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id="toc-The_H-field" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_H-field"> <div class="vector-toc-text"> 1.2 The H-field </div> </a> <ul id="toc-The_H-field-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Measurement" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Measurement"> <div class="vector-toc-text"> 1.3 Measurement </div> </a> <ul id="toc-Measurement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Visualization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Visualization"> <div class="vector-toc-text"> 1.4 Visualization </div> </a> <ul id="toc-Visualization-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Magnetic_field_of_permanent_magnets" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Magnetic_field_of_permanent_magnets"> <div class="vector-toc-text"> 2 Magnetic field of permanent magnets </div> </a> <button aria-controls="toc-Magnetic_field_of_permanent_magnets-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Magnetic field of permanent magnets subsection </button> <ul id="toc-Magnetic_field_of_permanent_magnets-sublist" class="vector-toc-list"> <li id="toc-Magnetic_pole_model" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Magnetic_pole_model"> <div class="vector-toc-text"> 2.1 Magnetic pole model </div> </a> <ul id="toc-Magnetic_pole_model-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Amperian_loop_model" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Amperian_loop_model"> <div class="vector-toc-text"> 2.2 Amperian loop model </div> </a> <ul id="toc-Amperian_loop_model-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Interactions_with_magnets" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Interactions_with_magnets"> <div class="vector-toc-text"> 3 Interactions with magnets </div> </a> <button aria-controls="toc-Interactions_with_magnets-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Interactions with magnets subsection </button> <ul id="toc-Interactions_with_magnets-sublist" class="vector-toc-list"> <li id="toc-Force_between_magnets" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Force_between_magnets"> <div class="vector-toc-text"> 3.1 Force between magnets </div> </a> <ul id="toc-Force_between_magnets-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Magnetic_torque_on_permanent_magnets" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Magnetic_torque_on_permanent_magnets"> <div class="vector-toc-text"> 3.2 Magnetic torque on permanent magnets </div> </a> <ul id="toc-Magnetic_torque_on_permanent_magnets-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Interactions_with_electric_currents" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Interactions_with_electric_currents"> <div class="vector-toc-text"> 4 Interactions with electric currents </div> </a> <button aria-controls="toc-Interactions_with_electric_currents-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Interactions with electric currents subsection </button> <ul id="toc-Interactions_with_electric_currents-sublist" class="vector-toc-list"> <li id="toc-Magnetic_field_due_to_moving_charges_and_electric_currents" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Magnetic_field_due_to_moving_charges_and_electric_currents"> <div class="vector-toc-text"> 4.1 Magnetic field due to moving charges and electric currents </div> </a> <ul id="toc-Magnetic_field_due_to_moving_charges_and_electric_currents-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Force_on_moving_charges_and_current" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Force_on_moving_charges_and_current"> <div class="vector-toc-text"> 4.2 Force on moving charges and current </div> </a> <ul id="toc-Force_on_moving_charges_and_current-sublist" class="vector-toc-list"> <li id="toc-Force_on_a_charged_particle" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Force_on_a_charged_particle"> <div class="vector-toc-text"> 4.2.1 Force on a charged particle </div> </a> <ul id="toc-Force_on_a_charged_particle-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Force_on_current-carrying_wire" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Force_on_current-carrying_wire"> <div class="vector-toc-text"> 4.2.2 Force on current-carrying wire </div> </a> <ul id="toc-Force_on_current-carrying_wire-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Relation_between_H_and_B" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Relation_between_H_and_B"> <div class="vector-toc-text"> 5 Relation between H and B </div> </a> <button aria-controls="toc-Relation_between_H_and_B-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Relation between H and B subsection </button> <ul id="toc-Relation_between_H_and_B-sublist" class="vector-toc-list"> <li id="toc-Magnetization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Magnetization"> <div class="vector-toc-text"> 5.1 Magnetization </div> </a> <ul id="toc-Magnetization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-H-field_and_magnetic_materials" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#H-field_and_magnetic_materials"> <div class="vector-toc-text"> 5.2 H-field and magnetic materials </div> </a> <ul id="toc-H-field_and_magnetic_materials-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Magnetism" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Magnetism"> <div class="vector-toc-text"> 5.3 Magnetism </div> </a> <ul id="toc-Magnetism-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Stored_energy" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Stored_energy"> <div class="vector-toc-text"> 6 Stored energy </div> </a> <ul id="toc-Stored_energy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Appearance_in_Maxwell's_equations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Appearance_in_Maxwell's_equations"> <div class="vector-toc-text"> 7 Appearance in Maxwell's equations </div> </a> <button aria-controls="toc-Appearance_in_Maxwell's_equations-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Appearance in Maxwell's equations subsection </button> <ul id="toc-Appearance_in_Maxwell's_equations-sublist" class="vector-toc-list"> <li id="toc-Gauss'_law_for_magnetism" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Gauss'_law_for_magnetism"> <div class="vector-toc-text"> 7.1 Gauss' law for magnetism </div> </a> <ul id="toc-Gauss'_law_for_magnetism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Faraday's_Law" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Faraday's_Law"> <div class="vector-toc-text"> 7.2 Faraday's Law </div> </a> <ul id="toc-Faraday's_Law-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ampère's_Law_and_Maxwell's_correction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ampère's_Law_and_Maxwell's_correction"> <div class="vector-toc-text"> 7.3 Ampère's Law and Maxwell's correction </div> </a> <ul id="toc-Ampère's_Law_and_Maxwell's_correction-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Formulation_in_special_relativity_and_quantum_electrodynamics" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Formulation_in_special_relativity_and_quantum_electrodynamics"> <div class="vector-toc-text"> 8 Formulation in special relativity and quantum electrodynamics </div> </a> <button aria-controls="toc-Formulation_in_special_relativity_and_quantum_electrodynamics-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Formulation in special relativity and quantum electrodynamics subsection </button> <ul id="toc-Formulation_in_special_relativity_and_quantum_electrodynamics-sublist" class="vector-toc-list"> <li id="toc-Relativistic_electrodynamics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relativistic_electrodynamics"> <div class="vector-toc-text"> 8.1 Relativistic electrodynamics </div> </a> <ul id="toc-Relativistic_electrodynamics-sublist" class="vector-toc-list"> <li id="toc-As_different_aspects_of_the_same_phenomenon" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#As_different_aspects_of_the_same_phenomenon"> <div class="vector-toc-text"> 8.1.1 As different aspects of the same phenomenon </div> </a> <ul id="toc-As_different_aspects_of_the_same_phenomenon-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Magnetic_vector_potential" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Magnetic_vector_potential"> <div class="vector-toc-text"> 8.1.2 Magnetic vector potential </div> </a> <ul id="toc-Magnetic_vector_potential-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Propagation_of_Electric_and_Magnetic_fields" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Propagation_of_Electric_and_Magnetic_fields"> <div class="vector-toc-text"> 8.1.3 Propagation of Electric and Magnetic fields </div> </a> <ul id="toc-Propagation_of_Electric_and_Magnetic_fields-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Magnetic_field_of_arbitrary_moving_point_charge" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Magnetic_field_of_arbitrary_moving_point_charge"> <div class="vector-toc-text"> 8.1.4 Magnetic field of arbitrary moving point charge </div> </a> <ul id="toc-Magnetic_field_of_arbitrary_moving_point_charge-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Quantum_electrodynamics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quantum_electrodynamics"> <div class="vector-toc-text"> 8.2 Quantum electrodynamics </div> </a> <ul id="toc-Quantum_electrodynamics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Uses_and_examples" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Uses_and_examples"> <div class="vector-toc-text"> 9 Uses and examples </div> </a> <button aria-controls="toc-Uses_and_examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Uses and examples subsection </button> <ul id="toc-Uses_and_examples-sublist" class="vector-toc-list"> <li id="toc-Earth's_magnetic_field" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Earth's_magnetic_field"> <div class="vector-toc-text"> 9.1 Earth's magnetic field </div> </a> <ul id="toc-Earth's_magnetic_field-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rotating_magnetic_fields" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rotating_magnetic_fields"> <div class="vector-toc-text"> 9.2 Rotating magnetic fields </div> </a> <ul id="toc-Rotating_magnetic_fields-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hall_effect" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hall_effect"> <div class="vector-toc-text"> 9.3 Hall effect </div> </a> <ul id="toc-Hall_effect-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Magnetic_circuits" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Magnetic_circuits"> <div class="vector-toc-text"> 9.4 Magnetic circuits </div> </a> <ul id="toc-Magnetic_circuits-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Largest_magnetic_fields" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Largest_magnetic_fields"> <div class="vector-toc-text"> 9.5 Largest magnetic fields </div> </a> <ul id="toc-Largest_magnetic_fields-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Common_formulæ" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Common_formulæ"> <div class="vector-toc-text"> 10 Common formulæ </div> </a> <ul id="toc-Common_formulæ-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> 11 History </div> </a> <button aria-controls="toc-History-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle History subsection </button> <ul id="toc-History-sublist" class="vector-toc-list"> <li id="toc-Early_developments" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Early_developments"> <div class="vector-toc-text"> 11.1 Early developments </div> </a> <ul id="toc-Early_developments-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mathematical_development" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mathematical_development"> <div class="vector-toc-text"> 11.2 Mathematical development </div> </a> <ul id="toc-Mathematical_development-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modern_developments" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Modern_developments"> <div class="vector-toc-text"> 11.3 Modern developments </div> </a> <ul id="toc-Modern_developments-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 12 See also </div> </a> <button aria-controls="toc-See_also-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle See also subsection </button> <ul id="toc-See_also-sublist" class="vector-toc-list"> <li id="toc-General" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General"> <div class="vector-toc-text"> 12.1 General </div> </a> <ul id="toc-General-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mathematics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Mathematics"> <div class="vector-toc-text"> 12.2 Mathematics </div> </a> <ul id="toc-Mathematics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> 12.3 Applications </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 13 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 14 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 15 Further reading </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 16 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </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">Magnetic field</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" 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Available in 113 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-113" aria-hidden="true" > 113 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Magneetveld" title="Magneetveld – Afrikaans" lang="af" hreflang="af" data-title="Magneetveld" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-als mw-list-item"><a href="https://als.wikipedia.org/wiki/Magnetfeld" title="Magnetfeld – Alemannic" lang="gsw" hreflang="gsw" data-title="Magnetfeld" data-language-autonym="Alemannisch" data-language-local-name="Alemannic" class="interlanguage-link-target">Alemannisch</a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%98%E1%8C%8D%E1%8A%90%E1%8C%A2%E1%88%B5_%E1%88%98%E1%88%B5%E1%8A%AD" title="መግነጢስ መስክ – Amharic" lang="am" hreflang="am" data-title="መግነጢስ መስክ" data-language-autonym="አማርኛ" data-language-local-name="Amharic" class="interlanguage-link-target">አማርኛ</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AD%D9%82%D9%84_%D9%85%D8%BA%D9%86%D8%A7%D8%B7%D9%8A%D8%B3%D9%8A" title="حقل مغناطيسي – Arabic" lang="ar" hreflang="ar" data-title="حقل مغناطيسي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-an mw-list-item"><a href="https://an.wikipedia.org/wiki/Campo_magnetico" title="Campo magnetico – Aragonese" lang="an" hreflang="an" data-title="Campo magnetico" data-language-autonym="Aragonés" data-language-local-name="Aragonese" class="interlanguage-link-target">Aragonés</a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%9A%E0%A7%81%E0%A6%AE%E0%A7%8D%E0%A6%AC%E0%A6%95%E0%A7%80%E0%A6%AF%E0%A6%BC_%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A7%87%E0%A6%A4%E0%A7%8D%E0%A7%B0" title="চুম্বকীয় ক্ষেত্ৰ – Assamese" lang="as" hreflang="as" data-title="চুম্বকীয় ক্ষেত্ৰ" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" class="interlanguage-link-target">অসমীয়া</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Campu_magn%C3%A9ticu" title="Campu magnéticu – Asturian" lang="ast" hreflang="ast" data-title="Campu magnéticu" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Maqnit_sah%C9%99si" title="Maqnit sahəsi – Azerbaijani" lang="az" hreflang="az" data-title="Maqnit sahəsi" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-azb mw-list-item"><a href="https://azb.wikipedia.org/wiki/%D9%85%D8%BA%D9%86%D8%A7%D8%B7%DB%8C%D8%B3_%D9%85%D8%A6%DB%8C%D8%AF%D8%A7%D9%86%DB%8C" title="مغناطیس مئیدانی – South Azerbaijani" lang="azb" hreflang="azb" data-title="مغناطیس مئیدانی" data-language-autonym="تۆرکجه" data-language-local-name="South Azerbaijani" class="interlanguage-link-target">تۆرکجه</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%9A%E0%A7%8C%E0%A6%AE%E0%A7%8D%E0%A6%AC%E0%A6%95_%E0%A6%95%E0%A7%8D%E0%A6%B7%E0%A7%87%E0%A6%A4%E0%A7%8D%E0%A6%B0" title="চৌম্বক ক্ষেত্র – Bangla" lang="bn" hreflang="bn" data-title="চৌম্বক ক্ষেত্র" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9C%D0%B0%D0%B3%D0%BD%D0%B8%D1%82_%D2%A1%D1%8B%D1%80%D1%8B" title="Магнит ҡыры – Bashkir" lang="ba" hreflang="ba" data-title="Магнит ҡыры" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9C%D0%B0%D0%B3%D0%BD%D1%96%D1%82%D0%BD%D0%B0%D0%B5_%D0%BF%D0%BE%D0%BB%D0%B5" title="Магнітнае поле – Belarusian" lang="be" hreflang="be" data-title="Магнітнае поле" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9C%D0%B0%D0%B3%D0%BD%D1%96%D1%82%D0%BD%D0%B0%D0%B5_%D0%BF%D0%BE%D0%BB%D0%B5" title="Магнітнае поле – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Магнітнае поле" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%B0%D0%B3%D0%BD%D0%B8%D1%82%D0%BD%D0%BE_%D0%BF%D0%BE%D0%BB%D0%B5" title="Магнитно поле – Bulgarian" lang="bg" hreflang="bg" data-title="Магнитно поле" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Magnetno_polje" title="Magnetno polje – Bosnian" lang="bs" hreflang="bs" data-title="Magnetno polje" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-bxr mw-list-item"><a href="https://bxr.wikipedia.org/wiki/%D0%A1%D1%83%D1%80%D0%B0%D0%BD%D0%B7%D0%B0%D0%BD_%D0%BE%D1%80%D0%BE%D0%BD" title="Суранзан орон – Russia Buriat" lang="bxr" hreflang="bxr" data-title="Суранзан орон" data-language-autonym="Буряад" data-language-local-name="Russia Buriat" class="interlanguage-link-target">Буряад</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Camp_magn%C3%A8tic" title="Camp magnètic – Catalan" lang="ca" hreflang="ca" data-title="Camp magnètic" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Magnetick%C3%A9_pole" title="Magnetické pole – Czech" lang="cs" hreflang="cs" data-title="Magnetické pole" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Maes_magnetig" title="Maes magnetig – Welsh" lang="cy" hreflang="cy" data-title="Maes magnetig" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Magnetfelt" title="Magnetfelt – Danish" lang="da" hreflang="da" data-title="Magnetfelt" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D9%85%D8%A7%D8%AC%D8%A7%D9%84_%D9%85%D8%BA%D9%86%D8%A7%D8%B7%D9%8A%D8%B3%D9%8A" title="ماجال مغناطيسي – Moroccan Arabic" lang="ary" hreflang="ary" data-title="ماجال مغناطيسي" data-language-autonym="الدارجة" data-language-local-name="Moroccan Arabic" class="interlanguage-link-target">الدارجة</a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Magnetfeld" title="Magnetfeld – German" lang="de" hreflang="de" data-title="Magnetfeld" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Magnetv%C3%A4li" title="Magnetväli – Estonian" lang="et" hreflang="et" data-title="Magnetväli" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9C%CE%B1%CE%B3%CE%BD%CE%B7%CF%84%CE%B9%CE%BA%CF%8C_%CF%80%CE%B5%CE%B4%CE%AF%CE%BF" title="Μαγνητικό πεδίο – Greek" lang="el" hreflang="el" data-title="Μαγνητικό πεδίο" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Campo_magn%C3%A9tico" title="Campo magnético – Spanish" lang="es" hreflang="es" data-title="Campo magnético" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Magneta_kampo" title="Magneta kampo – Esperanto" lang="eo" hreflang="eo" data-title="Magneta kampo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-ext mw-list-item"><a href="https://ext.wikipedia.org/wiki/Campu_man%C3%A9ticu" title="Campu manéticu – Extremaduran" lang="ext" hreflang="ext" data-title="Campu manéticu" data-language-autonym="Estremeñu" data-language-local-name="Extremaduran" class="interlanguage-link-target">Estremeñu</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Eremu_magnetiko" title="Eremu magnetiko – Basque" lang="eu" hreflang="eu" data-title="Eremu magnetiko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%DB%8C%D8%AF%D8%A7%D9%86_%D9%85%D8%BA%D9%86%D8%A7%D8%B7%DB%8C%D8%B3%DB%8C" title="میدان مغناطیسی – Persian" lang="fa" hreflang="fa" data-title="میدان مغناطیسی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-hif mw-list-item"><a href="https://hif.wikipedia.org/wiki/Magnetic_field" title="Magnetic field – Fiji Hindi" lang="hif" hreflang="hif" data-title="Magnetic field" data-language-autonym="Fiji Hindi" data-language-local-name="Fiji Hindi" class="interlanguage-link-target">Fiji Hindi</a></li><li class="interlanguage-link interwiki-fr badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://fr.wikipedia.org/wiki/Champ_magn%C3%A9tique" title="Champ magnétique – French" lang="fr" hreflang="fr" data-title="Champ magnétique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/R%C3%A9imse_maighn%C3%A9adach" title="Réimse maighnéadach – Irish" lang="ga" hreflang="ga" data-title="Réimse maighnéadach" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Campo_magn%C3%A9tico" title="Campo magnético – Galician" lang="gl" hreflang="gl" data-title="Campo magnético" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-gu mw-list-item"><a href="https://gu.wikipedia.org/wiki/%E0%AA%9A%E0%AB%81%E0%AA%82%E0%AA%AC%E0%AA%95%E0%AB%80%E0%AA%AF_%E0%AA%95%E0%AB%8D%E0%AA%B7%E0%AB%87%E0%AA%A4%E0%AB%8D%E0%AA%B0" title="ચુંબકીય ક્ષેત્ર – Gujarati" lang="gu" hreflang="gu" data-title="ચુંબકીય ક્ષેત્ર" data-language-autonym="ગુજરાતી" data-language-local-name="Gujarati" class="interlanguage-link-target">ગુજરાતી</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9E%90%EA%B8%B0%EC%9E%A5" title="자기장 – Korean" lang="ko" hreflang="ko" data-title="자기장" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%A1%D5%A3%D5%B6%D5%AB%D5%BD%D5%A1%D5%AF%D5%A1%D5%B6_%D5%A4%D5%A1%D5%B7%D5%BF" title="Մագնիսական դաշտ – Armenian" lang="hy" hreflang="hy" data-title="Մագնիսական դաշտ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%9A%E0%A5%81%E0%A4%AE%E0%A5%8D%E0%A4%AC%E0%A4%95%E0%A5%80%E0%A4%AF_%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0" title="चुम्बकीय क्षेत्र – Hindi" lang="hi" hreflang="hi" data-title="चुम्बकीय क्षेत्र" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Magnetsko_polje" title="Magnetsko polje – Croatian" lang="hr" hreflang="hr" data-title="Magnetsko polje" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-gor mw-list-item"><a href="https://gor.wikipedia.org/wiki/Medan_magnet" title="Medan magnet – Gorontalo" lang="gor" hreflang="gor" data-title="Medan magnet" data-language-autonym="Bahasa Hulontalo" data-language-local-name="Gorontalo" class="interlanguage-link-target">Bahasa Hulontalo</a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Magnetala_feldo" title="Magnetala feldo – Ido" lang="io" hreflang="io" data-title="Magnetala feldo" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target">Ido</a></li><li class="interlanguage-link interwiki-ig mw-list-item"><a href="https://ig.wikipedia.org/wiki/Igwe_magnetik" title="Igwe magnetik – Igbo" lang="ig" hreflang="ig" data-title="Igwe magnetik" data-language-autonym="Igbo" data-language-local-name="Igbo" class="interlanguage-link-target">Igbo</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Medan_magnet" title="Medan magnet – Indonesian" lang="id" hreflang="id" data-title="Medan magnet" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Campo_magnetic" title="Campo magnetic – Interlingua" lang="ia" hreflang="ia" data-title="Campo magnetic" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-ie mw-list-item"><a href="https://ie.wikipedia.org/wiki/Campe_magnetic" title="Campe magnetic – Interlingue" lang="ie" hreflang="ie" data-title="Campe magnetic" data-language-autonym="Interlingue" data-language-local-name="Interlingue" class="interlanguage-link-target">Interlingue</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Segulsvi%C3%B0" title="Segulsvið – Icelandic" lang="is" hreflang="is" data-title="Segulsvið" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Campo_magnetico" title="Campo magnetico – Italian" lang="it" hreflang="it" data-title="Campo magnetico" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A9%D7%93%D7%94_%D7%9E%D7%92%D7%A0%D7%98%D7%99" title="שדה מגנטי – Hebrew" lang="he" hreflang="he" data-title="שדה מגנטי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%90%E1%83%92%E1%83%9C%E1%83%98%E1%83%A2%E1%83%A3%E1%83%A0%E1%83%98_%E1%83%95%E1%83%94%E1%83%9A%E1%83%98" title="მაგნიტური ველი – Georgian" lang="ka" hreflang="ka" data-title="მაგნიტური ველი" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9C%D0%B0%D0%B3%D0%BD%D0%B8%D1%82_%D3%A9%D1%80%D1%96%D1%81%D1%96" title="Магнит өрісі – Kazakh" lang="kk" hreflang="kk" data-title="Магнит өрісі" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-sw mw-list-item"><a href="https://sw.wikipedia.org/wiki/Uga_sumaku" title="Uga sumaku – Swahili" lang="sw" hreflang="sw" data-title="Uga sumaku" data-language-autonym="Kiswahili" data-language-local-name="Swahili" class="interlanguage-link-target">Kiswahili</a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Chan_mayetik" title="Chan mayetik – Haitian Creole" lang="ht" hreflang="ht" data-title="Chan mayetik" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target">Kreyòl ayisyen</a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9C%D0%B0%D0%B3%D0%BD%D0%B8%D1%82_%D1%82%D0%B0%D0%BB%D0%B0%D0%B0%D1%81%D1%8B" title="Магнит талаасы – Kyrgyz" lang="ky" hreflang="ky" data-title="Магнит талаасы" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target">Кыргызча</a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%AA%E0%BA%B0%E0%BB%9C%E0%BA%B2%E0%BA%A1%E0%BB%81%E0%BA%A1%E0%BB%88%E0%BB%80%E0%BA%AB%E0%BA%BC%E0%BA%B1%E0%BA%81" title="ສະໜາມແມ່ເຫຼັກ – Lao" lang="lo" hreflang="lo" data-title="ສະໜາມແມ່ເຫຼັກ" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target">ລາວ</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Campus_magneticus" title="Campus magneticus – Latin" lang="la" hreflang="la" data-title="Campus magneticus" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Magn%C4%93tiskais_lauks" title="Magnētiskais lauks – Latvian" lang="lv" hreflang="lv" data-title="Magnētiskais lauks" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lez mw-list-item"><a href="https://lez.wikipedia.org/wiki/%D0%9C%D0%B0%D0%B3%D0%BD%D0%B8%D1%82%D0%B4%D0%B8%D0%BD_%D1%87%D1%83%D1%8C%D0%BB" title="Магнитдин чуьл – Lezghian" lang="lez" hreflang="lez" data-title="Магнитдин чуьл" data-language-autonym="Лезги" data-language-local-name="Lezghian" class="interlanguage-link-target">Лезги</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Magnetinis_laukas" title="Magnetinis laukas – Lithuanian" lang="lt" hreflang="lt" data-title="Magnetinis laukas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-li