'script', cache: true, url: '//www.britannica.com/webstats/mendelstats.js?v=1' } ) .done( function() { try {writeStat(null,EBStat);} catch(err){} } ); })(); </script> <div id="bc-fixed-dialogue"></div> </body> </html>

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