mw-list-item"><a href="https://li.wikipedia.org/wiki/Magnetisch_veldj" title="Magnetisch veldj – Limburgish" lang="li" hreflang="li" data-title="Magnetisch veldj" data-language-autonym="Limburgs" data-language-local-name="Limburgish" class="interlanguage-link-target">Limburgs</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/M%C3%A1gneses_mez%C5%91" title="Mágneses mező – Hungarian" lang="hu" hreflang="hu" data-title="Mágneses mező" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9C%D0%B0%D0%B3%D0%BD%D0%B5%D1%82%D0%BD%D0%BE_%D0%BF%D0%BE%D0%BB%D0%B5" title="Магнетно поле – Macedonian" lang="mk" hreflang="mk" data-title="Магнетно поле" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%95%E0%B4%BE%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B4%BF%E0%B4%95%E0%B4%95%E0%B5%8D%E0%B4%B7%E0%B5%87%E0%B4%A4%E0%B5%8D%E0%B4%B0%E0%B4%82" title="കാന്തികക്ഷേത്രം – Malayalam" lang="ml" hreflang="ml" data-title="കാന്തികക്ഷേത്രം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%9A%E0%A5%81%E0%A4%82%E0%A4%AC%E0%A4%95%E0%A5%80_%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0" title="चुंबकी क्षेत्र – Marathi" lang="mr" hreflang="mr" data-title="चुंबकी क्षेत्र" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Medan_magnet" title="Medan magnet – Malay" lang="ms" hreflang="ms" data-title="Medan magnet" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A1%D0%BE%D1%80%D0%BE%D0%BD%D0%B7%D0%BE%D0%BD_%D0%BE%D1%80%D0%BE%D0%BD" title="Соронзон орон – Mongolian" lang="mn" hreflang="mn" data-title="Соронзон орон" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%9E%E1%80%B6%E1%80%9C%E1%80%AD%E1%80%AF%E1%80%80%E1%80%BA%E1%80%85%E1%80%80%E1%80%BA%E1%80%80%E1%80%BD%E1%80%84%E1%80%BA%E1%80%B8" title="သံလိုက်စက်ကွင်း – Burmese" lang="my" hreflang="my" data-title="သံလိုက်စက်ကွင်း" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Magnetisch_veld" title="Magnetisch veld – Dutch" lang="nl" hreflang="nl" data-title="Magnetisch veld" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%9A%E0%A5%81%E0%A4%AE%E0%A5%8D%E0%A4%AC%E0%A4%95%E0%A5%80%E0%A4%AF_%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A5%87%E0%A4%A4%E0%A5%8D%E0%A4%B0" title="चुम्बकीय क्षेत्र – Nepali" lang="ne" hreflang="ne" data-title="चुम्बकीय क्षेत्र" data-language-autonym="नेपाली" data-language-local-name="Nepali" class="interlanguage-link-target">नेपाली</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%A3%81%E5%A0%B4" title="磁場 – Japanese" lang="ja" hreflang="ja" data-title="磁場" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Magneetisk_fial" title="Magneetisk fial – Northern Frisian" lang="frr" hreflang="frr" data-title="Magneetisk fial" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target">Nordfriisk</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Magnetfelt" title="Magnetfelt – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Magnetfelt" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Magnetfelt" title="Magnetfelt – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Magnetfelt" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Camp_magnetic" title="Camp magnetic – Occitan" lang="oc" hreflang="oc" data-title="Camp magnetic" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Dirree_maagneetii" title="Dirree maagneetii – Oromo" lang="om" hreflang="om" data-title="Dirree maagneetii" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target">Oromoo</a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Magnit_maydon" title="Magnit maydon – Uzbek" lang="uz" hreflang="uz" data-title="Magnit maydon" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target">Oʻzbekcha / ўзбекча</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%9A%E0%A9%81%E0%A9%B0%E0%A8%AC%E0%A8%95%E0%A9%80_%E0%A8%96%E0%A9%87%E0%A8%A4%E0%A8%B0" title="ਚੁੰਬਕੀ ਖੇਤਰ – Punjabi" lang="pa" hreflang="pa" data-title="ਚੁੰਬਕੀ ਖੇਤਰ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%85%D9%82%D9%86%D8%A7%D8%B7%DB%8C%D8%B3%DB%8C_%D9%85%DB%8C%D8%AF%D8%A7%D9%86" title="مقناطیسی میدان – Western Punjabi" lang="pnb" hreflang="pnb" data-title="مقناطیسی میدان" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target">پنجابی</a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D9%85%D9%82%D9%86%D8%A7%D8%B7%D9%8A%D8%B3%D9%8A_%D8%B3%D8%A7%D8%AD%D9%87" title="مقناطيسي ساحه – Pashto" lang="ps" hreflang="ps" data-title="مقناطيسي ساحه" data-language-autonym="پښتو" data-language-local-name="Pashto" class="interlanguage-link-target">پښتو</a></li><li class="interlanguage-link interwiki-nds mw-list-item"><a href="https://nds.wikipedia.org/wiki/Magneetfeld" title="Magneetfeld – Low German" lang="nds" hreflang="nds" data-title="Magneetfeld" data-language-autonym="Plattdüütsch" data-language-local-name="Low German" class="interlanguage-link-target">Plattdüütsch</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Pole_magnetyczne" title="Pole magnetyczne – Polish" lang="pl" hreflang="pl" data-title="Pole magnetyczne" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Campo_magn%C3%A9tico" title="Campo magnético – Portuguese" lang="pt" hreflang="pt" data-title="Campo magnético" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/C%C3%A2mp_magnetic" title="Câmp magnetic – Romanian" lang="ro" hreflang="ro" data-title="Câmp magnetic" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-rue mw-list-item"><a href="https://rue.wikipedia.org/wiki/%D0%9C%D0%B0%D2%91%D0%BD%D0%B5%D1%82%D1%96%D1%87%D0%BD%D0%B5_%D0%BF%D0%BE%D0%BB%D0%B5" title="Маґнетічне поле – Rusyn" lang="rue" hreflang="rue" data-title="Маґнетічне поле" data-language-autonym="Русиньскый" data-language-local-name="Rusyn" class="interlanguage-link-target">Русиньскый</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D0%B3%D0%BD%D0%B8%D1%82%D0%BD%D0%BE%D0%B5_%D0%BF%D0%BE%D0%BB%D0%B5" title="Магнитное поле – Russian" lang="ru" hreflang="ru" data-title="Магнитное поле" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Fusha_magnetike" title="Fusha magnetike – Albanian" lang="sq" hreflang="sq" data-title="Fusha magnetike" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Campu_magneticu_(art%C3%ACculu_%27n_calabbrisi)" title="Campu magneticu (artìculu 'n calabbrisi) – Sicilian" lang="scn" hreflang="scn" data-title="Campu magneticu (artìculu 'n calabbrisi)" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%A0%E0%B7%94%E0%B6%B8%E0%B7%8A%E0%B6%B6%E0%B6%9A_%E0%B6%9A%E0%B7%8A%E0%B7%82%E0%B7%9A%E0%B6%AD%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B6%BA" title="චුම්බක ක්ෂේත්‍රය – Sinhala" lang="si" hreflang="si" data-title="චුම්බක ක්ෂේත්‍රය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target">සිංහල</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Magnetic_field" title="Magnetic field – Simple English" lang="en-simple" hreflang="en-simple" data-title="Magnetic field" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Magnetick%C3%A9_pole" title="Magnetické pole – Slovak" lang="sk" hreflang="sk" data-title="Magnetické pole" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Magnetno_polje" title="Magnetno polje – Slovenian" lang="sl" hreflang="sl" data-title="Magnetno polje" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A8%D9%88%D8%A7%D8%B1%DB%8C_%D9%85%D9%88%DA%AF%D9%86%D8%A7%D8%AA%DB%8C%D8%B3%DB%8C" title="بواری موگناتیسی – Central Kurdish" lang="ckb" hreflang="ckb" data-title="بواری موگناتیسی" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9C%D0%B0%D0%B3%D0%BD%D0%B5%D1%82%D0%BD%D0%BE_%D0%BF%D0%BE%D1%99%D0%B5" title="Магнетно поље – Serbian" lang="sr" hreflang="sr" data-title="Магнетно поље" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Magnetno_polje" title="Magnetno polje – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Magnetno polje" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/M%C3%A9dan_magn%C3%A9tik" title="Médan magnétik – Sundanese" lang="su" hreflang="su" data-title="Médan magnétik" data-language-autonym="Sunda" data-language-local-name="Sundanese" class="interlanguage-link-target">Sunda</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Magneettikentt%C3%A4" title="Magneettikenttä – Finnish" lang="fi" hreflang="fi" data-title="Magneettikenttä" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Magnetf%C3%A4lt" title="Magnetfält – Swedish" lang="sv" hreflang="sv" data-title="Magnetfält" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AE%BE%E0%AE%A8%E0%AF%8D%E0%AE%A4%E0%AE%AA%E0%AF%8D_%E0%AE%AA%E0%AF%81%E0%AE%B2%E0%AE%AE%E0%AF%8D" title="காந்தப் புலம் – Tamil" lang="ta" hreflang="ta" data-title="காந்தப் புலம்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-tt mw-list-item"><a href="https://tt.wikipedia.org/wiki/%D0%9C%D0%B0%D0%B3%D0%BD%D0%B8%D1%82_%D0%BA%D1%8B%D1%80%D1%8B" title="Магнит кыры – Tatar" lang="tt" hreflang="tt" data-title="Магнит кыры" data-language-autonym="Татарча / tatarça" data-language-local-name="Tatar" class="interlanguage-link-target">Татарча / tatarça</a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%85%E0%B0%AF%E0%B0%B8%E0%B1%8D%E0%B0%95%E0%B0%BE%E0%B0%82%E0%B0%A4_%E0%B0%95%E0%B1%8D%E0%B0%B7%E0%B1%87%E0%B0%A4%E0%B1%8D%E0%B0%B0%E0%B0%82" title="అయస్కాంత క్షేత్రం – Telugu" lang="te" hreflang="te" data-title="అయస్కాంత క్షేత్రం" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target">తెలుగు</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AA%E0%B8%99%E0%B8%B2%E0%B8%A1%E0%B9%81%E0%B8%A1%E0%B9%88%E0%B9%80%E0%B8%AB%E0%B8%A5%E0%B9%87%E0%B8%81" title="สนามแม่เหล็ก – Thai" lang="th" hreflang="th" data-title="สนามแม่เหล็ก" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Manyetik_alan" title="Manyetik alan – Turkish" lang="tr" hreflang="tr" data-title="Manyetik alan" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-tyv mw-list-item"><a href="https://tyv.wikipedia.org/wiki/%D0%9C%D0%B0%D0%B3%D0%BD%D0%B8%D1%82%D1%82%D0%B8%D0%B3_%D1%88%D3%A9%D0%BB" title="Магниттиг шөл – Tuvinian" lang="tyv" hreflang="tyv" data-title="Магниттиг шөл" data-language-autonym="Тыва дыл" data-language-local-name="Tuvinian" class="interlanguage-link-target">Тыва дыл</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%B0%D0%B3%D0%BD%D1%96%D1%82%D0%BD%D0%B5_%D0%BF%D0%BE%D0%BB%D0%B5" title="Магнітне поле – Ukrainian" lang="uk" hreflang="uk" data-title="Магнітне поле" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%D9%82%D9%86%D8%A7%D8%B7%DB%8C%D8%B3%DB%8C_%D9%85%DB%8C%D8%AF%D8%A7%D9%86" title="مقناطیسی میدان – Urdu" lang="ur" hreflang="ur" data-title="مقناطیسی میدان" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-ug mw-list-item"><a href="https://ug.wikipedia.org/wiki/%D9%85%D8%A7%DA%AF%D9%86%D9%89%D8%AA_%D9%85%DB%95%D9%8A%D8%AF%D8%A7%D9%86%D9%89" title="ماگنىت مەيدانى – Uyghur" lang="ug" hreflang="ug" data-title="ماگنىت مەيدانى" data-language-autonym="ئۇيغۇرچە / Uyghurche" 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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> <div id="mw-indicator-pp-default" class="mw-indicator"><div class="mw-parser-output"><a href="/wiki/Wikipedia:Protection_policy#semi" title="This article is semi-protected."><img alt="Page semi-protected" src="//upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/20px-Semi-protection-shackle.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/30px-Semi-protection-shackle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/1/1b/Semi-protection-shackle.svg/40px-Semi-protection-shackle.svg.png 2x" data-file-width="512" data-file-height="512" /></a></div></div> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Distribution of magnetic force</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="/wiki/Magnetic_field_(disambiguation)" class="mw-disambig" title="Magnetic field (disambiguation)">Magnetic field (disambiguation)</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1237032888/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:292px;max-width:292px"><div class="trow"><div class="tsingle" style="width:110px;max-width:110px"><div class="thumbimage" style="height:175px;overflow:hidden"><a href="/wiki/File:Magnetic_field_of_horseshoe_magnet.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Magnetic_field_of_horseshoe_magnet.png/108px-Magnetic_field_of_horseshoe_magnet.png" decoding="async" width="108" height="176" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Magnetic_field_of_horseshoe_magnet.png/162px-Magnetic_field_of_horseshoe_magnet.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Magnetic_field_of_horseshoe_magnet.png/216px-Magnetic_field_of_horseshoe_magnet.png 2x" data-file-width="471" data-file-height="767" /></a></div><div class="thumbcaption">A <a href="/wiki/Permanent_magnet" class="mw-redirect" title="Permanent magnet">permanent magnet</a>, a piece of magnetized metal alloy</div></div><div class="tsingle" style="width:178px;max-width:178px"><div class="thumbimage" style="height:175px;overflow:hidden"><a href="/wiki/File:Magnetic_field_around_solenoid.jpg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Magnetic_field_around_solenoid.jpg/176px-Magnetic_field_around_solenoid.jpg" decoding="async" width="176" height="176" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/42/Magnetic_field_around_solenoid.jpg/264px-Magnetic_field_around_solenoid.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/42/Magnetic_field_around_solenoid.jpg/352px-Magnetic_field_around_solenoid.jpg 2x" data-file-width="3087" data-file-height="3087" /></a></div><div class="thumbcaption">A <a href="/wiki/Solenoid" title="Solenoid">solenoid</a> (<a href="/wiki/Electromagnet" title="Electromagnet">electromagnet</a>), a coil of wire with an electric current through it</div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">The shape of the magnetic fields of a permanent magnet and an electromagnet are revealed by the orientation of iron filings sprinkled on pieces of paper</div></div></div></div> A magnetic field (sometimes called B-field<a href="#cite_note-1">[1]</a>) is a <a href="/wiki/Physical_field" class="mw-redirect" title="Physical field">physical field</a> that describes the magnetic influence on moving <a href="/wiki/Electric_charge" title="Electric charge">electric charges</a>, <a href="/wiki/Electric_currents" class="mw-redirect" title="Electric currents">electric currents</a>,<a href="#cite_note-feynman-2">[2]</a>: ch1 <a href="#cite_note-3">[3]</a> and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to the magnetic field.<a href="#cite_note-feynman-2">[2]</a>: ch13 <a href="#cite_note-Purcell3ed-4">[4]</a>: 278  A <a href="/wiki/Permanent_magnet" class="mw-redirect" title="Permanent magnet">permanent magnet</a>'s magnetic field pulls on <a href="/wiki/Ferromagnetic_material" class="mw-redirect" title="Ferromagnetic material">ferromagnetic materials</a> such as <a href="/wiki/Iron" title="Iron">iron</a>, and attracts or repels other magnets. In addition, a nonuniform magnetic field exerts minuscule forces on "nonmagnetic" materials by three other magnetic effects: <a href="/wiki/Paramagnetism" title="Paramagnetism">paramagnetism</a>, <a href="/wiki/Diamagnetism" title="Diamagnetism">diamagnetism</a>, and <a href="/wiki/Antiferromagnetism" title="Antiferromagnetism">antiferromagnetism</a>, although these forces are usually so small they can only be detected by laboratory equipment. Magnetic fields surround magnetized materials, electric currents, and <a href="/wiki/Electric_field" title="Electric field">electric fields</a> varying in time. Since both strength and direction of a magnetic field may vary with location, it is described mathematically by a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> assigning a <a href="/wiki/Euclidean_vector" title="Euclidean vector">vector</a> to each point of space, called a <a href="/wiki/Vector_field" title="Vector field">vector field</a> (more precisely, a <a href="/wiki/Pseudovector" title="Pseudovector">pseudovector</a> field). In <a href="/wiki/Electromagnetics" class="mw-redirect" title="Electromagnetics">electromagnetics</a>, the term magnetic field is used for two distinct but closely related vector fields denoted by the symbols B and H. In the <a href="/wiki/International_System_of_Units" title="International System of Units">International System of Units</a>, the unit of B, <a href="/wiki/Magnetic_flux" title="Magnetic flux">magnetic flux</a> density, is the <a href="/wiki/Tesla_(unit)" title="Tesla (unit)">tesla</a> (in SI base units: kilogram per second squared per ampere),<a href="#cite_note-:1-5">[5]</a>: 21  which is equivalent to <a href="/wiki/Newton_(unit)" title="Newton (unit)">newton</a> per meter per ampere. The unit of H, magnetic field strength, is <a href="/wiki/Ampere" title="Ampere">ampere</a> per meter (A/m).<a href="#cite_note-:1-5">[5]</a>: 22  B and H differ in how they take the medium and/or magnetization into account. In <a href="/wiki/Vacuum" title="Vacuum">vacuum</a>, the two fields are related through the <a href="/wiki/Vacuum_permeability" title="Vacuum permeability">vacuum permeability</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} /\mu _{0}=\mathbf {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} /\mu _{0}=\mathbf {H} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/773a5eb3020e0a6c574f625c487a7e3d8eaec1bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.709ex; height:2.843ex;" alt="{\displaystyle \mathbf {B} /\mu _{0}=\mathbf {H} }">; in a magnetized material, the quantities on each side of this equation differ by the <a href="/wiki/Magnetization" title="Magnetization">magnetization</a> field of the material. Magnetic fields are produced by moving electric charges and the intrinsic <a href="/wiki/Magnetic_moment" title="Magnetic moment">magnetic moments</a> of <a href="/wiki/Elementary_particle" title="Elementary particle">elementary particles</a> associated with a fundamental quantum property, their <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a>.<a href="#cite_note-Jiles-6">[6]</a><a href="#cite_note-feynman-2">[2]</a>: ch1  Magnetic fields and <a href="/wiki/Electric_field" title="Electric field">electric fields</a> are interrelated and are both components of the <a href="/wiki/Electromagnetic_force" class="mw-redirect" title="Electromagnetic force">electromagnetic force</a>, one of the four <a href="/wiki/Fundamental_force" class="mw-redirect" title="Fundamental force">fundamental forces</a> of nature. Magnetic fields are used throughout modern technology, particularly in <a href="/wiki/Electrical_engineering" title="Electrical engineering">electrical engineering</a> and <a href="/wiki/Electromechanics" title="Electromechanics">electromechanics</a>. Rotating magnetic fields are used in both <a href="/wiki/Electric_motor" title="Electric motor">electric motors</a> and <a href="/wiki/Electric_generator" title="Electric generator">generators</a>. The interaction of magnetic fields in electric devices such as transformers is conceptualized and investigated as <a href="/wiki/Magnetic_circuit" title="Magnetic circuit">magnetic circuits</a>. Magnetic forces give information about the charge carriers in a material through the <a href="/wiki/Hall_effect" title="Hall effect">Hall effect</a>. The Earth produces <a href="/wiki/Earth%27s_magnetic_field" title="Earth's magnetic field">its own magnetic field</a>, which shields the Earth's ozone layer from the <a href="/wiki/Solar_wind" title="Solar wind">solar wind</a> and is important in <a href="/wiki/Navigation" title="Navigation">navigation</a> using a <a href="/wiki/Compass" title="Compass">compass</a>. <style data-mw-deduplicate="TemplateStyles:r886046785">.mw-parser-output .toclimit-2 .toclevel-1 ul,.mw-parser-output .toclimit-3 .toclevel-2 ul,.mw-parser-output .toclimit-4 .toclevel-3 ul,.mw-parser-output .toclimit-5 .toclevel-4 ul,.mw-parser-output .toclimit-6 .toclevel-5 ul,.mw-parser-output .toclimit-7 .toclevel-6 ul{display:none}</style><div class="toclimit-3"><meta property="mw:PageProp/toc" /></div> <div class="mw-heading mw-heading2"><h2 id="Description">Description</h2></div> The force on an electric charge depends on its location, speed, and direction; two vector fields are used to describe this force.<a href="#cite_note-feynman-2">[2]</a>: ch1  The first is the <a href="/wiki/Electric_field" title="Electric field">electric field</a>, which describes the force acting on a stationary charge and gives the component of the force that is independent of motion. The magnetic field, in contrast, describes the component of the force that is proportional to both the speed and direction of charged particles.<a href="#cite_note-feynman-2">[2]</a>: ch13  The field is defined by the <a href="/wiki/Lorentz_force_law" class="mw-redirect" title="Lorentz force law">Lorentz force law</a> and is, at each instant, perpendicular to both the motion of the charge and the force it experiences. There are two different, but closely related vector fields which are both sometimes called the "magnetic field" written B and H.<a href="#cite_note-7">[note 1]</a> While both the best names for these fields and exact interpretation of what these fields represent has been the subject of long running debate, there is wide agreement about how the underlying physics work.<a href="#cite_note-8">[7]</a> Historically, the term "magnetic field" was reserved for H while using other terms for B, but many recent textbooks use the term "magnetic field" to describe B as well as or in place of H.<a href="#cite_note-ex03-9">[note 2]</a> There are many alternative names for both (see sidebars). <div class="mw-heading mw-heading3"><h3 id="The_B-field">The B-field</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:392px;max-width:392px"><div class="trow"><div class="theader">Finding the magnetic force</div></div><div class="trow"><div class="tsingle" style="width:213px;max-width:213px"><div class="thumbimage" style="height:158px;overflow:hidden"><a href="/wiki/File:FuerzaCentripetaLorentzP.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/FuerzaCentripetaLorentzP.svg/211px-FuerzaCentripetaLorentzP.svg.png" decoding="async" width="211" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a9/FuerzaCentripetaLorentzP.svg/317px-FuerzaCentripetaLorentzP.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a9/FuerzaCentripetaLorentzP.svg/422px-FuerzaCentripetaLorentzP.svg.png 2x" data-file-width="416" data-file-height="313" /></a></div><div class="thumbcaption">A charged particle that is moving with velocity v in a magnetic field B will feel a magnetic force F. Since the magnetic force always pulls sideways to the direction of motion, the particle moves in a circle.</div></div><div class="tsingle" style="width:175px;max-width:175px"><div class="thumbimage" style="height:158px;overflow:hidden"><a href="/wiki/File:Mano-2.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Mano-2.svg/173px-Mano-2.svg.png" decoding="async" width="173" height="159" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Mano-2.svg/260px-Mano-2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Mano-2.svg/346px-Mano-2.svg.png 2x" data-file-width="313" data-file-height="288" /></a></div><div class="thumbcaption">Since these three vectors are related to each other by a <a href="/wiki/Cross_product" title="Cross product">cross product</a>, the direction of this force can be found using the <a href="/wiki/Right_hand_rule" class="mw-redirect" title="Right hand rule">right hand rule</a>.</div></div></div></div></div> <table class="wikitable" style="float:right;"> <tbody><tr> <th>Alternative names for B<a href="#cite_note-Electromagnetics-10">[8]</a> </th></tr> <tr> <td> <ul><li>Magnetic flux density<a href="#cite_note-:1-5">[5]</a>: 138 </li> <li>Magnetic induction<a href="#cite_note-Stratton-11">[9]</a></li> <li>Magnetic field (ambiguous)</li></ul> </td></tr></tbody></table> The magnetic field vector B at any point can be defined as the vector that, when used in the <a href="/wiki/Lorentz_force_law" class="mw-redirect" title="Lorentz force law">Lorentz force law</a>, correctly predicts the force on a charged particle at that point:<a href="#cite_note-purcell2ed-12">[10]</a><a href="#cite_note-Griffiths3ed-13">[11]</a>: 204  <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 5px; border-width:2px; border-style: solid; border-color: rgb(0,115,207); color: inherit;text-align: center; display: table">Lorentz force law (<a href="/wiki/Euclidean_vector" title="Euclidean vector">vector</a> form, <a href="/wiki/International_System_of_Units" title="International System of Units">SI units</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =q\mathbf {E} +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">E</mi> </mrow> <mo>+</mo> <mi>q</mi> <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">B</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =q\mathbf {E} +q(\mathbf {v} \times \mathbf {B} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f77eb69bd4696bfc996bb549b02d01cb20b2efd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.479ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} =q\mathbf {E} +q(\mathbf {v} \times \mathbf {B} )}"> </div> Here F is the force on the particle, q is the particle's <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a>, v, is the particle's <a href="/wiki/Velocity" title="Velocity">velocity</a>, and × denotes the <a href="/wiki/Cross_product" title="Cross product">cross product</a>. The direction of force on the charge can be determined by a <a href="/wiki/Mnemonic" title="Mnemonic">mnemonic</a> known as the right-hand rule (see the figure).<a href="#cite_note-14">[note 3]</a> Using the right hand, pointing the thumb in the direction of the current, and the fingers in the direction of the magnetic field, the resulting force on the charge points outwards from the palm. The force on a negatively charged particle is in the opposite direction. If both the speed and the charge are reversed then the direction of the force remains the same. For that reason a magnetic field measurement (by itself) cannot distinguish whether there is a positive charge moving to the right or a negative charge moving to the left. (Both of these cases produce the same current.) On the other hand, a magnetic field combined with an electric field can distinguish between these, see <a href="#Hall_effect">Hall effect</a> below. The first term in the Lorentz equation is from the theory of <a href="/wiki/Electrostatics" title="Electrostatics">electrostatics</a>, and says that a particle of charge q in an electric field E experiences an electric force: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{\text{electric}}=q\mathbf {E} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>electric</mtext> </mrow> </msub> <mo>=</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{\text{electric}}=q\mathbf {E} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d694aac8114153241a5251d7a3cb0279ad6a4b79" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.606ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} _{\text{electric}}=q\mathbf {E} .}"> The second term is the magnetic force:<a href="#cite_note-Griffiths3ed-13">[11]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} _{\text{magnetic}}=q(\mathbf {v} \times \mathbf {B} ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>magnetic</mtext> </mrow> </msub> <mo>=</mo> <mi>q</mi> <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">B</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} _{\text{magnetic}}=q(\mathbf {v} \times \mathbf {B} ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2865c0088b4b03798aef68322fbbd19307762a7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.175ex; height:3.009ex;" alt="{\displaystyle \mathbf {F} _{\text{magnetic}}=q(\mathbf {v} \times \mathbf {B} ).}"> Using the definition of the cross product, the magnetic force can also be written as a <a href="/wiki/Scalar_(physics)" title="Scalar (physics)">scalar</a> equation:<a href="#cite_note-purcell2ed-12">[10]</a>: 357  <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{\text{magnetic}}=qvB\sin(\theta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>magnetic</mtext> </mrow> </msub> <mo>=</mo> <mi>q</mi> <mi>v</mi> <mi>B</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{\text{magnetic}}=qvB\sin(\theta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b7c35be48f0f9efbcff642408d09df2c9e0ad58" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.413ex; height:3.009ex;" alt="{\displaystyle F_{\text{magnetic}}=qvB\sin(\theta )}"> where Fmagnetic, v, and B are the <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">scalar magnitude</a> of their respective vectors, and θ is the angle between the velocity of the particle and the magnetic field. The vector B is defined as the vector field necessary to make the Lorentz force law correctly describe the motion of a charged particle. In other words,<a href="#cite_note-purcell2ed-12">[10]</a>: <a rel="nofollow" class="external text" href="https://archive.org/details/electricitymagne00purc_621/page/n192">173</a>–4  <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote">[T]he command, "Measure the direction and magnitude of the vector B at such and such a place," calls for the following operations: Take a particle of known charge q. Measure the force on q at rest, to determine E. Then measure the force on the particle when its velocity is v; repeat with v in some other direction. Now find a B that makes the Lorentz force law fit all these results—that is the magnetic field at the place in question.</blockquote> The B field can also be defined by the torque on a magnetic dipole, m.<a href="#cite_note-jackson3ed-15">[12]</a>: 174  <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 5px; border-width:2px; border-style: solid; border-color: rgb(0,115,207); color: inherit;text-align: center; display: table">Magnetic torque (<a href="/wiki/Euclidean_vector" title="Euclidean vector">vector</a> form, <a href="/wiki/International_System_of_Units" title="International System of Units">SI units</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\tau }}=\mathbf {m} \times \mathbf {B} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">τ</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">m</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\tau }}=\mathbf {m} \times \mathbf {B} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d052375c134411da3ba1e4780670cb4399632d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.484ex; height:2.176ex;" alt="{\displaystyle {\boldsymbol {\tau }}=\mathbf {m} \times \mathbf {B} }"> </div> The <a href="/wiki/SI" class="mw-redirect" title="SI">SI</a> unit of B is <a href="/wiki/Tesla_(unit)" title="Tesla (unit)">tesla</a> (symbol: T).<a href="#cite_note-16">[note 4]</a> The <a href="/wiki/Gaussian_units" title="Gaussian units">Gaussian-cgs unit</a> of B is the <a href="/wiki/Gauss_(unit)" title="Gauss (unit)">gauss</a> (symbol: G). (The conversion is 1 T ≘ 10000 G.<a href="#cite_note-BIPMTab9-17">[13]</a><a href="#cite_note-KLang-18">[14]</a>) One nanotesla corresponds to 1 gamma (symbol: γ).<a href="#cite_note-KLang-18">[14]</a> <div class="mw-heading mw-heading3"><h3 id="The_H-field">The H-field</h3></div> <table class="wikitable" style="float:right;"> <tbody><tr> <th>Alternative names for H<a href="#cite_note-Electromagnetics-10">[8]</a> </th></tr> <tr> <td> <ul><li>Magnetic field intensity<a href="#cite_note-Stratton-11">[9]</a></li> <li>Magnetic field strength<a href="#cite_note-:1-5">[5]</a>: 139 </li> <li>Magnetic field</li> <li>Magnetizing field</li> <li>Auxiliary magnetic field</li></ul> </td></tr></tbody></table> The magnetic H field is defined:<a href="#cite_note-Griffiths3ed-13">[11]</a>: 269 <a href="#cite_note-jackson3ed-15">[12]</a>: 192 <a href="#cite_note-feynman-2">[2]</a>: ch36  <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 5px; border-width:2px; border-style: solid; border-color: rgb(0,115,207); color: inherit;text-align: center; display: table">Definition of the H field (<a href="/wiki/Euclidean_vector" title="Euclidean vector">vector</a> form, <a href="/wiki/International_System_of_Units" title="International System of Units">SI units</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} \equiv {\frac {1}{\mu _{0}}}\mathbf {B} -\mathbf {M} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {H} \equiv {\frac {1}{\mu _{0}}}\mathbf {B} -\mathbf {M} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3aef0bafe1332d17db417fa261092edfa266fdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.761ex; height:5.676ex;" alt="{\displaystyle \mathbf {H} \equiv {\frac {1}{\mu _{0}}}\mathbf {B} -\mathbf {M} }"> </div> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe2fd9b8decb38a3cd158e7b6c0c6e2d987fefcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.176ex;" alt="{\displaystyle \mu _{0}}"> is the <a href="/wiki/Vacuum_permeability" title="Vacuum permeability">vacuum permeability</a>, and M is the <a href="/wiki/Magnetization" title="Magnetization">magnetization vector</a>. In a vacuum, B and H are proportional to each other. Inside a material they are different (see <a href="#H-field_and_magnetic_materials">H and B inside and outside magnetic materials</a>). The SI unit of the H-field is the <a href="/wiki/Ampere" title="Ampere">ampere</a> per metre (A/m),<a href="#cite_note-19">[15]</a> and the CGS unit is the <a href="/wiki/Oersted" title="Oersted">oersted</a> (Oe).<a href="#cite_note-BIPMTab9-17">[13]</a><a href="#cite_note-purcell2ed-12">[10]</a>: <span title="Page: 286
Quotation: "Units: Tesla for describing a large magnetic force; gauss (tesla/10000) for describing a small magnetic force as that at the surface of earth."" class="tooltip tooltip-dashed" style="border-bottom: 1px dashed;">286  <div class="mw-heading mw-heading3"><h3 id="Measurement">Measurement</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Magnetometer" title="Magnetometer">Magnetometer</a></div> An instrument used to measure the local magnetic field is known as a <a href="/wiki/Magnetometer" title="Magnetometer">magnetometer</a>. Important classes of magnetometers include using <a href="/wiki/Search_coil" class="mw-redirect" title="Search coil">induction magnetometers</a> (or search-coil magnetometers) which measure only varying magnetic fields, <a href="/wiki/Magnetometer#Rotating_coil_magnetometer" title="Magnetometer">rotating coil magnetometers</a>, <a href="/wiki/Hall_effect" title="Hall effect">Hall effect</a> magnetometers, <a href="/wiki/Proton_magnetometer" title="Proton magnetometer">NMR magnetometers</a>, <a href="/wiki/SQUID" title="SQUID">SQUID magnetometers</a>, and <a href="/wiki/Magnetometer#Fluxgate_magnetometer" title="Magnetometer">fluxgate magnetometers</a>. The magnetic fields of distant <a href="/wiki/Astronomical_object" title="Astronomical object">astronomical objects</a> are measured through their effects on local charged particles. For instance, electrons spiraling around a field line produce <a href="/wiki/Synchrotron_radiation" title="Synchrotron radiation">synchrotron radiation</a> that is detectable in <a href="/wiki/Radio_waves" class="mw-redirect" title="Radio waves">radio waves</a>. The finest precision for a magnetic field measurement was attained by <a href="/wiki/Gravity_Probe_B" title="Gravity Probe B">Gravity Probe B</a> at 5 aT (5×10−18 T).<a href="#cite_note-20">[16]</a> <div class="mw-heading mw-heading3"><h3 id="Visualization">Visualization</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Field_line" title="Field line">Field line</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:342px;max-width:342px"><div class="trow"><div class="theader">Visualizing magnetic fields</div></div><div class="trow"><div class="tsingle" style="width:240px;max-width:240px"><div class="thumbimage" style="height:159px;overflow:hidden"><a href="/wiki/File:Magnet0873.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Magnet0873.png/238px-Magnet0873.png" decoding="async" width="238" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Magnet0873.png/357px-Magnet0873.png 1.5x, //upload.wikimedia.org/wikipedia/commons/5/57/Magnet0873.png 2x" data-file-width="444" data-file-height="298" /></a></div></div><div class="tsingle" style="width:98px;max-width:98px"><div class="thumbimage" style="height:159px;overflow:hidden"><a href="/wiki/File:Magnetic_field_near_pole.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Magnetic_field_near_pole.svg/96px-Magnetic_field_near_pole.svg.png" decoding="async" width="96" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Magnetic_field_near_pole.svg/144px-Magnetic_field_near_pole.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Magnetic_field_near_pole.svg/192px-Magnetic_field_near_pole.svg.png 2x" data-file-width="300" data-file-height="500" /></a></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Left: the direction of magnetic <a href="/wiki/Field_line" title="Field line">field lines</a> represented by <a href="/wiki/Iron_filings" title="Iron filings">iron filings</a> sprinkled on paper placed above a bar magnet. Right: <a href="/wiki/Compass" title="Compass">compass</a> needles point in the direction of the local magnetic field, towards a magnet's south pole and away from its north pole.</div></div></div></div> The field can be visualized by a set of magnetic field lines, that follow the direction of the field at each point. The lines can be constructed by measuring the strength and direction of the magnetic field at a large number of points (or at every point in space). Then, mark each location with an arrow (called a <a href="/wiki/Vector_(geometry)" class="mw-redirect" title="Vector (geometry)">vector</a>) pointing in the direction of the local magnetic field with its magnitude proportional to the strength of the magnetic field. Connecting these arrows then forms a set of magnetic field lines. The direction of the magnetic field at any point is parallel to the direction of nearby field lines, and the local density of field lines can be made proportional to its strength. Magnetic field lines are like <a href="/wiki/Streamlines,_streaklines,_and_pathlines" title="Streamlines, streaklines, and pathlines">streamlines</a> in <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid flow</a>, in that they represent a continuous distribution, and a different resolution would show more or fewer lines. An advantage of using magnetic field lines as a representation is that many laws of magnetism (and electromagnetism) can be stated completely and concisely using simple concepts such as the "number" of field lines through a surface. These concepts can be quickly "translated" to their mathematical form. For example, the number of field lines through a given surface is the <a href="/wiki/Surface_integral" title="Surface integral">surface integral</a> of the magnetic field.<a href="#cite_note-purcell2ed-12">[10]</a>: 237  Various phenomena "display" magnetic field lines as though the field lines were physical phenomena. For example, iron filings placed in a magnetic field form lines that correspond to "field lines".<a href="#cite_note-ex07-21">[note 5]</a> Magnetic field "lines" are also visually displayed in <a href="/wiki/Aurora_(astronomy)" class="mw-redirect" title="Aurora (astronomy)">polar auroras</a>, in which <a href="/wiki/Plasma_(physics)" title="Plasma (physics)">plasma</a> particle dipole interactions create visible streaks of light that line up with the local direction of Earth's magnetic field. Field lines can be used as a qualitative tool to visualize magnetic forces. In <a href="/wiki/Ferromagnetic" class="mw-redirect" title="Ferromagnetic">ferromagnetic</a> substances like <a href="/wiki/Iron" title="Iron">iron</a> and in plasmas, magnetic forces can be understood by imagining that the field lines exert a <a href="/wiki/Maxwell_stress_tensor" title="Maxwell stress tensor">tension</a>, (like a rubber band) along their length, and a pressure perpendicular to their length on neighboring field lines. "Unlike" poles of magnets attract because they are linked by many field lines; "like" poles repel because their field lines do not meet, but run parallel, pushing on each other. <div class="mw-heading mw-heading2"><h2 id="Magnetic_field_of_permanent_magnets">Magnetic field of permanent magnets</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Magnetic_moment#Models" title="Magnetic moment">Magnetic moment § Models</a></div> Permanent magnets are objects that produce their own persistent magnetic fields. They are made of <a href="/wiki/Ferromagnetism" title="Ferromagnetism">ferromagnetic</a> materials, such as iron and <a href="/wiki/Nickel" title="Nickel">nickel</a>, that have been magnetized, and they have both a north and a south pole. The magnetic field of permanent magnets can be quite complicated, especially near the magnet. The magnetic field of a small<a href="#cite_note-ex05-22">[note 6]</a> straight magnet is proportional to the magnet's strength (called its <a href="/wiki/Magnetic_dipole_moment" class="mw-redirect" title="Magnetic dipole moment">magnetic dipole moment</a> m). The <a href="/wiki/Dipole#Field_of_a_static_magnetic_dipole" title="Dipole">equations</a> are non-trivial and depend on the distance from the magnet and the orientation of the magnet. For simple magnets, m points in the direction of a line drawn from the south to the north pole of the magnet. Flipping a bar magnet is equivalent to rotating its m by 180 degrees. The magnetic field of larger magnets can be obtained by modeling them as a collection of a large number of small magnets called <a href="/wiki/Dipole" title="Dipole">dipoles</a> each having their own m. The magnetic field produced by the magnet then is the net magnetic field of these dipoles; any net force on the magnet is a result of adding up the forces on the individual dipoles. There are two simplified models for the nature of these dipoles: the <a class="mw-selflink-fragment" href="#Magnetic_pole_model">magnetic pole model</a> and the <a class="mw-selflink-fragment" href="#Amperian_loop_model">Amperian loop model</a>. These two models produce two different magnetic fields, H and B. Outside a material, though, the two are identical (to a multiplicative constant) so that in many cases the distinction can be ignored. This is particularly true for magnetic fields, such as those due to electric currents, that are not generated by magnetic materials. A realistic model of magnetism is more complicated than either of these models; neither model fully explains why materials are magnetic. The monopole model has no experimental support. The Amperian loop model explains some, but not all of a material's magnetic moment. The model predicts that the motion of electrons within an atom are connected to those electrons' <a href="/wiki/Electron_magnetic_moment#Orbital_magnetic_dipole_moment" title="Electron magnetic moment">orbital magnetic dipole moment</a>, and these orbital moments do contribute to the magnetism seen at the macroscopic level. However, the motion of electrons is not classical, and the <a href="/wiki/Spin_magnetic_moment" class="mw-redirect" title="Spin magnetic moment">spin magnetic moment</a> of electrons (which is not explained by either model) is also a significant contribution to the total moment of magnets. <div class="mw-heading mw-heading3"><h3 id="Magnetic_pole_model">Magnetic pole model</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Magnetic_monopole" title="Magnetic monopole">Magnetic monopole</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:VFPt_dipole_electric.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/VFPt_dipole_electric.svg/200px-VFPt_dipole_electric.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/VFPt_dipole_electric.svg/300px-VFPt_dipole_electric.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/VFPt_dipole_electric.svg/400px-VFPt_dipole_electric.svg.png 2x" data-file-width="600" data-file-height="600" /></a><figcaption>The magnetic pole model: two opposing poles, North (+) and South (−), separated by a distance d produce a H-field (lines).</figcaption></figure> Historically, early physics textbooks would model the force and torques between two magnets as due to magnetic poles repelling or attracting each other in the same manner as the <a href="/wiki/Coulomb_force" class="mw-redirect" title="Coulomb force">Coulomb force</a> between electric charges. At the microscopic level, this model contradicts the experimental evidence, and the pole model of magnetism is no longer the typical way to introduce the concept.<a href="#cite_note-Griffiths3ed-13">[11]</a>: 258  However, it is still sometimes used as a macroscopic model for ferromagnetism due to its mathematical simplicity.<a href="#cite_note-23">[17]</a> In this model, a magnetic H-field is produced by fictitious magnetic charges that are spread over the surface of each pole. These magnetic charges are in fact related to the magnetization field M. The H-field, therefore, is analogous to the <a href="/wiki/Electric_field" title="Electric field">electric field</a> E, which starts at a positive <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a> and ends at a negative electric charge. Near the north pole, therefore, all H-field lines point away from the north pole (whether inside the magnet or out) while near the south pole all H-field lines point toward the south pole (whether inside the magnet or out). Too, a north pole feels a force in the direction of the H-field while the force on the south pole is opposite to the H-field. In the magnetic pole model, the elementary magnetic dipole m is formed by two opposite magnetic poles of pole strength qm separated by a small distance vector d, such that m = qm d. The magnetic pole model predicts correctly the field H both inside and outside magnetic materials, in particular the fact that H is opposite to the magnetization field M inside a permanent magnet. Since it is based on the fictitious idea of a magnetic charge density, the pole model has limitations. Magnetic poles cannot exist apart from each other as electric charges can, but always come in north–south pairs. If a magnetized object is divided in half, a new pole appears on the surface of each piece, so each has a pair of complementary poles. The magnetic pole model does not account for magnetism that is produced by electric currents, nor the inherent connection between <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> and magnetism. The pole model usually treats magnetic charge as a mathematical abstraction, rather than a physical property of particles. However, a <a href="/wiki/Magnetic_monopole" title="Magnetic monopole">magnetic monopole</a> is a hypothetical particle (or class of particles) that physically has only one magnetic pole (either a north pole or a south pole). In other words, it would possess a "magnetic charge" analogous to an electric charge. Magnetic field lines would start or end on magnetic monopoles, so if they exist, they would give exceptions to the rule that magnetic field lines neither start nor end. Some theories (such as <a href="/wiki/Grand_Unified_Theory" title="Grand Unified Theory">Grand Unified Theories</a>) have predicted the existence of magnetic monopoles, but so far, none have been observed. <div class="mw-heading mw-heading3"><h3 id="Amperian_loop_model">Amperian loop model</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Magnetic_dipole" title="Magnetic dipole">Magnetic dipole</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Spin_magnetic_moment" class="mw-redirect" title="Spin magnetic moment">Spin magnetic moment</a> and <a href="/wiki/Micromagnetism" class="mw-redirect" title="Micromagnetism">Micromagnetism</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:312px;max-width:312px"><div class="trow"><div class="theader">The Amperian loop model</div></div><div class="trow"><div class="tsingle" style="width:154px;max-width:154px"><div class="thumbimage" style="height:152px;overflow:hidden"><a href="/wiki/File:VFPt_dipole_magnetic3.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/VFPt_dipole_magnetic3.svg/152px-VFPt_dipole_magnetic3.svg.png" decoding="async" width="152" height="152" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/56/VFPt_dipole_magnetic3.svg/228px-VFPt_dipole_magnetic3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/56/VFPt_dipole_magnetic3.svg/304px-VFPt_dipole_magnetic3.svg.png 2x" data-file-width="600" data-file-height="600" /></a></div></div><div class="tsingle" style="width:154px;max-width:154px"><div class="thumbimage" style="height:152px;overflow:hidden"><a href="/wiki/File:VFPt_dipole_animation_magnetic.gif" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3b/VFPt_dipole_animation_magnetic.gif/152px-VFPt_dipole_animation_magnetic.gif" decoding="async" width="152" height="152" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/3/3b/VFPt_dipole_animation_magnetic.gif 1.5x" data-file-width="220" data-file-height="220" /></a></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">A current loop (ring) that goes into the page at the x and comes out at the dot produces a B-field (lines). As the radius of the current loop shrinks, the fields produced become identical to an abstract "magnetostatic dipole" (represented by an arrow pointing to the right).</div></div></div></div> In the model developed by <a href="/wiki/Andr%C3%A9-Marie_Amp%C3%A8re" title="André-Marie Ampère">Ampere</a>, the elementary magnetic dipole that makes up all magnets is a sufficiently small Amperian loop with current I and loop area A. The dipole moment of this loop is m = IA. These magnetic dipoles produce a magnetic B-field. The magnetic field of a magnetic dipole is depicted in the figure. From outside, the ideal magnetic dipole is identical to that of an ideal electric dipole of the same strength. Unlike the electric dipole, a magnetic dipole is properly modeled as a current loop having a current I and an area a. Such a current loop has a magnetic moment of <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=Ia,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mi>I</mi> <mi>a</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=Ia,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bed6ffe7528c65af396ecbdf02f278b9f2484806" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.187ex; height:2.509ex;" alt="{\displaystyle m=Ia,}"> where the direction of m is perpendicular to the area of the loop and depends on the direction of the current using the right-hand rule. An ideal magnetic dipole is modeled as a real magnetic dipole whose area a has been reduced to zero and its current I increased to infinity such that the product m = Ia is finite. This model clarifies the connection between angular momentum and magnetic moment, which is the basis of the <a href="/wiki/Einstein%E2%80%93de_Haas_effect" title="Einstein–de Haas effect">Einstein–de Haas effect</a> rotation by magnetization and its inverse, the <a href="/wiki/Barnett_effect" title="Barnett effect">Barnett effect</a> or magnetization by rotation.<a href="#cite_note-Graham-24">[18]</a> Rotating the loop faster (in the same direction) increases the current and therefore the magnetic moment, for example. <div class="mw-heading mw-heading2"><h2 id="Interactions_with_magnets">Interactions with magnets</h2></div> <div class="mw-heading mw-heading3"><h3 id="Force_between_magnets">Force between magnets</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Force_between_magnets" title="Force between magnets">Force between magnets</a></div> Specifying the <a href="/wiki/Magnetic_moment#Forces_between_two_magnetic_dipoles" title="Magnetic moment">force between two small magnets</a> is quite complicated because it depends on the strength and <a href="/wiki/Orientation_(geometry)" title="Orientation (geometry)">orientation</a> of both magnets and their distance and direction relative to each other. The force is particularly sensitive to rotations of the magnets due to magnetic torque. The force on each magnet depends on its magnetic moment and the magnetic field<a href="#cite_note-ex04-25">[note 7]</a> of the other. To understand the force between magnets, it is useful to examine the magnetic pole model given above. In this model, the H-field of one magnet pushes and pulls on both poles of a second magnet. If this H-field is the same at both poles of the second magnet then there is no net force on that magnet since the force is opposite for opposite poles. If, however, the magnetic field of the first magnet is nonuniform (such as the H near one of its poles), each pole of the second magnet sees a different field and is subject to a different force. This difference in the two forces moves the magnet in the direction of increasing magnetic field and may also cause a net torque. This is a specific example of a general rule that magnets are attracted (or repulsed depending on the orientation of the magnet) into regions of higher magnetic field. Any non-uniform magnetic field, whether caused by permanent magnets or electric currents, exerts a force on a small magnet in this way. The details of the Amperian loop model are different and more complicated but yield the same result: that magnetic dipoles are attracted/repelled into regions of higher magnetic field. Mathematically, the force on a small magnet having a magnetic moment m due to a magnetic field B is:<a href="#cite_note-Trigg-26">[19]</a>: Eq. 11.42  <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} ={\boldsymbol {\nabla }}\left(\mathbf {m} \cdot \mathbf {B} \right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">m</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} ={\boldsymbol {\nabla }}\left(\mathbf {m} \cdot \mathbf {B} \right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2000ebec9a4d95147f858853f908ec09e1c0c23b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.044ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} ={\boldsymbol {\nabla }}\left(\mathbf {m} \cdot \mathbf {B} \right),}"> where the <a href="/wiki/Gradient" title="Gradient">gradient</a> ∇ is the change of the quantity m · B per unit distance and the direction is that of maximum increase of m · B. The <a href="/wiki/Dot_product" title="Dot product">dot product</a> m · B = mBcos(θ), where m and B represent the <a href="/wiki/Magnitude_(vector)" class="mw-redirect" title="Magnitude (vector)">magnitude</a> of the m and B vectors and θ is the angle between them. If m is in the same direction as B then the dot product is positive and the gradient points "uphill" pulling the magnet into regions of higher B-field (more strictly larger m · B). This equation is strictly only valid for magnets of zero size, but is often a good approximation for not too large magnets. The magnetic force on larger magnets is determined by dividing them into smaller regions each having their own m then <a href="/wiki/Integral" title="Integral">summing up the forces on each of these very small regions</a>. <div class="mw-heading mw-heading3"><h3 id="Magnetic_torque_on_permanent_magnets">Magnetic torque on permanent magnets</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Magnetic_torque" class="mw-redirect" title="Magnetic torque">Magnetic torque</a></div> If two like poles of two separate magnets are brought near each other, and one of the magnets is allowed to turn, it promptly rotates to align itself with the first. In this example, the magnetic field of the stationary magnet creates a magnetic torque on the magnet that is free to rotate. This magnetic torque τ tends to align a magnet's poles with the magnetic field lines. A compass, therefore, turns to align itself with Earth's magnetic field. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1237032888/mw-parser-output/.tmulti"><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:312px;max-width:312px"><div class="trow"><div class="theader">Torque on a dipole</div></div><div class="trow"><div class="tsingle" style="width:166px;max-width:166px"><div class="thumbimage" style="height:73px;overflow:hidden"><a href="/wiki/File:Dipole_in_uniform_H_field.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Dipole_in_uniform_H_field.svg/164px-Dipole_in_uniform_H_field.svg.png" decoding="async" width="164" height="74" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Dipole_in_uniform_H_field.svg/246px-Dipole_in_uniform_H_field.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Dipole_in_uniform_H_field.svg/328px-Dipole_in_uniform_H_field.svg.png 2x" data-file-width="465" data-file-height="210" /></a></div><div class="thumbcaption">In the pole model of a dipole, an H field (to right) causes equal but opposite forces on a N pole (+q) and a S pole (−q) creating a torque.</div></div><div class="tsingle" style="width:142px;max-width:142px"><div class="thumbimage" style="height:73px;overflow:hidden"><a href="/wiki/File:Torque-current-loop.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Torque-current-loop.svg/140px-Torque-current-loop.svg.png" decoding="async" width="140" height="74" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Torque-current-loop.svg/210px-Torque-current-loop.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f8/Torque-current-loop.svg/280px-Torque-current-loop.svg.png 2x" data-file-width="1346" data-file-height="709" /></a></div><div class="thumbcaption">Equivalently, a B field induces the same torque on a current loop with the same magnetic dipole moment.</div></div></div></div></div> In terms of the pole model, two equal and opposite magnetic charges experiencing the same H also experience equal and opposite forces. Since these equal and opposite forces are in different locations, this produces a torque proportional to the distance (perpendicular to the force) between them. With the definition of m as the pole strength times the distance between the poles, this leads to τ = μ0 m H sin θ, where μ0 is a constant called the <a href="/wiki/Vacuum_permeability" title="Vacuum permeability">vacuum permeability</a>, measuring 4π×10−7 <a href="/wiki/Volt" title="Volt">V</a>·<a href="/wiki/Second" title="Second">s</a>/(<a href="/wiki/Ampere" title="Ampere">A</a>·<a href="/wiki/Meter" class="mw-redirect" title="Meter">m</a>) and θ is the angle between H and m. Mathematically, the torque τ on a small magnet is proportional both to the applied magnetic field and to the magnetic moment m of the magnet: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\tau }}=\mathbf {m} \times \mathbf {B} =\mu _{0}\mathbf {m} \times \mathbf {H} ,\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">τ</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">m</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">m</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\tau }}=\mathbf {m} \times \mathbf {B} =\mu _{0}\mathbf {m} \times \mathbf {H} ,\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b2df4e74d1520e0bfd55c8f844c33c2d77c41bb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.23ex; height:2.676ex;" alt="{\displaystyle {\boldsymbol {\tau }}=\mathbf {m} \times \mathbf {B} =\mu _{0}\mathbf {m} \times \mathbf {H} ,\,}"> where × represents the vector <a href="/wiki/Cross_product" title="Cross product">cross product</a>. This equation includes all of the qualitative information included above. There is no torque on a magnet if m is in the same direction as the magnetic field, since the cross product is zero for two vectors that are in the same direction. Further, all other orientations feel a torque that twists them toward the direction of magnetic field. <div class="mw-heading mw-heading2"><h2 id="Interactions_with_electric_currents">Interactions with electric currents</h2></div> Currents of electric charges both generate a magnetic field and feel a force due to magnetic B-fields. <div class="mw-heading mw-heading3"><h3 id="Magnetic_field_due_to_moving_charges_and_electric_currents">Magnetic field due to moving charges and electric currents</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Electromagnet" title="Electromagnet">Electromagnet</a>, <a href="/wiki/Biot%E2%80%93Savart_law" title="Biot–Savart law">Biot–Savart law</a>, and <a href="/wiki/Amp%C3%A8re%27s_circuital_law" title="Ampère's circuital law">Ampère's law</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Manoderecha.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Manoderecha.svg/220px-Manoderecha.svg.png" decoding="async" width="220" height="163" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Manoderecha.svg/330px-Manoderecha.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Manoderecha.svg/440px-Manoderecha.svg.png 2x" data-file-width="407" data-file-height="301" /></a><figcaption><a href="/wiki/Right_hand_grip_rule" class="mw-redirect" title="Right hand grip rule">Right hand grip rule</a>: a current flowing in the direction of the white arrow produces a magnetic field shown by the red arrows.</figcaption></figure> All moving charged particles produce magnetic fields. Moving <a href="/wiki/Point_particle" title="Point particle">point</a> charges, such as <a href="/wiki/Electron" title="Electron">electrons</a>, produce complicated but well known magnetic fields that depend on the charge, velocity, and acceleration of the particles.<a href="#cite_note-27">[20]</a> Magnetic field lines form in <a href="/wiki/Concentric" class="mw-redirect" title="Concentric">concentric</a> circles around a <a href="/wiki/Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">cylindrical</a> current-carrying conductor, such as a length of wire. The direction of such a magnetic field can be determined by using the "<a href="/wiki/Right-hand_grip_rule" class="mw-redirect" title="Right-hand grip rule">right-hand grip rule</a>" (see figure at right). The strength of the magnetic field decreases with distance from the wire. (For an infinite length wire the strength is inversely proportional to the distance.) <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:VFPt_cylindrical_tightly-wound_coil-and-bar-magnet-comparison_stacked.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/VFPt_cylindrical_tightly-wound_coil-and-bar-magnet-comparison_stacked.svg/150px-VFPt_cylindrical_tightly-wound_coil-and-bar-magnet-comparison_stacked.svg.png" decoding="async" width="150" height="233" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/VFPt_cylindrical_tightly-wound_coil-and-bar-magnet-comparison_stacked.svg/225px-VFPt_cylindrical_tightly-wound_coil-and-bar-magnet-comparison_stacked.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d5/VFPt_cylindrical_tightly-wound_coil-and-bar-magnet-comparison_stacked.svg/300px-VFPt_cylindrical_tightly-wound_coil-and-bar-magnet-comparison_stacked.svg.png 2x" data-file-width="512" data-file-height="795" /></a><figcaption>A <a href="/wiki/Solenoid" title="Solenoid">Solenoid</a> with electric current running through it behaves like a magnet.</figcaption></figure> Bending a current-carrying wire into a loop concentrates the magnetic field inside the loop while weakening it outside. Bending a wire into multiple closely spaced loops to form a coil or "<a href="/wiki/Solenoid" title="Solenoid">solenoid</a>" enhances this effect. A device so formed around an iron <a href="/wiki/Magnetic_core" title="Magnetic core">core</a> may act as an electromagnet, generating a strong, well-controlled magnetic field. An infinitely long cylindrical electromagnet has a uniform magnetic field inside, and no magnetic field outside. A finite length electromagnet produces a magnetic field that looks similar to that produced by a uniform permanent magnet, with its strength and polarity determined by the current flowing through the coil. The magnetic field generated by a steady current I (a constant flow of electric charges, in which charge neither accumulates nor is depleted at any point)<a href="#cite_note-ex12-29">[note 8]</a> is described by the <a href="/wiki/Biot%E2%80%93Savart_law" title="Biot–Savart law">Biot–Savart law</a>:<a href="#cite_note-Griffiths4ed-28">[21]</a>: 224  <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} ={\frac {\mu _{0}I}{4\pi }}\int _{\mathrm {wire} }{\frac {\mathrm {d} {\boldsymbol {\ell }}\times \mathbf {\hat {r}} }{r^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>I</mi> </mrow> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">w</mi> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ℓ</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">r</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> </mrow> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} ={\frac {\mu _{0}I}{4\pi }}\int _{\mathrm {wire} }{\frac {\mathrm {d} {\boldsymbol {\ell }}\times \mathbf {\hat {r}} }{r^{2}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95838c9bca9c99ce5d48fd7282eb55b4c679abfb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.835ex; height:5.843ex;" alt="{\displaystyle \mathbf {B} ={\frac {\mu _{0}I}{4\pi }}\int _{\mathrm {wire} }{\frac {\mathrm {d} {\boldsymbol {\ell }}\times \mathbf {\hat {r}} }{r^{2}}},}"> where the integral sums over the wire length where vector dℓ is the vector <a href="/wiki/Line_element" title="Line element">line element</a> with direction in the same sense as the current I, μ0 is the <a href="/wiki/Magnetic_constant" class="mw-redirect" title="Magnetic constant">magnetic constant</a>, r is the distance between the location of dℓ and the location where the magnetic field is calculated, and r̂ is a unit vector in the direction of r. For example, in the case of a sufficiently long, straight wire, this becomes: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\mathbf {B} |={\frac {\mu _{0}}{2\pi r}}I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mn>2</mn> <mi>π</mi> <mi>r</mi> </mrow> </mfrac> </mrow> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\mathbf {B} |={\frac {\mu _{0}}{2\pi r}}I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c44da9834dea33f37e49df0e5abc5a15cfcb9d3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.844ex; height:4.843ex;" alt="{\displaystyle |\mathbf {B} |={\frac {\mu _{0}}{2\pi r}}I}"> where r = |r|. The direction is tangent to a circle perpendicular to the wire according to the right hand rule.<a href="#cite_note-Griffiths4ed-28">[21]</a>: 225  A slightly more general<a href="#cite_note-30">[22]</a><a href="#cite_note-ex13-31">[note 9]</a> way of relating the current <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> to the B-field is through <a href="/wiki/Amp%C3%A8re%27s_circuital_law" title="Ampère's circuital law">Ampère's law</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint \mathbf {B} \cdot \mathrm {d} {\boldsymbol {\ell }}=\mu _{0}I_{\mathrm {enc} },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ℓ</mi> </mrow> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint \mathbf {B} \cdot \mathrm {d} {\boldsymbol {\ell }}=\mu _{0}I_{\mathrm {enc} },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaffba3a24a5ddbffecba6df92dadec61d58048d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.386ex; height:5.676ex;" alt="{\displaystyle \oint \mathbf {B} \cdot \mathrm {d} {\boldsymbol {\ell }}=\mu _{0}I_{\mathrm {enc} },}"> where the <a href="/wiki/Line_integral" title="Line integral">line integral</a> is over any arbitrary loop and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{\text{enc}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>enc</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{\text{enc}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08c4588b9529bb549347f50049e6adac33d798d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.629ex; height:2.509ex;" alt="{\displaystyle I_{\text{enc}}}"> is the current enclosed by that loop. Ampère's law is always valid for steady currents and can be used to calculate the B-field for certain highly symmetric situations such as an infinite wire or an infinite solenoid. In a modified form that accounts for time varying electric fields, Ampère's law is one of four <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a> that describe electricity and magnetism. <div class="mw-heading mw-heading3"><h3 id="Force_on_moving_charges_and_current">Force on moving charges and current</h3></div> <div class="mw-heading mw-heading4"><h4 id="Force_on_a_charged_particle">Force on a charged particle</h4></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Lorentz_force" title="Lorentz force">Lorentz force</a></div> A <a href="/wiki/Charged_particle" title="Charged particle">charged particle</a> moving in a B-field experiences a sideways force that is proportional to the strength of the magnetic field, the component of the velocity that is perpendicular to the magnetic field and the charge of the particle. This force is known as the Lorentz force, and is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} =q\mathbf {E} +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">E</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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {F} =q\mathbf {E} +q\mathbf {v} \times \mathbf {B} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daba70054e9cc1b66449c534969a45695657700f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.317ex; height:2.509ex;" alt="{\displaystyle \mathbf {F} =q\mathbf {E} +q\mathbf {v} \times \mathbf {B} ,}"> where F is the <a href="/wiki/Force" title="Force">force</a>, q is the <a href="/wiki/Electric_charge" title="Electric charge">electric charge</a> of the particle, v is the instantaneous <a href="/wiki/Velocity" title="Velocity">velocity</a> of the particle, and B is the magnetic field (in <a href="/wiki/Tesla_(unit)" title="Tesla (unit)">teslas</a>). The Lorentz force is always perpendicular to both the velocity of the particle and the magnetic field that created it. When a charged particle moves in a static magnetic field, it traces a helical path in which the helix axis is parallel to the magnetic field, and in which the speed of the particle remains constant. Because the magnetic force is always perpendicular to the motion, the magnetic field can do no <a href="/wiki/Mechanical_work" class="mw-redirect" title="Mechanical work">work</a> on an isolated charge.<a href="#cite_note-32">[23]</a><a href="#cite_note-33">[24]</a> It can only do work indirectly, via the electric field generated by a changing magnetic field. It is often claimed that the magnetic force can do work to a non-elementary <a href="/wiki/Magnetic_dipole" title="Magnetic dipole">magnetic dipole</a>, or to charged particles whose motion is constrained by other forces, but this is incorrect<a href="#cite_note-Deissler-34">[25]</a> because the work in those cases is performed by the electric forces of the charges deflected by the magnetic field. <div class="mw-heading mw-heading4"><h4 id="Force_on_current-carrying_wire">Force on current-carrying wire</h4></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Laplace_force" class="mw-redirect" title="Laplace force">Laplace force</a></div> The force on a current carrying wire is similar to that of a moving charge as expected since a current carrying wire is a collection of moving charges. A current-carrying wire feels a force in the presence of a magnetic field. The Lorentz force on a macroscopic current is often referred to as the Laplace force. Consider a conductor of length ℓ, cross section A, and charge q due to electric current i. If this conductor is placed in a magnetic field of magnitude B that makes an angle θ with the velocity of charges in the conductor, the force exerted on a single charge q is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=qvB\sin \theta ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mi>q</mi> <mi>v</mi> <mi>B</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=qvB\sin \theta ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/579e409eb2d96b820933438392094864fdc6dae2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.168ex; height:2.509ex;" alt="{\displaystyle F=qvB\sin \theta ,}"> so, for N charges where <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=n\ell A,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mi>n</mi> <mi>ℓ</mi> <mi>A</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=n\ell A,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cd359cf87c3e5bec6f04da11f5c849bcdbc0ba5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.916ex; height:2.509ex;" alt="{\displaystyle N=n\ell A,}"> the force exerted on the conductor is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=FN=qvBn\ell A\sin \theta =Bi\ell \sin \theta ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mi>F</mi> <mi>N</mi> <mo>=</mo> <mi>q</mi> <mi>v</mi> <mi>B</mi> <mi>n</mi> <mi>ℓ</mi> <mi>A</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>=</mo> <mi>B</mi> <mi>i</mi> <mi>ℓ</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=FN=qvBn\ell A\sin \theta =Bi\ell \sin \theta ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dc5e41331151813e3e9c827ed97983668d98d56" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:36.071ex; height:2.509ex;" alt="{\displaystyle f=FN=qvBn\ell A\sin \theta =Bi\ell \sin \theta ,}"> where i = nqvA. <div class="mw-heading mw-heading2"><h2 id="Relation_between_H_and_B">Relation between H and B</h2></div> The formulas derived for the magnetic field above are correct when dealing with the entire current. A magnetic material placed inside a magnetic field, though, generates its own <a href="/wiki/Bound_current" class="mw-redirect" title="Bound current">bound current</a>, which can be a challenge to calculate. (This bound current is due to the sum of atomic sized current loops and the <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a> of the subatomic particles such as electrons that make up the material.) The H-field as defined above helps factor out this bound current; but to see how, it helps to introduce the concept of magnetization first. <div class="mw-heading mw-heading3"><h3 id="Magnetization">Magnetization</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Magnetization" title="Magnetization">Magnetization</a></div> The magnetization vector field M represents how strongly a region of material is magnetized. It is defined as the net <a href="/wiki/Magnetic_dipole_moment" class="mw-redirect" title="Magnetic dipole moment">magnetic dipole moment</a> per unit volume of that region. The magnetization of a uniform magnet is therefore a material constant, equal to the magnetic moment m of the magnet divided by its volume. Since the SI unit of magnetic moment is A⋅m2, the SI unit of magnetization M is ampere per meter, identical to that of the H-field. The magnetization M field of a region points in the direction of the average magnetic dipole moment in that region. Magnetization field lines, therefore, begin near the magnetic south pole and ends near the magnetic north pole. (Magnetization does not exist outside the magnet.) In the Amperian loop model, the magnetization is due to combining many tiny Amperian loops to form a resultant current called <a href="/wiki/Bound_current" class="mw-redirect" title="Bound current">bound current</a>. This bound current, then, is the source of the magnetic B field due to the magnet. Given the definition of the magnetic dipole, the magnetization field follows a similar law to that of Ampere's law:<a href="#cite_note-35">[26]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint \mathbf {M} \cdot \mathrm {d} {\boldsymbol {\ell }}=I_{\mathrm {b} },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ℓ</mi> </mrow> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">b</mi> </mrow> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint \mathbf {M} \cdot \mathrm {d} {\boldsymbol {\ell }}=I_{\mathrm {b} },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8617b3ca2fd2f45af47797bd11b89dbaf75f14c9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.107ex; height:5.676ex;" alt="{\displaystyle \oint \mathbf {M} \cdot \mathrm {d} {\boldsymbol {\ell }}=I_{\mathrm {b} },}"> where the integral is a line integral over any closed loop and Ib is the bound current enclosed by that closed loop. In the magnetic pole model, magnetization begins at and ends at magnetic poles. If a given region, therefore, has a net positive "magnetic pole strength" (corresponding to a north pole) then it has more magnetization field lines entering it than leaving it. Mathematically this is equivalent to: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{S}\mu _{0}\mathbf {M} \cdot \mathrm {d} \mathbf {A} =-q_{\mathrm {M} },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mo>−</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{S}\mu _{0}\mathbf {M} \cdot \mathrm {d} \mathbf {A} =-q_{\mathrm {M} },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50803d43052cc495a33b72b65efb2c383fc40cb4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.286ex; height:5.676ex;" alt="{\displaystyle \oint _{S}\mu _{0}\mathbf {M} \cdot \mathrm {d} \mathbf {A} =-q_{\mathrm {M} },}"> where the integral is a closed surface integral over the closed surface S and qM is the "magnetic charge" (in units of <a href="/wiki/Magnetic_flux" title="Magnetic flux">magnetic flux</a>) enclosed by S. (A closed surface completely surrounds a region with no holes to let any field lines escape.) The negative sign occurs because the magnetization field moves from south to north. <div class="mw-heading mw-heading3"><h3 id="H-field_and_magnetic_materials">H-field and magnetic materials</h3></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:VFPt_magnets_BHM.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/VFPt_magnets_BHM.svg/220px-VFPt_magnets_BHM.svg.png" decoding="async" width="220" height="396" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a7/VFPt_magnets_BHM.svg/330px-VFPt_magnets_BHM.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a7/VFPt_magnets_BHM.svg/440px-VFPt_magnets_BHM.svg.png 2x" data-file-width="600" data-file-height="1080" /></a><figcaption>Comparison of B, H and M inside and outside a cylindrical bar magnet.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Demagnetizing_field" title="Demagnetizing field">Demagnetizing field</a></div> In SI units, the H-field is related to the B-field by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} \ \equiv \ {\frac {\mathbf {B} }{\mu _{0}}}-\mathbf {M} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mtext> </mtext> <mo>≡</mo> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {H} \ \equiv \ {\frac {\mathbf {B} }{\mu _{0}}}-\mathbf {M} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c33309a5805d55e2693d904f3c1c96e6ffa5f38a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.668ex; height:5.843ex;" alt="{\displaystyle \mathbf {H} \ \equiv \ {\frac {\mathbf {B} }{\mu _{0}}}-\mathbf {M} .}"> In terms of the H-field, Ampere's law is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint \mathbf {H} \cdot \mathrm {d} {\boldsymbol {\ell }}=\oint \left({\frac {\mathbf {B} }{\mu _{0}}}-\mathbf {M} \right)\cdot \mathrm {d} {\boldsymbol {\ell }}=I_{\mathrm {tot} }-I_{\mathrm {b} }=I_{\mathrm {f} },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ℓ</mi> </mrow> <mo>=</mo> <mo>∮</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ℓ</mi> </mrow> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">t</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">t</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">b</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> </mrow> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint \mathbf {H} \cdot \mathrm {d} {\boldsymbol {\ell }}=\oint \left({\frac {\mathbf {B} }{\mu _{0}}}-\mathbf {M} \right)\cdot \mathrm {d} {\boldsymbol {\ell }}=I_{\mathrm {tot} }-I_{\mathrm {b} }=I_{\mathrm {f} },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14c07877a2da356deb28317727d27e1e111899ef" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.667ex; height:6.176ex;" alt="{\displaystyle \oint \mathbf {H} \cdot \mathrm {d} {\boldsymbol {\ell }}=\oint \left({\frac {\mathbf {B} }{\mu _{0}}}-\mathbf {M} \right)\cdot \mathrm {d} {\boldsymbol {\ell }}=I_{\mathrm {tot} }-I_{\mathrm {b} }=I_{\mathrm {f} },}"> where If represents the 'free current' enclosed by the loop so that the line integral of H does not depend at all on the bound currents.<a href="#cite_note-Slater-36">[27]</a> For the differential equivalent of this equation see <a href="#Maxwell's_equations">Maxwell's equations</a>. Ampere's law leads to the boundary condition <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\mathbf {H_{1}^{\parallel }} -\mathbf {H_{2}^{\parallel }} \right)=\mathbf {K} _{\mathrm {f} }\times {\hat {\mathbf {n} }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi mathvariant="bold">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∥</mo> </mrow> </msubsup> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi mathvariant="bold">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∥</mo> </mrow> </msubsup> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">K</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> </mrow> </mrow> </msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\mathbf {H_{1}^{\parallel }} -\mathbf {H_{2}^{\parallel }} \right)=\mathbf {K} _{\mathrm {f} }\times {\hat {\mathbf {n} }},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2df175bdc4ade6ce983a59f98b074cd4478f0b96" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.163ex; height:4.843ex;" alt="{\displaystyle \left(\mathbf {H_{1}^{\parallel }} -\mathbf {H_{2}^{\parallel }} \right)=\mathbf {K} _{\mathrm {f} }\times {\hat {\mathbf {n} }},}"> where Kf is the surface free current density and the unit normal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {\mathbf {n} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {n} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aae87b164ba005e99b51066c46d1eacc7f56564a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.343ex;" alt="{\displaystyle {\hat {\mathbf {n} }}}"> points in the direction from medium 2 to medium 1.<a href="#cite_note-37">[28]</a> Similarly, a <a href="/wiki/Surface_integral" title="Surface integral">surface integral</a> of H over any <a href="/wiki/Closed_surface" class="mw-redirect" title="Closed surface">closed surface</a> is independent of the free currents and picks out the "magnetic charges" within that closed surface: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{S}\mu _{0}\mathbf {H} \cdot \mathrm {d} \mathbf {A} =\oint _{S}(\mathbf {B} -\mu _{0}\mathbf {M} )\cdot \mathrm {d} \mathbf {A} =0-(-q_{\mathrm {M} })=q_{\mathrm {M} },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">M</mi> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mn>0</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">M</mi> </mrow> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{S}\mu _{0}\mathbf {H} \cdot \mathrm {d} \mathbf {A} =\oint _{S}(\mathbf {B} -\mu _{0}\mathbf {M} )\cdot \mathrm {d} \mathbf {A} =0-(-q_{\mathrm {M} })=q_{\mathrm {M} },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41357ab097d4ed6e748bffbd834c0db69fb26810" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:54.745ex; height:5.676ex;" alt="{\displaystyle \oint _{S}\mu _{0}\mathbf {H} \cdot \mathrm {d} \mathbf {A} =\oint _{S}(\mathbf {B} -\mu _{0}\mathbf {M} )\cdot \mathrm {d} \mathbf {A} =0-(-q_{\mathrm {M} })=q_{\mathrm {M} },}"> which does not depend on the free currents. The H-field, therefore, can be separated into two<a href="#cite_note-38">[note 10]</a> independent parts: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {H} =\mathbf {H} _{0}+\mathbf {H} _{\mathrm {d} },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {H} =\mathbf {H} _{0}+\mathbf {H} _{\mathrm {d} },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0945fc1eb7c8628d744df0cfd2f181efe3db0a77" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.061ex; height:2.509ex;" alt="{\displaystyle \mathbf {H} =\mathbf {H} _{0}+\mathbf {H} _{\mathrm {d} },}"> where H0 is the applied magnetic field due only to the free currents and Hd is the <a href="/wiki/Demagnetizing_field" title="Demagnetizing field">demagnetizing field</a> due only to the bound currents. The magnetic H-field, therefore, re-factors the bound current in terms of "magnetic charges". The H field lines loop only around "free current" and, unlike the magnetic B field, begins and ends near magnetic poles as well. <div class="mw-heading mw-heading3"><h3 id="Magnetism">Magnetism</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Magnetism" title="Magnetism">Magnetism</a></div> Most materials respond to an applied B-field by producing their own magnetization M and therefore their own B-fields. Typically, the response is weak and exists only when the magnetic field is applied. The term magnetism describes how materials respond on the microscopic level to an applied magnetic field and is used to categorize the magnetic <a href="/wiki/Phase_(matter)" title="Phase (matter)">phase</a> of a material. Materials are divided into groups based upon their magnetic behavior: <ul><li><a href="/wiki/Diamagnetism" title="Diamagnetism">Diamagnetic materials</a><a href="#cite_note-Tilley-39">[29]</a> produce a magnetization that opposes the magnetic field.</li> <li><a href="/wiki/Paramagnetism" title="Paramagnetism">Paramagnetic materials</a><a href="#cite_note-Tilley-39">[29]</a> produce a magnetization in the same direction as the applied magnetic field.</li> <li><a href="/wiki/Ferromagnetism" title="Ferromagnetism">Ferromagnetic materials</a> and the closely related <a href="/wiki/Ferrimagnetism" title="Ferrimagnetism">ferrimagnetic materials</a> and <a href="/wiki/Antiferromagnetism" title="Antiferromagnetism">antiferromagnetic materials</a><a href="#cite_note-Chikazumi-40">[30]</a><a href="#cite_note-Aharoni-41">[31]</a> can have a magnetization independent of an applied B-field with a complex relationship between the two fields.</li> <li><a href="/wiki/Superconductor" class="mw-redirect" title="Superconductor">Superconductors</a> (and <a href="/wiki/Ferromagnetic_superconductor" title="Ferromagnetic superconductor">ferromagnetic superconductors</a>)<a href="#cite_note-Bennemann-42">[32]</a><a href="#cite_note-Lewis-43">[33]</a> are materials that are characterized by perfect conductivity below a critical temperature and magnetic field. They also are highly magnetic and can be perfect diamagnets below a lower critical magnetic field. Superconductors often have a broad range of temperatures and magnetic fields (the so-named <a href="/wiki/Type_II_superconductor#Mixed_state" class="mw-redirect" title="Type II superconductor">mixed state</a>) under which they exhibit a complex hysteretic dependence of M on B.</li></ul> In the case of paramagnetism and diamagnetism, the magnetization M is often proportional to the applied magnetic field such that: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} =\mu \mathbf {H} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} =\mu \mathbf {H} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a575e79a4a2ae086cb8e2db0b14d3ddbd466df2d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.14ex; height:2.676ex;" alt="{\displaystyle \mathbf {B} =\mu \mathbf {H} ,}"> where μ is a material dependent parameter called the <a href="/wiki/Permeability_(electromagnetism)" title="Permeability (electromagnetism)">permeability</a>. In some cases the permeability may be a second rank <a href="/wiki/Tensor" title="Tensor">tensor</a> so that H may not point in the same direction as B. These relations between B and H are examples of <a href="/wiki/Constitutive_equation" title="Constitutive equation">constitutive equations</a>. However, superconductors and ferromagnets have a more complex B-to-H relation; see <a href="/wiki/Hysteresis#Magnetic_hysteresis" title="Hysteresis">magnetic hysteresis</a>. <div class="mw-heading mw-heading2"><h2 id="Stored_energy">Stored energy</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Magnetic_energy" title="Magnetic energy">Magnetic energy</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Magnetic_hysteresis" title="Magnetic hysteresis">Magnetic hysteresis</a></div> Energy is needed to generate a magnetic field both to work against the electric field that a changing magnetic field creates and to change the magnetization of any material within the magnetic field. For non-dispersive materials, this same energy is released when the magnetic field is destroyed so that the energy can be modeled as being stored in the magnetic field. For linear, non-dispersive, materials (such that B = μH where μ is frequency-independent), the <a href="/wiki/Energy_density" title="Energy density">energy density</a> is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u={\frac {\mathbf {B} \cdot \mathbf {H} }{2}}={\frac {\mathbf {B} \cdot \mathbf {B} }{2\mu }}={\frac {\mu \mathbf {H} \cdot \mathbf {H} }{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mrow> <mn>2</mn> <mi>μ</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u={\frac {\mathbf {B} \cdot \mathbf {H} }{2}}={\frac {\mathbf {B} \cdot \mathbf {B} }{2\mu }}={\frac {\mu \mathbf {H} \cdot \mathbf {H} }{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56d2a975eb0ed09b79a50affd4cfe3deb0f238fd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.197ex; height:5.843ex;" alt="{\displaystyle u={\frac {\mathbf {B} \cdot \mathbf {H} }{2}}={\frac {\mathbf {B} \cdot \mathbf {B} }{2\mu }}={\frac {\mu \mathbf {H} \cdot \mathbf {H} }{2}}.}"> If there are no magnetic materials around then μ can be replaced by μ0. The above equation cannot be used for nonlinear materials, though; a more general expression given below must be used. In general, the incremental amount of work per unit volume δW needed to cause a small change of magnetic field δB is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta W=\mathbf {H} \cdot \delta \mathbf {B} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>W</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo>⋅</mo> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta W=\mathbf {H} \cdot \delta \mathbf {B} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38cbbaed7511fa5fcbe6fd25a9c11a643ca16027" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.949ex; height:2.343ex;" alt="{\displaystyle \delta W=\mathbf {H} \cdot \delta \mathbf {B} .}"> Once the relationship between H and B is known this equation is used to determine the work needed to reach a given magnetic state. For <a href="/wiki/Hysteresis" title="Hysteresis">hysteretic materials</a> such as ferromagnets and superconductors, the work needed also depends on how the magnetic field is created. For linear non-dispersive materials, though, the general equation leads directly to the simpler energy density equation given above. <div class="mw-heading mw-heading2"><h2 id="Appearance_in_Maxwell's_equations">Appearance in Maxwell's equations</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Electromagnetism" title="Electromagnetism">Electromagnetism</a></div> Like all vector fields, a magnetic field has two important mathematical properties that relates it to its sources. (For B the sources are currents and changing electric fields.) These two properties, along with the two corresponding properties of the electric field, make up Maxwell's Equations. Maxwell's Equations together with the Lorentz force law form a complete description of <a href="/wiki/Classical_electromagnetism" title="Classical electromagnetism">classical electrodynamics</a> including both electricity and magnetism. The first property is the <a href="/wiki/Divergence" title="Divergence">divergence</a> of a vector field A, ∇ · A, which represents how A "flows" outward from a given point. As discussed above, a B-field line never starts or ends at a point but instead forms a complete loop. This is mathematically equivalent to saying that the divergence of B is zero. (Such vector fields are called <a href="/wiki/Solenoidal_vector_field" title="Solenoidal vector field">solenoidal vector fields</a>.) This property is called <a href="/wiki/Gauss%27s_law_for_magnetism" title="Gauss's law for magnetism">Gauss's law for magnetism</a> and is equivalent to the statement that there are no isolated magnetic poles or <a href="/wiki/Magnetic_monopole" title="Magnetic monopole">magnetic monopoles</a>. The second mathematical property is called the <a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">curl</a>, such that ∇ × A represents how A curls or "circulates" around a given point. The result of the curl is called a "circulation source". The equations for the curl of B and of E are called the <a href="/wiki/Amp%C3%A8re%E2%80%93Maxwell_equation" class="mw-redirect" title="Ampère–Maxwell equation">Ampère–Maxwell equation</a> and <a href="/wiki/Faraday%27s_law_of_induction" title="Faraday's law of induction">Faraday's law</a> respectively. <div class="mw-heading mw-heading3"><h3 id="Gauss'_law_for_magnetism">Gauss' law for magnetism</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Gauss%27s_law_for_magnetism" title="Gauss's law for magnetism">Gauss's law for magnetism</a></div> One important property of the B-field produced this way is that magnetic B-field lines neither start nor end (mathematically, B is a <a href="/wiki/Solenoidal_vector_field" title="Solenoidal vector field">solenoidal vector field</a>); a field line may only extend to infinity, or wrap around to form a closed curve, or follow a never-ending (possibly chaotic) path.<a href="#cite_note-lieb-44">[34]</a> Magnetic field lines exit a magnet near its north pole and enter near its south pole, but inside the magnet B-field lines continue through the magnet from the south pole back to the north.<a href="#cite_note-ex08-45">[note 11]</a> If a B-field line enters a magnet somewhere it has to leave somewhere else; it is not allowed to have an end point. More formally, since all the magnetic field lines that enter any given region must also leave that region, subtracting the "number"<a href="#cite_note-ex09-46">[note 12]</a> of field lines that enter the region from the number that exit gives identically zero. Mathematically this is equivalent to <a href="/wiki/Gauss%27s_law_for_magnetism" title="Gauss's law for magnetism">Gauss's law for magnetism</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \oint _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {A} =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \oint _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {A} =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79e91c0569752b6bf7ad9b43ad36a846c1940a72" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.125ex; height:5.676ex;" alt="{\displaystyle \oint _{S}\mathbf {B} \cdot \mathrm {d} \mathbf {A} =0}"> where the integral is a <a href="/wiki/Surface_integral" title="Surface integral">surface integral</a> over the <a href="/wiki/Closed_surface" class="mw-redirect" title="Closed surface">closed surface</a> S (a closed surface is one that completely surrounds a region with no holes to let any field lines escape). Since dA points outward, the dot product in the integral is positive for B-field pointing out and negative for B-field pointing in. <div class="mw-heading mw-heading3"><h3 id="Faraday's_Law">Faraday's Law</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Faraday%27s_law_of_induction" title="Faraday's law of induction">Faraday's law of induction</a></div> A changing magnetic field, such as a magnet moving through a conducting coil, generates an <a href="/wiki/Electric_field" title="Electric field">electric field</a> (and therefore tends to drive a current in such a coil). This is known as Faraday's law and forms the basis of many <a href="/wiki/Electrical_generator" class="mw-redirect" title="Electrical generator">electrical generators</a> and <a href="/wiki/Electric_motor" title="Electric motor">electric motors</a>. Mathematically, Faraday's law is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}=-{\frac {\mathrm {d} \Phi }{\mathrm {d} t}}}"> <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> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}=-{\frac {\mathrm {d} \Phi }{\mathrm {d} t}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcc4e063e3475deb0df6e0f1f05a750c4349c1ba" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.024ex; height:5.509ex;" alt="{\displaystyle {\mathcal {E}}=-{\frac {\mathrm {d} \Phi }{\mathrm {d} t}}}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c298ed828ff778065aeb5f0f305097f55bb9ae0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.311ex; height:2.176ex;" alt="{\displaystyle {\mathcal {E}}}"> is the <a href="/wiki/Electromotive_force" title="Electromotive force">electromotive force</a> (or EMF, the <a href="/wiki/Voltage" title="Voltage">voltage</a> generated around a closed loop) and Φ is the <a href="/wiki/Magnetic_flux" title="Magnetic flux">magnetic flux</a>—the product of the area times the magnetic field <a href="/wiki/Tangential_and_normal_components" title="Tangential and normal components">normal</a> to that area. (This definition of magnetic flux is why B is often referred to as magnetic flux density.)<a href="#cite_note-47">[35]</a>: 210  The negative sign represents the fact that any current generated by a changing magnetic field in a coil produces a magnetic field that opposes the change in the magnetic field that induced it. This phenomenon is known as <a href="/wiki/Lenz%27s_law" title="Lenz's law">Lenz's law</a>. This integral formulation of Faraday's law can be converted<a href="#cite_note-ex14-48">[note 13]</a> into a differential form, which applies under slightly different conditions. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eb118e22c941e34f5537dbbdcaa3d7ba23603e0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.495ex; height:5.509ex;" alt="{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}"> <div class="mw-heading mw-heading3"><h3 id="Ampère's_Law_and_Maxwell's_correction">Ampère's Law and Maxwell's correction</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Amp%C3%A8re%27s_circuital_law" title="Ampère's circuital law">Ampère's circuital law</a></div> Similar to the way that a changing magnetic field generates an electric field, a changing electric field generates a magnetic field. This fact is known as Maxwell's correction to Ampère's law and is applied as an additive term to Ampere's law as given above. This additional term is proportional to the time rate of change of the electric flux and is similar to Faraday's law above but with a different and positive constant out front. (The electric flux through an area is proportional to the area times the perpendicular part of the electric field.) The full law including the correction term is known as the Maxwell–Ampère equation. It is not commonly given in integral form because the effect is so small that it can typically be ignored in most cases where the integral form is used. The Maxwell term is critically important in the creation and propagation of electromagnetic waves. Maxwell's correction to Ampère's Law together with Faraday's law of induction describes how mutually changing electric and magnetic fields interact to sustain each other and thus to form <a href="/wiki/Electromagnetic_waves" class="mw-redirect" title="Electromagnetic waves">electromagnetic waves</a>, such as light: a changing electric field generates a changing magnetic field, which generates a changing electric field again. These, though, are usually described using the differential form of this equation given below. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>+</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36a7ef1851c4d9440f2adb07b0cd02a48e41481b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:24.958ex; height:5.509ex;" alt="{\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}}"> where J is the complete microscopic <a href="/wiki/Current_density" title="Current density">current density</a>, and ε0 is the <a href="/wiki/Vacuum_permittivity" title="Vacuum permittivity">vacuum permittivity</a>. As discussed above, materials respond to an applied electric E field and an applied magnetic B field by producing their own internal "bound" charge and current distributions that contribute to E and B but are difficult to calculate. To circumvent this problem, H and D fields are used to re-factor Maxwell's equations in terms of the free current density Jf: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \times \mathbf {H} =\mathbf {J} _{\mathrm {f} }+{\frac {\partial \mathbf {D} }{\partial t}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">f</mi> </mrow> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">D</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \times \mathbf {H} =\mathbf {J} _{\mathrm {f} }+{\frac {\partial \mathbf {D} }{\partial t}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/380d5f3b42edfb0c7dcb8d6b4b4a060b9b5e126a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.235ex; height:5.509ex;" alt="{\displaystyle \nabla \times \mathbf {H} =\mathbf {J} _{\mathrm {f} }+{\frac {\partial \mathbf {D} }{\partial t}}}"> These equations are not any more general than the original equations (if the "bound" charges and currents in the material are known). They also must be supplemented by the relationship between B and H as well as that between E and D. On the other hand, for simple relationships between these quantities this form of Maxwell's equations can circumvent the need to calculate the bound charges and currents. <div class="mw-heading mw-heading2"><h2 id="Formulation_in_special_relativity_and_quantum_electrodynamics">Formulation in special relativity and quantum electrodynamics</h2></div> <div class="mw-heading mw-heading3"><h3 id="Relativistic_electrodynamics">Relativistic electrodynamics</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Relativistic_electromagnetism" title="Relativistic electromagnetism">Relativistic electromagnetism</a></div> <div class="mw-heading mw-heading4"><h4 id="As_different_aspects_of_the_same_phenomenon">As different aspects of the same phenomenon</h4></div> According to <a href="/wiki/Special_relativity" title="Special relativity">the special theory of relativity</a>, the partition of the <a href="/wiki/Electromagnetic_force" class="mw-redirect" title="Electromagnetic force">electromagnetic force</a> into separate electric and magnetic components is not fundamental, but varies with the <a href="/wiki/Frame_of_reference#Observational_frames_of_reference" title="Frame of reference">observational frame of reference</a>: An electric force perceived by one observer may be perceived by another (in a different frame of reference) as a magnetic force, or a mixture of electric and magnetic forces. The magnetic field existing as electric field in other frames can be shown by consistency of equations obtained from <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a> of four force from <a href="/wiki/Coulomb%27s_law" title="Coulomb's law">Coulomb's Law</a> in particle's rest frame with <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's laws</a> considering definition of fields from <a href="/wiki/Lorentz_force#Lorentz_force_law_as_the_definition_of_E_and_B" title="Lorentz force">Lorentz force</a> and for non accelerating condition. The form of magnetic field hence obtained by <a href="/wiki/Lorentz_transformation" title="Lorentz transformation">Lorentz transformation</a> of <a href="/wiki/Four-force" title="Four-force">four-force</a> from the form of <a href="/wiki/Coulomb%27s_law" title="Coulomb's law">Coulomb's law</a> in source's initial frame is given by:<a href="#cite_note-49">[36]</a><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} ={\frac {q}{4\pi \varepsilon _{0}r^{3}}}{\frac {1-\beta ^{2}}{(1-\beta ^{2}\sin ^{2}\theta )^{3/2}}}{\frac {\mathbf {v} \times \mathbf {r} }{c^{2}}}={\frac {\mathbf {v} \times \mathbf {E} }{c^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>q</mi> <mrow> <mn>4</mn> <mi>π</mi> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>sin</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⁡</mo> <mi>θ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mrow> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} ={\frac {q}{4\pi \varepsilon _{0}r^{3}}}{\frac {1-\beta ^{2}}{(1-\beta ^{2}\sin ^{2}\theta )^{3/2}}}{\frac {\mathbf {v} \times \mathbf {r} }{c^{2}}}={\frac {\mathbf {v} \times \mathbf {E} }{c^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c084d54172e9e0130140fad562dc6800a8f49162" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:46.216ex; height:6.676ex;" alt="{\displaystyle \mathbf {B} ={\frac {q}{4\pi \varepsilon _{0}r^{3}}}{\frac {1-\beta ^{2}}{(1-\beta ^{2}\sin ^{2}\theta )^{3/2}}}{\frac {\mathbf {v} \times \mathbf {r} }{c^{2}}}={\frac {\mathbf {v} \times \mathbf {E} }{c^{2}}}}">where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"> is the charge of the point source, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varepsilon _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acb0a8377db20e42274444cb181d51b5532b5844" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{0}}"> is the <a href="/wiki/Vacuum_permittivity" title="Vacuum permittivity">vacuum permittivity</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"> is the position vector from the point source to the point in space, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"> is the velocity vector of the charged particle, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"> is the ratio of speed of the charged particle divided by the speed of light and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> is the angle between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }">. This form of magnetic field can be shown to satisfy maxwell's laws within the constraint of particle being non accelerating.<a href="#cite_note-50">[37]</a> The above reduces to <a href="/wiki/Biot%E2%80%93Savart_law" title="Biot–Savart law">Biot-Savart law</a> for non relativistic stream of current (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \ll 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>≪</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta \ll 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a023c5789a518613947d223437d76e178e8fd607" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.109ex; height:2.509ex;" alt="{\displaystyle \beta \ll 1}">). Formally, special relativity combines the electric and magnetic fields into a rank-2 <a href="/wiki/Tensor" title="Tensor">tensor</a>, called the <a href="/wiki/Electromagnetic_tensor" title="Electromagnetic tensor">electromagnetic tensor</a>. Changing reference frames mixes these components. This is analogous to the way that special relativity mixes space and time into <a href="/wiki/Spacetime" title="Spacetime">spacetime</a>, and mass, momentum, and energy into <a href="/wiki/Four-momentum" title="Four-momentum">four-momentum</a>.<a href="#cite_note-51">[38]</a> Similarly, the <a href="/wiki/Magnetic_energy" title="Magnetic energy">energy stored in a magnetic field</a> is mixed with the energy stored in an electric field in the <a href="/wiki/Electromagnetic_stress%E2%80%93energy_tensor" title="Electromagnetic stress–energy tensor">electromagnetic stress–energy tensor</a>. <div class="mw-heading mw-heading4"><h4 id="Magnetic_vector_potential">Magnetic vector potential</h4></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Magnetic_vector_potential" title="Magnetic vector potential">Magnetic vector potential</a></div> In advanced topics such as <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> and <a href="/wiki/Theory_of_relativity" title="Theory of relativity">relativity</a> it is often easier to work with a potential formulation of electrodynamics rather than in terms of the electric and magnetic fields. In this representation, the <a href="/wiki/Magnetic_vector_potential" title="Magnetic vector potential">magnetic vector potential</a> A, and the <a href="/wiki/Electric_potential" title="Electric potential">electric scalar potential</a> φ, are defined using <a href="/wiki/Gauge_fixing" title="Gauge fixing">gauge fixing</a> such that: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {B} &=\nabla \times \mathbf {A} ,\\\mathbf {E} &=-\nabla \varphi -{\frac {\partial \mathbf {A} }{\partial t}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mi>φ</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {B} &=\nabla \times \mathbf {A} ,\\\mathbf {E} &=-\nabla \varphi -{\frac {\partial \mathbf {A} }{\partial t}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/727409f6151fc24d6f89c41cb664b4538cc45771" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.427ex; margin-bottom: -0.244ex; width:18.676ex; height:8.509ex;" alt="{\displaystyle {\begin{aligned}\mathbf {B} &=\nabla \times \mathbf {A} ,\\\mathbf {E} &=-\nabla \varphi -{\frac {\partial \mathbf {A} }{\partial t}}.\end{aligned}}}"> The vector potential, A given by this form may be interpreted as a generalized potential <a href="/wiki/Momentum" title="Momentum">momentum</a> per unit charge <a href="#cite_note-52">[39]</a> just as φ is interpreted as a generalized <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a> per unit charge. There are multiple choices one can make for the potential fields that satisfy the above condition. However, the choice of potentials is represented by its respective gauge condition. Maxwell's equations when expressed in terms of the potentials in <a href="/wiki/Lorenz_gauge_condition" title="Lorenz gauge condition">Lorenz gauge</a> can be cast into a form that agrees with <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a>.<a href="#cite_note-53">[40]</a> In relativity, A together with φ forms a <a href="/wiki/Four-potential" class="mw-redirect" title="Four-potential">four-potential</a> regardless of the gauge condition, analogous to the <a href="/wiki/Four-vector#Four-momentum" title="Four-vector">four-momentum</a> that combines the momentum and energy of a particle. Using the four potential instead of the electromagnetic tensor has the advantage of being much simpler—and it can be easily modified to work with quantum mechanics. <div class="mw-heading mw-heading4"><h4 id="Propagation_of_Electric_and_Magnetic_fields">Propagation of Electric and Magnetic fields</h4></div> <a href="/wiki/Special_relativity" title="Special relativity">Special theory of relativity</a> imposes the condition for events related by <a href="/wiki/Causality" title="Causality">cause and effect</a> to be time-like separated, that is that causal efficacy propagates no faster than light.<a href="#cite_note-54">[41]</a> <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a> for electromagnetism are found to be in favor of this as electric and magnetic disturbances are found to travel at the speed of light in space. Electric and magnetic fields from classical electrodynamics obey the <a href="/wiki/Principle_of_locality" title="Principle of locality">principle of locality</a> in physics and are expressed in terms of retarded time or the time at which the cause of a measured field originated given that the influence of field travelled at speed of light. The retarded time for a point particle is given as solution of: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{r}=\mathbf {t} -{\frac {|\mathbf {r} -\mathbf {r} _{s}(t_{r})|}{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> <mi>c</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{r}=\mathbf {t} -{\frac {|\mathbf {r} -\mathbf {r} _{s}(t_{r})|}{c}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7f6016227a77a9ad998fd70ea885614095faba8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:20.592ex; height:5.676ex;" alt="{\displaystyle t_{r}=\mathbf {t} -{\frac {|\mathbf {r} -\mathbf {r} _{s}(t_{r})|}{c}}}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {t_{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {t_{r}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cdd506a85017825151ed5d92eb1923d604a58d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.813ex; height:2.343ex;" alt="{\textstyle {t_{r}}}"> is <a href="/wiki/Retarded_time" title="Retarded time">retarded time</a> or the time at which the source's contribution of the field originated, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {r}_{s}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {r}_{s}(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eedcd3e403f83fb4588ca7d684e2d02b8033f851" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.701ex; height:2.843ex;" alt="{\textstyle {r}_{s}(t)}"> is the position vector of the particle as function of time, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mathbf {r} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5a91b11214869eaa0f737b2da837b549978c26d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\textstyle \mathbf {r} }"> is the point in space, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {t} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mathbf {t} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ec78f6d5f7bb3ff55803ad7271dc35c65598232" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.039ex; height:2.009ex;" alt="{\textstyle \mathbf {t} }"> is the time at which fields are measured and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d411ca19645ddd4fff0704de95ec770681093bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\textstyle c}"> is the speed of light. The equation subtracts the time taken for light to travel from particle to the point in space from the time of measurement to find time of origin of the fields. The uniqueness of solution for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {t_{r}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {t_{r}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cdd506a85017825151ed5d92eb1923d604a58d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.813ex; height:2.343ex;" alt="{\textstyle {t_{r}}}"> for given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {t} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {t} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16ff63cc74a931900e79b3caacbae3fa8cc66845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.039ex; height:2.009ex;" alt="{\displaystyle \mathbf {t} }">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.102ex; height:1.676ex;" alt="{\displaystyle \mathbf {r} }"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{s}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{s}(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ff13e3015a6d8b182882d741b5ae0499a7d3562" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.701ex; height:2.843ex;" alt="{\displaystyle r_{s}(t)}"> is valid for charged particles moving slower than speed of light.<a href="#cite_note-55">[42]</a> <div class="mw-heading mw-heading4"><h4 id="Magnetic_field_of_arbitrary_moving_point_charge">Magnetic field of arbitrary moving point charge</h4></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Li%C3%A9nard%E2%80%93Wiechert_potential" title="Liénard–Wiechert potential">Liénard–Wiechert potential</a></div> The solution of maxwell's equations for electric and magnetic field of a point charge is expressed in terms of <a href="/wiki/Retarded_time" title="Retarded time">retarded time</a> or the time at which the particle in the past causes the field at the point, given that the influence travels across space at the speed of light. Any arbitrary motion of point charge causes electric and magnetic fields found by solving maxwell's equations using green's function for retarded potentials and hence finding the fields to be as follows: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} (\mathbf {r} ,\mathbf {t} )={\frac {\mu _{0}c}{4\pi }}\left({\frac {q{\boldsymbol {\beta }}_{s}}{(1-\mathbf {n} _{s}\cdot {\boldsymbol {\beta }}_{s})|\mathbf {r} -\mathbf {r} _{s}|}}\right)_{t=t_{r}}={\frac {{\boldsymbol {\beta }}_{s}(t_{r})}{c}}\varphi (\mathbf {r} ,\mathbf {t} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>c</mi> </mrow> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mi>c</mi> </mfrac> </mrow> <mi>φ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} (\mathbf {r} ,\mathbf {t} )={\frac {\mu _{0}c}{4\pi }}\left({\frac {q{\boldsymbol {\beta }}_{s}}{(1-\mathbf {n} _{s}\cdot {\boldsymbol {\beta }}_{s})|\mathbf {r} -\mathbf {r} _{s}|}}\right)_{t=t_{r}}={\frac {{\boldsymbol {\beta }}_{s}(t_{r})}{c}}\varphi (\mathbf {r} ,\mathbf {t} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8528f0775100a844d976311882054d1de02978e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:58.582ex; height:6.843ex;" alt="{\displaystyle \mathbf {A} (\mathbf {r} ,\mathbf {t} )={\frac {\mu _{0}c}{4\pi }}\left({\frac {q{\boldsymbol {\beta }}_{s}}{(1-\mathbf {n} _{s}\cdot {\boldsymbol {\beta }}_{s})|\mathbf {r} -\mathbf {r} _{s}|}}\right)_{t=t_{r}}={\frac {{\boldsymbol {\beta }}_{s}(t_{r})}{c}}\varphi (\mathbf {r} ,\mathbf {t} )}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} (\mathbf {r} ,\mathbf {t} )={\frac {\mu _{0}}{4\pi }}\left({\frac {qc({\boldsymbol {\beta }}_{s}\times \mathbf {n} _{s})}{\gamma ^{2}(1-\mathbf {n} _{s}\cdot {\boldsymbol {\beta }}_{s})^{3}|\mathbf {r} -\mathbf {r} _{s}|^{2}}}+{\frac {q\mathbf {n} _{s}\times {\Big (}\mathbf {n} _{s}\times {\big (}(\mathbf {n} _{s}-{\boldsymbol {\beta }}_{s})\times {\dot {{\boldsymbol {\beta }}_{s}}}{\big )}{\Big )}}{(1-\mathbf {n} _{s}\cdot {\boldsymbol {\beta }}_{s})^{3}|\mathbf {r} -\mathbf {r} _{s}|}}\right)_{t=t_{r}}={\frac {\mathbf {n} _{s}(t_{r})}{c}}\times \mathbf {E} (\mathbf {r} ,\mathbf {t} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <mi>c</mi> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>×</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <msup> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>q</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mi>c</mi> </mfrac> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} (\mathbf {r} ,\mathbf {t} )={\frac {\mu _{0}}{4\pi }}\left({\frac {qc({\boldsymbol {\beta }}_{s}\times \mathbf {n} _{s})}{\gamma ^{2}(1-\mathbf {n} _{s}\cdot {\boldsymbol {\beta }}_{s})^{3}|\mathbf {r} -\mathbf {r} _{s}|^{2}}}+{\frac {q\mathbf {n} _{s}\times {\Big (}\mathbf {n} _{s}\times {\big (}(\mathbf {n} _{s}-{\boldsymbol {\beta }}_{s})\times {\dot {{\boldsymbol {\beta }}_{s}}}{\big )}{\Big )}}{(1-\mathbf {n} _{s}\cdot {\boldsymbol {\beta }}_{s})^{3}|\mathbf {r} -\mathbf {r} _{s}|}}\right)_{t=t_{r}}={\frac {\mathbf {n} _{s}(t_{r})}{c}}\times \mathbf {E} (\mathbf {r} ,\mathbf {t} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24e9c2728ab3cdbe544a4d0c2932b8c0bd61d15b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:100.975ex; height:10.509ex;" alt="{\displaystyle \mathbf {B} (\mathbf {r} ,\mathbf {t} )={\frac {\mu _{0}}{4\pi }}\left({\frac {qc({\boldsymbol {\beta }}_{s}\times \mathbf {n} _{s})}{\gamma ^{2}(1-\mathbf {n} _{s}\cdot {\boldsymbol {\beta }}_{s})^{3}|\mathbf {r} -\mathbf {r} _{s}|^{2}}}+{\frac {q\mathbf {n} _{s}\times {\Big (}\mathbf {n} _{s}\times {\big (}(\mathbf {n} _{s}-{\boldsymbol {\beta }}_{s})\times {\dot {{\boldsymbol {\beta }}_{s}}}{\big )}{\Big )}}{(1-\mathbf {n} _{s}\cdot {\boldsymbol {\beta }}_{s})^{3}|\mathbf {r} -\mathbf {r} _{s}|}}\right)_{t=t_{r}}={\frac {\mathbf {n} _{s}(t_{r})}{c}}\times \mathbf {E} (\mathbf {r} ,\mathbf {t} )}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \varphi (\mathbf {r} ,\mathbf {t} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>φ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \varphi (\mathbf {r} ,\mathbf {t} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4aae5d1d32c43f85a7dc7afe5c3d6da062dce0e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.505ex; height:2.843ex;" alt="{\textstyle \varphi (\mathbf {r} ,\mathbf {t} )}">and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {A} (\mathbf {r} ,\mathbf {t} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \mathbf {A} (\mathbf {r} ,\mathbf {t} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2394415d492ac57022e002f3ed484639748ea371" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.004ex; height:2.843ex;" alt="{\textstyle \mathbf {A} (\mathbf {r} ,\mathbf {t} )}"> are electric scalar potential and magnetic vector potential in Lorentz gauge, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06809d64fa7c817ffc7e323f85997f783dbdf71d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}"> is the charge of the point source, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {n}_{s}(\mathbf {r} ,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {n}_{s}(\mathbf {r} ,t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9425135a7e38334ed3ca8dc835ff02d0018fa49c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.183ex; height:2.843ex;" alt="{\textstyle {n}_{s}(\mathbf {r} ,t)}"> is a unit vector pointing from charged particle to the point in space, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\boldsymbol {\beta }}_{s}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">β</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\boldsymbol {\beta }}_{s}(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/199aee04208761da6a8d8281b23cf57b6c6e2662" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.186ex; height:2.843ex;" alt="{\textstyle {\boldsymbol {\beta }}_{s}(t)}"> is the velocity of the particle divided by the speed of light and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \gamma (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \gamma (t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23fbfda4c2fb66c456639061f5585f7d3e0e2f9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.911ex; height:2.843ex;" alt="{\textstyle \gamma (t)}"> is the corresponding <a href="/wiki/Lorentz_factor" title="Lorentz factor">Lorentz factor</a>. Hence by the <a href="/wiki/Superposition_principle" title="Superposition principle">principle of superposition</a>, the fields of a system of charges also obey <a href="/wiki/Principle_of_locality" title="Principle of locality">principle of locality</a>. <div class="mw-heading mw-heading3"><h3 id="Quantum_electrodynamics">Quantum electrodynamics</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Standard_Model" title="Standard Model">Standard Model</a> and <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a></div> The classical electromagnetic field incorporated into quantum mechanics forms what is known as the semi-classical theory of radiation. However, it is not able to make experimentally observed predictions such as <a href="/wiki/Spontaneous_emission" title="Spontaneous emission">spontaneous emission process</a> or <a href="/wiki/Lamb_shift" title="Lamb shift">Lamb shift</a> implying the need for quantization of fields. In modern physics, the electromagnetic field is understood to be not a <a href="/wiki/Classical_physics" title="Classical physics">classical</a> <a href="/wiki/Field_(physics)" title="Field (physics)">field</a>, but rather a <a href="/wiki/Quantum_field" class="mw-redirect" title="Quantum field">quantum field</a>; it is represented not as a vector of three <a href="/wiki/Real_number" title="Real number">numbers</a> at each point, but as a vector of three <a href="/wiki/Operator_(physics)" title="Operator (physics)">quantum operators</a> at each point. The most accurate modern description of the electromagnetic interaction (and much else) is quantum electrodynamics (QED),<a href="#cite_note-56">[43]</a> which is incorporated into a more complete theory known as the Standard Model of particle physics. In QED, the magnitude of the electromagnetic interactions between charged particles (and their <a href="/wiki/Antiparticle" title="Antiparticle">antiparticles</a>) is computed using <a href="/wiki/Perturbation_theory_(quantum_mechanics)" title="Perturbation theory (quantum mechanics)">perturbation theory</a>. These rather complex formulas produce a remarkable pictorial representation as <a href="/wiki/Feynman_diagram" title="Feynman diagram">Feynman diagrams</a> in which <a href="/wiki/Virtual_photon" title="Virtual photon">virtual photons</a> are exchanged. Predictions of QED agree with experiments to an extremely high degree of accuracy: currently about 10−12 (and limited by experimental errors); for details see <a href="/wiki/Precision_tests_of_QED" title="Precision tests of QED">precision tests of QED</a>. This makes QED one of the most accurate physical theories constructed thus far. All equations in this article are in the <a href="/wiki/Classical_electromagnetism" title="Classical electromagnetism">classical approximation</a>, which is less accurate than the quantum description mentioned here. However, under most everyday circumstances, the difference between the two theories is negligible. <div class="mw-heading mw-heading2"><h2 id="Uses_and_examples">Uses and examples</h2></div> <div class="mw-heading mw-heading3"><h3 id="Earth's_magnetic_field">Earth's magnetic field</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Earth%27s_magnetic_field" title="Earth's magnetic field">Earth's magnetic field</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:VFPt_Earths_Magnetic_Field_Confusion.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/VFPt_Earths_Magnetic_Field_Confusion.svg/220px-VFPt_Earths_Magnetic_Field_Confusion.svg.png" decoding="async" width="220" height="206" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/VFPt_Earths_Magnetic_Field_Confusion.svg/330px-VFPt_Earths_Magnetic_Field_Confusion.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/ff/VFPt_Earths_Magnetic_Field_Confusion.svg/440px-VFPt_Earths_Magnetic_Field_Confusion.svg.png 2x" data-file-width="320" data-file-height="300" /></a><figcaption>A sketch of Earth's magnetic field representing the source of the field as a magnet. The south pole of the magnetic field is near the geographic north pole of the Earth.</figcaption></figure> The Earth's magnetic field is produced by <a href="/wiki/Convection" title="Convection">convection</a> of a liquid iron alloy in the <a href="/wiki/Outer_core" class="mw-redirect" title="Outer core">outer core</a>. In a <a href="/wiki/Dynamo_theory" title="Dynamo theory">dynamo process</a>, the movements drive a feedback process in which electric currents create electric and magnetic fields that in turn act on the currents.<a href="#cite_note-Weiss-57">[44]</a> The field at the surface of the Earth is approximately the same as if a giant bar magnet were positioned at the center of the Earth and tilted at an angle of about 11° off the rotational axis of the Earth (see the figure).<a href="#cite_note-58">[45]</a> The north pole of a magnetic compass needle points roughly north, toward the <a href="/wiki/North_Magnetic_Pole" class="mw-redirect" title="North Magnetic Pole">North Magnetic Pole</a>. However, because a magnetic pole is attracted to its opposite, the North Magnetic Pole is actually the south pole of the geomagnetic field. This confusion in terminology arises because the pole of a magnet is defined by the geographical direction it points.<a href="#cite_note-59">[46]</a> Earth's magnetic field is not constant—the strength of the field and the location of its poles vary.<a href="#cite_note-60">[47]</a> Moreover, the poles periodically reverse their orientation in a process called <a href="/wiki/Geomagnetic_reversal" title="Geomagnetic reversal">geomagnetic reversal</a>. The <a href="/wiki/Brunhes%E2%80%93Matuyama_reversal" title="Brunhes–Matuyama reversal">most recent reversal</a> occurred 780,000 years ago.<a href="#cite_note-61">[48]</a> <div class="mw-heading mw-heading3"><h3 id="Rotating_magnetic_fields">Rotating magnetic fields</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Rotating_magnetic_field" title="Rotating magnetic field">Rotating magnetic field</a> and <a href="/wiki/Alternator" title="Alternator">Alternator</a></div> The rotating magnetic field is a key principle in the operation of <a href="/wiki/Electric_motor#AC_motors" title="Electric motor">alternating-current motors</a>. A permanent magnet in such a field rotates so as to maintain its alignment with the external field. Magnetic torque is used to drive <a href="/wiki/Electric_motor" title="Electric motor">electric motors</a>. In one simple motor design, a magnet is fixed to a freely rotating shaft and subjected to a magnetic field from an array of <a href="/wiki/Electromagnet" title="Electromagnet">electromagnets</a>. By continuously switching the electric current through each of the electromagnets, thereby flipping the polarity of their magnetic fields, like poles are kept next to the rotor; the resultant torque is transferred to the shaft. A rotating magnetic field can be constructed using two orthogonal coils with 90 degrees phase difference in their AC currents. However, in practice such a system would be supplied through a three-wire arrangement with unequal currents. This inequality would cause serious problems in standardization of the conductor size and so, to overcome it, <a href="/wiki/Three-phase_electric_power" title="Three-phase electric power">three-phase</a> systems are used where the three currents are equal in magnitude and have 120 degrees phase difference. Three similar coils having mutual geometrical angles of 120 degrees create the rotating magnetic field in this case. The ability of the three-phase system to create a rotating field, utilized in electric motors, is one of the main reasons why three-phase systems dominate the world's <a href="/wiki/Electrical_power" class="mw-redirect" title="Electrical power">electrical power</a> supply systems. <a href="/wiki/Synchronous_motor" title="Synchronous motor">Synchronous motors</a> use DC-voltage-fed rotor windings, which lets the excitation of the machine be controlled—and <a href="/wiki/Induction_motor" title="Induction motor">induction motors</a> use short-circuited <a href="/wiki/Rotor_(electric)" title="Rotor (electric)">rotors</a> (instead of a magnet) following the rotating magnetic field of a multicoiled <a href="/wiki/Stator" title="Stator">stator</a>. The short-circuited turns of the rotor develop <a href="/wiki/Eddy_current" title="Eddy current">eddy currents</a> in the rotating field of the stator, and these currents in turn move the rotor by the Lorentz force. The Italian physicist <a href="/wiki/Galileo_Ferraris" title="Galileo Ferraris">Galileo Ferraris</a> and the Serbian-American <a href="/wiki/Electrical_engineer" class="mw-redirect" title="Electrical engineer">electrical engineer</a> <a href="/wiki/Nikola_Tesla" title="Nikola Tesla">Nikola Tesla</a> independently researched the use of rotating magnetic fields in electric motors. In 1888, Ferraris published his research in a paper to the Royal Academy of Sciences in <a href="/wiki/Turin" title="Turin">Turin</a> and Tesla gained <a rel="nofollow" class="external text" href="https://patents.google.com/patent/US381968">U.S. patent 381,968</a> for his work. <div class="mw-heading mw-heading3"><h3 id="Hall_effect">Hall effect</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Hall_effect" title="Hall effect">Hall effect</a></div> The charge carriers of a current-carrying conductor placed in a transverse magnetic field experience a sideways Lorentz force; this results in a charge separation in a direction perpendicular to the current and to the magnetic field. The resultant voltage in that direction is proportional to the applied magnetic field. This is known as the Hall effect. The Hall effect is often used to measure the magnitude of a magnetic field. It is used as well to find the sign of the dominant charge carriers in materials such as semiconductors (negative electrons or positive holes). <div class="mw-heading mw-heading3"><h3 id="Magnetic_circuits">Magnetic circuits</h3></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Magnetic_circuit" title="Magnetic circuit">Magnetic circuit</a></div> An important use of H is in magnetic circuits where B = μH inside a linear material. Here, μ is the <a href="/wiki/Magnetic_permeability" class="mw-redirect" title="Magnetic permeability">magnetic permeability</a> of the material. This result is similar in form to <a href="/wiki/Ohm%27s_law" title="Ohm's law">Ohm's law</a> J = σE, where J is the current density, σ is the conductance and E is the electric field. Extending this analogy, the counterpart to the macroscopic Ohm's law (I = V⁄R) is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi ={\frac {F}{R}}_{\mathrm {m} },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>F</mi> <mi>R</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi ={\frac {F}{R}}_{\mathrm {m} },}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18a0f4b5e7c42f2f7ba26fca357bac5ed9a2cf6a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.625ex; height:5.343ex;" alt="{\displaystyle \Phi ={\frac {F}{R}}_{\mathrm {m} },}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Phi =\int \mathbf {B} \cdot \mathrm {d} \mathbf {A} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \Phi =\int \mathbf {B} \cdot \mathrm {d} \mathbf {A} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/741ae04e7f140c9dbc036abcfe61329eb5ef192c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.474ex; height:3.176ex;" alt="{\textstyle \Phi =\int \mathbf {B} \cdot \mathrm {d} \mathbf {A} }"> is the magnetic flux in the circuit, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle F=\int \mathbf {H} \cdot \mathrm {d} {\boldsymbol {\ell }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ℓ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle F=\int \mathbf {H} \cdot \mathrm {d} {\boldsymbol {\ell }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eaf4a3fd61dd5f4825fa3ae0452aa129020f1827" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.809ex; height:3.176ex;" alt="{\textstyle F=\int \mathbf {H} \cdot \mathrm {d} {\boldsymbol {\ell }}}"> is the <a href="/wiki/Magnetomotive_force" title="Magnetomotive force">magnetomotive force</a> applied to the circuit, and Rm is the <a href="/wiki/Reluctance" class="mw-redirect" title="Reluctance">reluctance</a> of the circuit. Here the reluctance Rm is a quantity similar in nature to <a href="/wiki/Electrical_resistance" class="mw-redirect" title="Electrical resistance">resistance</a> for the flux. Using this analogy it is straightforward to calculate the magnetic flux of complicated magnetic field geometries, by using all the available techniques of <a href="/wiki/Circuit_theory" class="mw-redirect" title="Circuit theory">circuit theory</a>. <div class="mw-heading mw-heading3"><h3 id="Largest_magnetic_fields">Largest magnetic fields</h3></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Update plainlinks metadata ambox ambox-content ambox-Update" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Ambox_current_red_Americas.svg/42px-Ambox_current_red_Americas.svg.png" decoding="async" width="42" height="34" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/53/Ambox_current_red_Americas.svg/63px-Ambox_current_red_Americas.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/53/Ambox_current_red_Americas.svg/84px-Ambox_current_red_Americas.svg.png 2x" data-file-width="360" data-file-height="290" /></div></td><td class="mbox-text"><div class="mbox-text-span">This section needs to be updated. Please help update this article to reflect recent events or newly available information. Last update: October 2018 (July 2021)</div></td></tr></tbody></table> As of October 2018<a class="external text" href="https://en.wikipedia.org/w/index.php?title=Magnetic_field&action=edit">[update]</a>, the largest magnetic field produced over a macroscopic volume outside a lab setting is 2.8 kT (<a href="/wiki/VNIIEF" class="mw-redirect" title="VNIIEF">VNIIEF</a> in <a href="/wiki/Sarov" title="Sarov">Sarov</a>, <a href="/wiki/Russia" title="Russia">Russia</a>, 1998).<a href="#cite_note-62">[49]</a><a href="#cite_note-smithsonianMagnetRecord-63">[50]</a> As of October 2018, the largest magnetic field produced in a laboratory over a macroscopic volume was 1.2 kT by researchers at the <a href="/wiki/University_of_Tokyo" title="University of Tokyo">University of Tokyo</a> in 2018.<a href="#cite_note-smithsonianMagnetRecord-63">[50]</a> The largest magnetic fields produced in a laboratory occur in particle accelerators, such as <a href="/wiki/Relativistic_Heavy_Ion_Collider" title="Relativistic Heavy Ion Collider">RHIC</a>, inside the collisions of heavy ions, where microscopic fields reach 1014 T.<a href="#cite_note-64">[51]</a><a href="#cite_note-65">[52]</a> <a href="/wiki/Magnetar" title="Magnetar">Magnetars</a> have the strongest known magnetic fields of any naturally occurring object, ranging from 0.1 to 100 GT (108 to 1011 T).<a href="#cite_note-66">[53]</a> <div class="mw-heading mw-heading2"><h2 id="Common_formulæ">Common formulæ</h2></div> <table class="wikitable"> <caption> </caption> <tbody><tr> <th>Current configuration </th> <th>Figure </th> <th colspan="2">Magnetic field </th></tr> <tr> <td>Finite beam of current </td> <td><a href="/wiki/File:Finite_beam_of_current.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Finite_beam_of_current.svg/235px-Finite_beam_of_current.svg.png" decoding="async" width="235" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Finite_beam_of_current.svg/353px-Finite_beam_of_current.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/Finite_beam_of_current.svg/470px-Finite_beam_of_current.svg.png 2x" data-file-width="567" data-file-height="567" /></a> </td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} ={\mu _{0}I \over 4\pi x}(\cos \theta _{1}+\cos \theta _{2}){\hat {\mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>I</mi> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} ={\mu _{0}I \over 4\pi x}(\cos \theta _{1}+\cos \theta _{2}){\hat {\mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6d936180333561c0dac5c26c8a6a21d14eb0f5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.006ex; height:5.343ex;" alt="{\displaystyle \mathbf {B} ={\mu _{0}I \over 4\pi x}(\cos \theta _{1}+\cos \theta _{2}){\hat {\mathbf {x} }}}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> is the uniform current throughout the beam, with the direction of magnetic field as shown. </td></tr> <tr> <td>Infinite wire </td> <td><a href="/wiki/File:Infinite_current_carrying_wire.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Infinite_current_carrying_wire.svg/235px-Infinite_current_carrying_wire.svg.png" decoding="async" width="235" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Infinite_current_carrying_wire.svg/353px-Infinite_current_carrying_wire.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Infinite_current_carrying_wire.svg/470px-Infinite_current_carrying_wire.svg.png 2x" data-file-width="567" data-file-height="567" /></a> </td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} ={\mu _{0}I \over 2\pi x}{\hat {\mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>I</mi> </mrow> <mrow> <mn>2</mn> <mi>π</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} ={\mu _{0}I \over 2\pi x}{\hat {\mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e796921f94308e43672b0b33c6eeb38ccf3dc8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.071ex; height:5.343ex;" alt="{\displaystyle \mathbf {B} ={\mu _{0}I \over 2\pi x}{\hat {\mathbf {x} }}}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> is the uniform current flowing through the wire with the direction of magnetic field as shown. </td></tr> <tr> <td>Infinite cylindrical wire </td> <td><a href="/wiki/File:Infinite_current_carrying_cylinder.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Infinite_current_carrying_cylinder.svg/235px-Infinite_current_carrying_cylinder.svg.png" decoding="async" width="235" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Infinite_current_carrying_cylinder.svg/353px-Infinite_current_carrying_cylinder.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Infinite_current_carrying_cylinder.svg/470px-Infinite_current_carrying_cylinder.svg.png 2x" data-file-width="567" data-file-height="567" /></a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} ={\mu _{0}I \over 2\pi x}{\hat {\mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>I</mi> </mrow> <mrow> <mn>2</mn> <mi>π</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} ={\mu _{0}I \over 2\pi x}{\hat {\mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e796921f94308e43672b0b33c6eeb38ccf3dc8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.071ex; height:5.343ex;" alt="{\displaystyle \mathbf {B} ={\mu _{0}I \over 2\pi x}{\hat {\mathbf {x} }}}"> outside the wire carrying a current <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> uniformly, with the direction of magnetic field as shown. </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} ={\mu _{0}Ix \over 2\pi R^{2}}{\hat {\mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>I</mi> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <mi>π</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} ={\mu _{0}Ix \over 2\pi R^{2}}{\hat {\mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7953ec61d0484a124593183eb8b48165ec4ad67f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.559ex; height:5.676ex;" alt="{\displaystyle \mathbf {B} ={\mu _{0}Ix \over 2\pi R^{2}}{\hat {\mathbf {x} }}}"> inside the wire carrying a current <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> uniformly, with the direction of magnetic field as shown. </td></tr> <tr> <td>Circular loop </td> <td><a href="/wiki/File:Current_carrying_ring.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Current_carrying_ring.svg/235px-Current_carrying_ring.svg.png" decoding="async" width="235" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/54/Current_carrying_ring.svg/353px-Current_carrying_ring.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/54/Current_carrying_ring.svg/470px-Current_carrying_ring.svg.png 2x" data-file-width="567" data-file-height="567" /></a> </td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} ={\mu _{0}IR^{2} \over 2(x^{2}+R^{2})^{3/2}}{\hat {\mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>I</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} ={\mu _{0}IR^{2} \over 2(x^{2}+R^{2})^{3/2}}{\hat {\mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a7c2995055bcaeb2d3f8c115c97810d80ff1c02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:20.959ex; height:6.676ex;" alt="{\displaystyle \mathbf {B} ={\mu _{0}IR^{2} \over 2(x^{2}+R^{2})^{3/2}}{\hat {\mathbf {x} }}}"> along the axis of the loop, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> is the uniform current flowing through the loop. </td></tr> <tr> <td>Solenoid </td> <td><a href="/wiki/File:Solenoid_segment.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Solenoid_segment.svg/235px-Solenoid_segment.svg.png" decoding="async" width="235" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Solenoid_segment.svg/353px-Solenoid_segment.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Solenoid_segment.svg/470px-Solenoid_segment.svg.png 2x" data-file-width="567" data-file-height="567" /></a> </td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} ={\mu _{0}nI \over 2}(\cos \theta _{1}+\cos \theta _{2}){\hat {\mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>n</mi> <mi>I</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} ={\mu _{0}nI \over 2}(\cos \theta _{1}+\cos \theta _{2}){\hat {\mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed55e8fa4f3783ec209de373b14474710f2109ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.204ex; height:5.343ex;" alt="{\displaystyle \mathbf {B} ={\mu _{0}nI \over 2}(\cos \theta _{1}+\cos \theta _{2}){\hat {\mathbf {x} }}}"> along the axis of the solenoid carrying current <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, uniform number of loops of currents per length of solenoid; and the direction of magnetic field as shown. </td></tr> <tr> <td>Infinite solenoid </td> <td><a href="/wiki/File:Infinite_solenoid.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Infinite_solenoid.svg/235px-Infinite_solenoid.svg.png" decoding="async" width="235" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Infinite_solenoid.svg/353px-Infinite_solenoid.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Infinite_solenoid.svg/470px-Infinite_solenoid.svg.png 2x" data-file-width="567" data-file-height="567" /></a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/312c45fb45e8efef089963119c8e68eab464cf95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.162ex; height:2.176ex;" alt="{\displaystyle \mathbf {B} =0}"> outside the solenoid carrying current <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, uniform number of loops of currents per length of solenoid. </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} =\mu _{0}nI{\hat {\mathbf {x} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>n</mi> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} =\mu _{0}nI{\hat {\mathbf {x} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/841fa2a14b1c14b178815cca739d9f0eb95b0ef1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.433ex; height:2.843ex;" alt="{\displaystyle \mathbf {B} =\mu _{0}nI{\hat {\mathbf {x} }}}"> inside the solenoid carrying current <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, uniform number of loops of currents per length of solenoid, with the direction of magnetic field as shown. </td></tr> <tr> <td>Circular Toroid </td> <td><a href="/wiki/File:Circular_toroidal_inductor.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Circular_toroidal_inductor.svg/235px-Circular_toroidal_inductor.svg.png" decoding="async" width="235" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Circular_toroidal_inductor.svg/353px-Circular_toroidal_inductor.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Circular_toroidal_inductor.svg/470px-Circular_toroidal_inductor.svg.png 2x" data-file-width="567" data-file-height="567" /></a> </td> <td colspan="2"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B={\frac {\mu _{0}NI}{2\pi R}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>N</mi> <mi>I</mi> </mrow> <mrow> <mn>2</mn> <mi>π</mi> <mi>R</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B={\frac {\mu _{0}NI}{2\pi R}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6f43b1b3ea41fef9f4162683244fded0c2688e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.39ex; height:5.509ex;" alt="{\displaystyle B={\frac {\mu _{0}NI}{2\pi R}}}"> along the bulk of the circular toroid carrying uniform current <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> through <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"> number of uniformly distributed poloidal loops, with the direction of magnetic field as indicated. </td></tr> <tr> <td>Magnetic Dipole </td> <td><a href="/wiki/File:Magnetic_dipole.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Magnetic_dipole.svg/235px-Magnetic_dipole.svg.png" decoding="async" width="235" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Magnetic_dipole.svg/353px-Magnetic_dipole.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Magnetic_dipole.svg/470px-Magnetic_dipole.svg.png 2x" data-file-width="567" data-file-height="567" /></a> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} =-{\frac {\mu _{0}\mathbf {m} }{4\pi r^{3}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">m</mi> </mrow> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} =-{\frac {\mu _{0}\mathbf {m} }{4\pi r^{3}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f918577070fae238f6e8e6c692c57c66d3ec70e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:12.973ex; height:5.176ex;" alt="{\displaystyle \mathbf {B} =-{\frac {\mu _{0}\mathbf {m} }{4\pi r^{3}}},}"> on the equatorial plane, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {m} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">m</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {m} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58d48fbdcdf404353d3d53cdc2b5968c35fa5125" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.226ex; height:1.676ex;" alt="{\displaystyle \mathbf {m} }"> is the <a href="/wiki/Magnetic_dipole_moment" class="mw-redirect" title="Magnetic dipole moment">magnetic dipole moment</a>. </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {B} ={\frac {\mu _{0}\mathbf {m} }{2\pi x^{3}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">m</mi> </mrow> </mrow> <mrow> <mn>2</mn> <mi>π</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {B} ={\frac {\mu _{0}\mathbf {m} }{2\pi x^{3}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/251864748ab8f09eda3a8c7f7db5c3c0405b8749" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:11.361ex; height:5.176ex;" alt="{\displaystyle \mathbf {B} ={\frac {\mu _{0}\mathbf {m} }{2\pi x^{3}}},}"> on the axial plane (given that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\gg R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>≫</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\gg R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebbbe773c88f3e02fbce397ab44bedb39e0518e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.708ex; height:2.176ex;" alt="{\displaystyle x\gg R}">),where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> can also be negative to indicate position at the opposite direction on the axis, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {m} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">m</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {m} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58d48fbdcdf404353d3d53cdc2b5968c35fa5125" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.226ex; height:1.676ex;" alt="{\displaystyle \mathbf {m} }"> is the <a href="/wiki/Magnetic_dipole_moment" class="mw-redirect" title="Magnetic dipole moment">magnetic dipole moment</a>. </td></tr></tbody></table> Additional magnetic field values can be found through the magnetic field of a finite beam, for example, that the magnetic field of an arc of angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }"> and radius <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> at the center is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B={\mu _{0}\theta I \over 4\pi R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>θ</mi> <mi>I</mi> </mrow> <mrow> <mn>4</mn> <mi>π</mi> <mi>R</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B={\mu _{0}\theta I \over 4\pi R}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16dc7337cc2bc7ec3abb32ab8608f4cf390a1699" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.417ex; height:5.676ex;" alt="{\displaystyle B={\mu _{0}\theta I \over 4\pi R}}">, or that the magnetic field at the center of a N-sided regular polygon of side <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B={\mu _{0}NI \over \pi a}\sin {\pi \over N}\tan {\pi \over N}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>N</mi> <mi>I</mi> </mrow> <mrow> <mi>π</mi> <mi>a</mi> </mrow> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mi>N</mi> </mfrac> </mrow> <mi>tan</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mi>N</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B={\mu _{0}NI \over \pi a}\sin {\pi \over N}\tan {\pi \over N}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d45c0c953da4b30f5f276340af280147fc9b8eb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.953ex; height:5.343ex;" alt="{\displaystyle B={\mu _{0}NI \over \pi a}\sin {\pi \over N}\tan {\pi \over N}}">, both outside of the plane with proper directions as inferred by right hand thumb rule. <div class="mw-heading mw-heading2"><h2 id="History">History</h2></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/History_of_electromagnetic_theory" title="History of electromagnetic theory">History of electromagnetic theory</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Timeline_of_electromagnetism_and_classical_optics" title="Timeline of electromagnetism and classical optics">Timeline of electromagnetism and classical optics</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Descartes_magnetic_field.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Descartes_magnetic_field.jpg/260px-Descartes_magnetic_field.jpg" decoding="async" width="260" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Descartes_magnetic_field.jpg/390px-Descartes_magnetic_field.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Descartes_magnetic_field.jpg/520px-Descartes_magnetic_field.jpg 2x" data-file-width="1372" data-file-height="1165" /></a><figcaption>One of the first drawings of a magnetic field, by <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a>, 1644, showing the Earth attracting <a href="/wiki/Lodestone" title="Lodestone">lodestones</a>. It illustrated his theory that magnetism was caused by the circulation of tiny helical particles, "threaded parts", through threaded pores in magnets.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Early_developments">Early developments</h3></div> While magnets and some properties of magnetism were known to ancient societies, the research of magnetic fields began in 1269 when French scholar <a href="/wiki/Petrus_Peregrinus_de_Maricourt" title="Petrus Peregrinus de Maricourt">Petrus Peregrinus de Maricourt</a> mapped out the magnetic field on the surface of a spherical magnet using iron needles. Noting the resulting field lines crossed at two points he named those points "poles" in analogy to Earth's poles. He also articulated the principle that magnets always have both a north and south pole, no matter how finely one slices them.<a href="#cite_note-67">[54]</a><a href="#cite_note-68">[note 14]</a> Almost three centuries later, <a href="/wiki/William_Gilbert_(astronomer)" class="mw-redirect" title="William Gilbert (astronomer)">William Gilbert</a> of <a href="/wiki/Colchester" title="Colchester">Colchester</a> replicated Petrus Peregrinus' work and was the first to state explicitly that Earth is a magnet.<a href="#cite_note-Whittaker1910-69">[55]</a>: 34  Published in 1600, Gilbert's work, <a href="/wiki/De_Magnete" title="De Magnete">De Magnete</a>, helped to establish magnetism as a science. <div class="mw-heading mw-heading3"><h3 id="Mathematical_development">Mathematical development</h3></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Hans_Christian_%C3%98rsted,_Der_Geist_in_der_Natur,_1854.tiff" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Hans_Christian_%C3%98rsted%2C_Der_Geist_in_der_Natur%2C_1854.tiff/lossy-page1-220px-Hans_Christian_%C3%98rsted%2C_Der_Geist_in_der_Natur%2C_1854.tiff.jpg" decoding="async" width="220" height="337" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Hans_Christian_%C3%98rsted%2C_Der_Geist_in_der_Natur%2C_1854.tiff/lossy-page1-330px-Hans_Christian_%C3%98rsted%2C_Der_Geist_in_der_Natur%2C_1854.tiff.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/Hans_Christian_%C3%98rsted%2C_Der_Geist_in_der_Natur%2C_1854.tiff/lossy-page1-440px-Hans_Christian_%C3%98rsted%2C_Der_Geist_in_der_Natur%2C_1854.tiff.jpg 2x" data-file-width="1936" data-file-height="2968" /></a><figcaption><a href="/wiki/Hans_Christian_%C3%98rsted" title="Hans Christian Ørsted">Hans Christian Ørsted</a>, Der Geist in der Natur, 1854</figcaption></figure> In 1750, <a href="/wiki/John_Michell" title="John Michell">John Michell</a> stated that magnetic poles attract and repel in accordance with an <a href="/wiki/Inverse_square_law" class="mw-redirect" title="Inverse square law">inverse square law</a><a href="#cite_note-Whittaker1910-69">[55]</a>: 56  <a href="/wiki/Charles-Augustin_de_Coulomb" title="Charles-Augustin de Coulomb">Charles-Augustin de Coulomb</a> experimentally verified this in 1785 and stated explicitly that north and south poles cannot be separated.<a href="#cite_note-Whittaker1910-69">[55]</a>: 59  Building on this force between poles, <a href="/wiki/Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Siméon Denis Poisson</a> (1781–1840) created the first successful model of the magnetic field, which he presented in 1824.<a href="#cite_note-Whittaker1910-69">[55]</a>: 64  In this model, a magnetic H-field is produced by magnetic poles and magnetism is due to small pairs of north–south magnetic poles. Three discoveries in 1820 challenged this foundation of magnetism. <a href="/wiki/Hans_Christian_%C3%98rsted" title="Hans Christian Ørsted">Hans Christian Ørsted</a> demonstrated that a current-carrying wire is surrounded by a circular magnetic field.<a href="#cite_note-70">[note 15]</a><a href="#cite_note-71">[56]</a> Then <a href="/wiki/Andr%C3%A9-Marie_Amp%C3%A8re" title="André-Marie Ampère">André-Marie Ampère</a> showed that parallel wires with currents attract one another if the currents are in the same direction and repel if they are in opposite directions.<a href="#cite_note-Whittaker1910-69">[55]</a>: 87 <a href="#cite_note-72">[57]</a> Finally, <a href="/wiki/Jean-Baptiste_Biot" title="Jean-Baptiste Biot">Jean-Baptiste Biot</a> and <a href="/wiki/F%C3%A9lix_Savart" title="Félix Savart">Félix Savart</a> announced empirical results about the forces that a current-carrying long, straight wire exerted on a small magnet, determining the forces were inversely proportional to the perpendicular distance from the wire to the magnet.<a href="#cite_note-Tricker23-73">[58]</a><a href="#cite_note-Whittaker1910-69">[55]</a>: 86  <a href="/wiki/Laplace" class="mw-redirect" title="Laplace">Laplace</a> later deduced a law of force based on the differential action of a differential section of the wire,<a href="#cite_note-Tricker23-73">[58]</a><a href="#cite_note-74">[59]</a> which became known as the <a href="/wiki/Biot%E2%80%93Savart_law" title="Biot–Savart law">Biot–Savart law</a>, as Laplace did not publish his findings.<a href="#cite_note-75">[60]</a> Extending these experiments, Ampère published his own successful model of magnetism in 1825. In it, he showed the equivalence of electrical currents to magnets<a href="#cite_note-Whittaker1910-69">[55]</a>: 88  and proposed that magnetism is due to perpetually flowing loops of current instead of the dipoles of magnetic charge in Poisson's model.<a href="#cite_note-76">[note 16]</a> Further, Ampère derived both <a href="/wiki/Amp%C3%A8re%27s_force_law" title="Ampère's force law">Ampère's force law</a> describing the force between two currents and <a href="/wiki/Amp%C3%A8re%27s_law" class="mw-redirect" title="Ampère's law">Ampère's law</a>, which, like the Biot–Savart law, correctly described the magnetic field generated by a steady current. Also in this work, Ampère introduced the term <a href="/wiki/Electrodynamics" class="mw-redirect" title="Electrodynamics">electrodynamics</a> to describe the relationship between electricity and magnetism.<a href="#cite_note-Whittaker1910-69">[55]</a>: 88–92  In 1831, <a href="/wiki/Michael_Faraday" title="Michael Faraday">Michael Faraday</a> discovered <a href="/wiki/Electromagnetic_induction" title="Electromagnetic induction">electromagnetic induction</a> when he found that a changing magnetic field generates an encircling electric field, formulating what is now known as <a href="/wiki/Faraday%27s_law_of_induction" title="Faraday's law of induction">Faraday's law of induction</a>.<a href="#cite_note-Whittaker1910-69">[55]</a>: 189–192  Later, <a href="/wiki/Franz_Ernst_Neumann" title="Franz Ernst Neumann">Franz Ernst Neumann</a> proved that, for a moving conductor in a magnetic field, induction is a consequence of Ampère's force law.<a href="#cite_note-Whittaker1910-69">[55]</a>: 222  In the process, he introduced the magnetic vector potential, which was later shown to be equivalent to the underlying mechanism proposed by Faraday.<a href="#cite_note-Whittaker1910-69">[55]</a>: 225  In 1850, <a href="/wiki/Lord_Kelvin" title="Lord Kelvin">Lord Kelvin</a>, then known as William Thomson, distinguished between two magnetic fields now denoted H and B. The former applied to Poisson's model and the latter to Ampère's model and induction.<a href="#cite_note-Whittaker1910-69">[55]</a>: 224  Further, he derived how H and B relate to each other and coined the term permeability.<a href="#cite_note-Whittaker1910-69">[55]</a>: 245 <a href="#cite_note-77">[61]</a> Between 1861 and 1865, <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a> developed and published <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a>, which explained and united all of <a href="/wiki/Classical_theory" class="mw-redirect" title="Classical theory">classical</a> electricity and magnetism. The first set of these equations was published in a paper entitled <a href="/w/index.php?title=CCommons:File:On_Physical_Lines_of_Force.pdf&action=edit&redlink=1" class="new" title="CCommons:File:On Physical Lines of Force.pdf (page does not exist)">On Physical Lines of Force</a> in 1861. These equations were valid but incomplete. Maxwell completed his set of equations in his later 1865 paper <a href="/wiki/A_Dynamical_Theory_of_the_Electromagnetic_Field" title="A Dynamical Theory of the Electromagnetic Field">A Dynamical Theory of the Electromagnetic Field</a> and demonstrated the fact that light is an <a href="/wiki/Electromagnetic_wave" class="mw-redirect" title="Electromagnetic wave">electromagnetic wave</a>. <a href="/wiki/Heinrich_Hertz" title="Heinrich Hertz">Heinrich Hertz</a> published papers in 1887 and 1888 experimentally confirming this fact.<a href="#cite_note-h202-78">[62]</a><a href="#cite_note-79">[63]</a> <div class="mw-heading mw-heading3"><h3 id="Modern_developments">Modern developments</h3></div> In 1887, Tesla developed an <a href="/wiki/Induction_motor" title="Induction motor">induction motor</a> that ran on <a href="/wiki/Alternating_current" title="Alternating current">alternating current</a>. The motor used <a href="/wiki/Polyphase_system" title="Polyphase system">polyphase</a> current, which generated a <a href="/wiki/Rotating_magnetic_field" title="Rotating magnetic field">rotating magnetic field</a> to turn the motor (a principle that Tesla claimed to have conceived in 1882).<a href="#cite_note-80">[64]</a><a href="#cite_note-81">[65]</a><a href="#cite_note-82">[66]</a> Tesla received a patent for his electric motor in May 1888.<a href="#cite_note-83">[67]</a><a href="#cite_note-84">[68]</a> In 1885, <a href="/wiki/Galileo_Ferraris" title="Galileo Ferraris">Galileo Ferraris</a> independently researched rotating magnetic fields and subsequently published his research in a paper to the Royal Academy of Sciences in <a href="/wiki/Turin" title="Turin">Turin</a>, just two months before Tesla was awarded his patent, in March 1888.<a href="#cite_note-85">[69]</a> The twentieth century showed that classical electrodynamics is already consistent with special relativity, and extended classical electrodynamics to work with quantum mechanics. <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a>, in his paper of 1905 that established relativity, showed that both the electric and magnetic fields are part of the same phenomena viewed from different reference frames. Finally, the emergent field of <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> was merged with electrodynamics to form <a href="/wiki/Quantum_electrodynamics" title="Quantum electrodynamics">quantum electrodynamics</a>, which first formalized the notion that electromagnetic field energy is quantized in the form of photons. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div> <div class="mw-heading mw-heading3"><h3 id="General">General</h3></div> <ul><li><a href="/wiki/Magnetohydrodynamics" title="Magnetohydrodynamics">Magnetohydrodynamics</a> – the study of the dynamics of electrically conducting fluids</li> <li><a href="/wiki/Magnetic_hysteresis" title="Magnetic hysteresis">Magnetic hysteresis</a> – application to <a href="/wiki/Ferromagnetism" title="Ferromagnetism">ferromagnetism</a></li> <li><a href="/wiki/Magnetic_nanoparticles" title="Magnetic nanoparticles">Magnetic nanoparticles</a> – extremely small magnetic particles that are tens of atoms wide</li> <li><a href="/wiki/Magnetic_reconnection" title="Magnetic reconnection">Magnetic reconnection</a> – an effect that causes <a href="/wiki/Solar_flare" title="Solar flare">solar flares</a> and auroras</li> <li><a href="/wiki/Magnetic_scalar_potential" title="Magnetic scalar potential">Magnetic scalar potential</a></li> <li><a href="/wiki/SI_electromagnetism_units" class="mw-redirect" title="SI electromagnetism units">SI electromagnetism units</a> – common units used in electromagnetism</li> <li><a href="/wiki/Orders_of_magnitude_(magnetic_field)" title="Orders of magnitude (magnetic field)">Orders of magnitude (magnetic field)</a> – list of magnetic field sources and measurement devices from smallest magnetic fields to largest detected</li> <li><a href="/wiki/Upward_continuation" title="Upward continuation">Upward continuation</a></li> <li><a href="/wiki/Moses_Effect" class="mw-redirect" title="Moses Effect">Moses Effect</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Mathematics">Mathematics</h3></div> <ul><li><a href="/wiki/Magnetic_helicity" title="Magnetic helicity">Magnetic helicity</a> – extent to which a magnetic field wraps around itself</li></ul> <div class="mw-heading mw-heading3"><h3 id="Applications">Applications</h3></div> <ul><li><a href="/wiki/Dynamo_theory" title="Dynamo theory">Dynamo theory</a> – a proposed mechanism for the creation of the Earth's magnetic field</li> <li><a href="/wiki/Helmholtz_coil" title="Helmholtz coil">Helmholtz coil</a> – a device for producing a region of nearly uniform magnetic field</li> <li><a href="/wiki/Magnetic_field_viewing_film" title="Magnetic field viewing film">Magnetic field viewing film</a> – Film used to view the magnetic field of an area</li> <li><a href="/wiki/Magnetic_pistol" title="Magnetic pistol">Magnetic pistol</a> – a device on torpedoes or naval mines that detect the magnetic field of their target</li> <li><a href="/wiki/Maxwell_coil" title="Maxwell coil">Maxwell coil</a> – a device for producing a large volume of an almost constant magnetic field</li> <li><a href="/wiki/Stellar_magnetic_field" title="Stellar magnetic field">Stellar magnetic field</a> – a discussion of the magnetic field of stars</li> <li><a href="/wiki/Teltron_tube" title="Teltron tube">Teltron tube</a> – device used to display an electron beam and demonstrates effect of electric and magnetic fields on moving charges</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-7"><a href="#cite_ref-7">^</a> The letters B and H were originally chosen by Maxwell in his <a href="/wiki/Treatise_on_Electricity_and_Magnetism" class="mw-redirect" title="Treatise on Electricity and Magnetism">Treatise on Electricity and Magnetism</a> (Vol. II, pp. 236–237). For many quantities, he simply started choosing letters from the beginning of the alphabet. See <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFRalph_Baierlein2000" class="citation journal cs1">Ralph Baierlein (2000). "Answer to Question #73. S is for entropy, Q is for charge". American Journal of Physics. 68 (8): 691. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2000AmJPh..68..691B">2000AmJPh..68..691B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.19524">10.1119/1.19524</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Physics&rft.atitle=Answer+to+Question+%2373.+S+is+for+entropy%2C+Q+is+for+charge&rft.volume=68&rft.issue=8&rft.pages=691&rft.date=2000&rft_id=info%3Adoi%2F10.1119%2F1.19524&rft_id=info%3Abibcode%2F2000AmJPh..68..691B&rft.au=Ralph+Baierlein&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-ex03-9"><a href="#cite_ref-ex03_9-0">^</a> <a href="/wiki/Edward_Mills_Purcell" title="Edward Mills Purcell">Edward Purcell</a>, in Electricity and Magnetism, McGraw-Hill, 1963, writes, Even some modern writers who treat B as the primary field feel obliged to call it the magnetic induction because the name magnetic field was historically preempted by H. This seems clumsy and pedantic. If you go into the laboratory and ask a physicist what causes the pion trajectories in his bubble chamber to curve, he'll probably answer "magnetic field", not "magnetic induction." You will seldom hear a geophysicist refer to the Earth's magnetic induction, or an astrophysicist talk about the magnetic induction of the galaxy. We propose to keep on calling B the magnetic field. As for H, although other names have been invented for it, we shall call it "the field H" or even "the magnetic field H." In a similar vein, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFM_Gerloch1983" class="citation book cs1">M Gerloch (1983). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Ovo8AAAAIAAJ&pg=PA110">Magnetism and Ligand-field Analysis</a>. Cambridge University Press. p. 110. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-24939-3" title="Special:BookSources/978-0-521-24939-3"><bdi>978-0-521-24939-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Magnetism+and+Ligand-field+Analysis&rft.pages=110&rft.pub=Cambridge+University+Press&rft.date=1983&rft.isbn=978-0-521-24939-3&rft.au=M+Gerloch&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DOvo8AAAAIAAJ%26pg%3DPA110&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> says: "So we may think of both B and H as magnetic fields, but drop the word 'magnetic' from H so as to maintain the distinction ... As Purcell points out, 'it is only the names that give trouble, not the symbols'." </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> An alternative mnemonic to the right hand rule is <a href="/wiki/Fleming%27s_left-hand_rule_for_motors" title="Fleming's left-hand rule for motors">Fleming's left-hand rule</a>. </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> The SI unit of ΦB (<a href="/wiki/Magnetic_flux" title="Magnetic flux">magnetic flux</a>) is the <a href="/wiki/Weber_(unit)" title="Weber (unit)">weber</a> (symbol: Wb), related to the <a href="/wiki/Tesla_(unit)" title="Tesla (unit)">tesla</a> by 1 Wb/m2 = 1 T. The SI unit tesla is equal to (<a href="/wiki/Newton_(unit)" title="Newton (unit)">newton</a>·<a href="/wiki/Second" title="Second">second</a>)/(<a href="/wiki/Coulomb_(unit)" class="mw-redirect" title="Coulomb (unit)">coulomb</a>·<a href="/wiki/Metre" title="Metre">metre</a>). This can be seen from the magnetic part of the Lorentz force law. </li> <li id="cite_note-ex07-21"><a href="#cite_ref-ex07_21-0">^</a> The use of iron filings to display a field presents something of an exception to this picture; the filings alter the magnetic field so that it is much larger along the "lines" of iron, because of the large <a href="/wiki/Magnetic_permeability" class="mw-redirect" title="Magnetic permeability">permeability</a> of iron relative to air. </li> <li id="cite_note-ex05-22"><a href="#cite_ref-ex05_22-0">^</a> Here, "small" means that the observer is sufficiently far away from the magnet, so that the magnet can be considered as infinitesimally small. "Larger" magnets need to include more complicated terms in the mathematical expression of the magnetic field and depend on the entire geometry of the magnet not just m. </li> <li id="cite_note-ex04-25"><a href="#cite_ref-ex04_25-0">^</a> Either B or H may be used for the magnetic field outside the magnet. </li> <li id="cite_note-ex12-29"><a href="#cite_ref-ex12_29-0">^</a> In practice, the Biot–Savart law and other laws of magnetostatics are often used even when a current change in time, as long as it does not change too quickly. It is often used, for instance, for standard household currents, which oscillate sixty times per second.<a href="#cite_note-Griffiths4ed-28">[21]</a>: 223  </li> <li id="cite_note-ex13-31"><a href="#cite_ref-ex13_31-0">^</a> The Biot–Savart law contains the additional restriction (boundary condition) that the B-field must go to zero fast enough at infinity. It also depends on the divergence of B being zero, which is always valid. (There are no magnetic charges.) </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> A third term is needed for changing electric fields and polarization currents; this displacement current term is covered in Maxwell's equations below. </li> <li id="cite_note-ex08-45"><a href="#cite_ref-ex08_45-0">^</a> To see that this must be true imagine placing a compass inside a magnet. There, the north pole of the compass points toward the north pole of the magnet since magnets stacked on each other point in the same direction. </li> <li id="cite_note-ex09-46"><a href="#cite_ref-ex09_46-0">^</a> As discussed above, magnetic field lines are primarily a conceptual tool used to represent the mathematics behind magnetic fields. The total "number" of field lines is dependent on how the field lines are drawn. In practice, integral equations such as the one that follows in the main text are used instead. </li> <li id="cite_note-ex14-48"><a href="#cite_ref-ex14_48-0">^</a> A complete expression for Faraday's law of induction in terms of the electric E and magnetic fields can be written as: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}=-{\frac {d\Phi }{dt}}=\oint _{\partial \Sigma (t)}\left(\mathbf {E} (\mathbf {r} ,\ t)+\mathbf {v} \times \mathbf {B} (\mathbf {r} ,\ t)\right)\cdot d{\boldsymbol {\ell }}\ =-{\frac {d}{dt}}\iint _{\Sigma (t)}d{\boldsymbol {A}}\cdot \mathbf {B} (\mathbf {r} ,\ t)}"> <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> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <msub> <mo>∮</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mtext> </mtext> <mi>t</mi> <mo stretchy="false">)</mo> <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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mtext> </mtext> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ℓ</mi> </mrow> <mtext> </mtext> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Σ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">A</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">B</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo>,</mo> <mtext> </mtext> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {E}}=-{\frac {d\Phi }{dt}}=\oint _{\partial \Sigma (t)}\left(\mathbf {E} (\mathbf {r} ,\ t)+\mathbf {v} \times \mathbf {B} (\mathbf {r} ,\ t)\right)\cdot d{\boldsymbol {\ell }}\ =-{\frac {d}{dt}}\iint _{\Sigma (t)}d{\boldsymbol {A}}\cdot \mathbf {B} (\mathbf {r} ,\ t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e8abb40ebcc28ab6cc1fe51d9138984481e8827" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:73.382ex; height:6.176ex;" alt="{\displaystyle {\mathcal {E}}=-{\frac {d\Phi }{dt}}=\oint _{\partial \Sigma (t)}\left(\mathbf {E} (\mathbf {r} ,\ t)+\mathbf {v} \times \mathbf {B} (\mathbf {r} ,\ t)\right)\cdot d{\boldsymbol {\ell }}\ =-{\frac {d}{dt}}\iint _{\Sigma (t)}d{\boldsymbol {A}}\cdot \mathbf {B} (\mathbf {r} ,\ t)}"> where ∂Σ(t) is the moving closed path bounding the moving surface Σ(t), and dA is an element of surface area of Σ(t). The first integral calculates the work done moving a charge a distance dℓ based upon the Lorentz force law. In the case where the bounding surface is stationary, the <a href="/wiki/Kelvin%E2%80%93Stokes_theorem" class="mw-redirect" title="Kelvin–Stokes theorem">Kelvin–Stokes theorem</a> can be used to show this equation is equivalent to the Maxwell–Faraday equation. </li> <li id="cite_note-68"><a href="#cite_ref-68">^</a> His Epistola Petri Peregrini de Maricourt ad Sygerum de Foucaucourt Militem de Magnete, which is often shortened to Epistola de magnete, is dated 1269 C.E. </li> <li id="cite_note-70"><a href="#cite_ref-70">^</a> During a lecture demonstration on the effects of a current on a campus needle, Ørsted showed that when a current-carrying wire is placed at a right angle with the compass, nothing happens. When he tried to orient the wire parallel to the compass needle, however, it produced a pronounced deflection of the compass needle. By placing the compass on different sides of the wire, he was able to determine the field forms perfect circles around the wire.<a href="#cite_note-Whittaker1910-69">[55]</a>: 85  </li> <li id="cite_note-76"><a href="#cite_ref-76">^</a> From the outside, the field of a dipole of magnetic charge has exactly the same form as a current loop when both are sufficiently small. Therefore, the two models differ only for magnetism inside magnetic material. </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNave" class="citation web cs1">Nave, Rod. <a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magfie.html">"Magnetic Field"</a>. HyperPhysics. Retrieved 20 May 2024.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=HyperPhysics&rft.atitle=Magnetic+Field&rft.aulast=Nave&rft.aufirst=Rod&rft_id=http%3A%2F%2Fhyperphysics.phy-astr.gsu.edu%2Fhbase%2Fmagnetic%2Fmagfie.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-feynman-2">^ <a href="#cite_ref-feynman_2-0">a</a> <a href="#cite_ref-feynman_2-1">b</a> <a href="#cite_ref-feynman_2-2">c</a> <a href="#cite_ref-feynman_2-3">d</a> <a href="#cite_ref-feynman_2-4">e</a> <a href="#cite_ref-feynman_2-5">f</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFeynmanLeightonSands1963" class="citation book cs1">Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (1963). <a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu">The Feynman Lectures on Physics</a>. Vol. 2. California Institute of Technology. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780465040858" title="Special:BookSources/9780465040858"><bdi>9780465040858</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Feynman+Lectures+on+Physics&rft.pub=California+Institute+of+Technology&rft.date=1963&rft.isbn=9780465040858&rft.aulast=Feynman&rft.aufirst=Richard+P.&rft.au=Leighton%2C+Robert+B.&rft.au=Sands%2C+Matthew&rft_id=https%3A%2F%2Ffeynmanlectures.caltech.edu&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYoungFreedmanFord2008" class="citation book cs1">Young, Hugh D.; Freedman, Roger A.; Ford, A. Lewis (2008). Sears and Zemansky's university physics : with modern physics. Vol. 2. Pearson Addison-Wesley. pp. 918–919. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780321501219" title="Special:BookSources/9780321501219"><bdi>9780321501219</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Sears+and+Zemansky%27s+university+physics+%3A+with+modern+physics&rft.pages=918-919&rft.pub=Pearson+Addison-Wesley&rft.date=2008&rft.isbn=9780321501219&rft.aulast=Young&rft.aufirst=Hugh+D.&rft.au=Freedman%2C+Roger+A.&rft.au=Ford%2C+A.+Lewis&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-Purcell3ed-4"><a href="#cite_ref-Purcell3ed_4-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPurcellMorin2013" class="citation book cs1"><a href="/wiki/Edward_Mills_Purcell" title="Edward Mills Purcell">Purcell, Edward M.</a>; Morin, David J. (2013). <a href="/wiki/Electricity_and_Magnetism_(book)" title="Electricity and Magnetism (book)">Electricity and Magnetism</a> (3rd ed.). Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781107014022" title="Special:BookSources/9781107014022"><bdi>9781107014022</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electricity+and+Magnetism&rft.edition=3rd&rft.pub=Cambridge+University+Press&rft.date=2013&rft.isbn=9781107014022&rft.aulast=Purcell&rft.aufirst=Edward+M.&rft.au=Morin%2C+David+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-:1-5">^ <a href="#cite_ref-:1_5-0">a</a> <a href="#cite_ref-:1_5-1">b</a> <a href="#cite_ref-:1_5-2">c</a> <a href="#cite_ref-:1_5-3">d</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.bipm.org/documents/20126/41483022/SI-Brochure-9-EN.pdf">The International System of Units</a> (PDF) (9th ed.), International Bureau of Weights and Measures, December 2022, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-92-822-2272-0" title="Special:BookSources/978-92-822-2272-0"><bdi>978-92-822-2272-0</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+International+System+of+Units&rft.edition=9th&rft.pub=International+Bureau+of+Weights+and+Measures&rft.date=2022-12&rft.isbn=978-92-822-2272-0&rft_id=https%3A%2F%2Fwww.bipm.org%2Fdocuments%2F20126%2F41483022%2FSI-Brochure-9-EN.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-Jiles-6"><a href="#cite_ref-Jiles_6-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJiles1998" class="citation book cs1">Jiles, David C. (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=axyWXjsdorMC&pg=PA3">Introduction to Magnetism and Magnetic Materials</a> (2 ed.). CRC. p. 3. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0412798603" title="Special:BookSources/978-0412798603"><bdi>978-0412798603</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Magnetism+and+Magnetic+Materials&rft.pages=3&rft.edition=2&rft.pub=CRC&rft.date=1998&rft.isbn=978-0412798603&rft.aulast=Jiles&rft.aufirst=David+C.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DaxyWXjsdorMC%26pg%3DPA3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_J._Roche2000" class="citation journal cs1">John J. Roche (2000). "B and H, the intensity vectors of magnetism: A new approach to resolving a century-old controversy". American Journal of Physics. 68 (5): 438. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2000AmJPh..68..438R">2000AmJPh..68..438R</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.19459">10.1119/1.19459</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Physics&rft.atitle=B+and+H%2C+the+intensity+vectors+of+magnetism%3A+A+new+approach+to+resolving+a+century-old+controversy&rft.volume=68&rft.issue=5&rft.pages=438&rft.date=2000&rft_id=info%3Adoi%2F10.1119%2F1.19459&rft_id=info%3Abibcode%2F2000AmJPh..68..438R&rft.au=John+J.+Roche&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-Electromagnetics-10">^ <a href="#cite_ref-Electromagnetics_10-0">a</a> <a href="#cite_ref-Electromagnetics_10-1">b</a> E. J. Rothwell and M. J. Cloud (2010) <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7AHLBQAAQBAJ&pg=PA23">Electromagnetics</a>. Taylor & Francis. p. 23. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1420058266" title="Special:BookSources/1420058266">1420058266</a>. </li> <li id="cite_note-Stratton-11">^ <a href="#cite_ref-Stratton_11-0">a</a> <a href="#cite_ref-Stratton_11-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStratton1941" class="citation book cs1">Stratton, Julius Adams (1941). Electromagnetic Theory (1st ed.). McGraw-Hill. p. 1. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0070621503" title="Special:BookSources/978-0070621503"><bdi>978-0070621503</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electromagnetic+Theory&rft.pages=1&rft.edition=1st&rft.pub=McGraw-Hill&rft.date=1941&rft.isbn=978-0070621503&rft.aulast=Stratton&rft.aufirst=Julius+Adams&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-purcell2ed-12">^ <a href="#cite_ref-purcell2ed_12-0">a</a> <a href="#cite_ref-purcell2ed_12-1">b</a> <a href="#cite_ref-purcell2ed_12-2">c</a> <a href="#cite_ref-purcell2ed_12-3">d</a> <a href="#cite_ref-purcell2ed_12-4">e</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPurcell2011" class="citation book cs1">Purcell, E. (2011). <a rel="nofollow" class="external text" href="https://archive.org/details/electricitymagne00purc_621">Electricity and Magnetism</a> (2nd ed.). Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1107013605" title="Special:BookSources/978-1107013605"><bdi>978-1107013605</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electricity+and+Magnetism&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=2011&rft.isbn=978-1107013605&rft.aulast=Purcell&rft.aufirst=E.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Felectricitymagne00purc_621&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-Griffiths3ed-13">^ <a href="#cite_ref-Griffiths3ed_13-0">a</a> <a href="#cite_ref-Griffiths3ed_13-1">b</a> <a href="#cite_ref-Griffiths3ed_13-2">c</a> <a href="#cite_ref-Griffiths3ed_13-3">d</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGriffiths1999" class="citation book cs1"><a href="/wiki/David_J._Griffiths" title="David J. Griffiths">Griffiths, David J.</a> (1999). Introduction to Electrodynamics (3rd ed.). Pearson. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-805326-X" title="Special:BookSources/0-13-805326-X"><bdi>0-13-805326-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Electrodynamics&rft.edition=3rd&rft.pub=Pearson&rft.date=1999&rft.isbn=0-13-805326-X&rft.aulast=Griffiths&rft.aufirst=David+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-jackson3ed-15">^ <a href="#cite_ref-jackson3ed_15-0">a</a> <a href="#cite_ref-jackson3ed_15-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJackson1998" class="citation book cs1"><a href="/wiki/John_David_Jackson_(physicist)" title="John David Jackson (physicist)">Jackson, John David</a> (1998). Classical electrodynamics (3rd ed.). New York: Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-30932-X" title="Special:BookSources/0-471-30932-X"><bdi>0-471-30932-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+electrodynamics&rft.place=New+York&rft.edition=3rd&rft.pub=Wiley&rft.date=1998&rft.isbn=0-471-30932-X&rft.aulast=Jackson&rft.aufirst=John+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-BIPMTab9-17">^ <a href="#cite_ref-BIPMTab9_17-0">a</a> <a href="#cite_ref-BIPMTab9_17-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20190608123210/https://www.bipm.org/en/publications/si-brochure/table9.html">"Non-SI units accepted for use with the SI, and units based on fundamental constants (contd.)"</a>. SI Brochure: The International System of Units (SI) [8th edition, 2006; updated in 2014]. Bureau International des Poids et Mesures. Archived from <a rel="nofollow" class="external text" href="https://www.bipm.org/en/publications/si-brochure/table9.html">the original</a> on 8 June 2019. Retrieved 19 April 2018.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=SI+Brochure%3A+The+International+System+of+Units+%28SI%29+%5B8th+edition%2C+2006%3B+updated+in+2014%5D&rft.atitle=Non-SI+units+accepted+for+use+with+the+SI%2C+and+units+based+on+fundamental+constants+%28contd.%29&rft_id=https%3A%2F%2Fwww.bipm.org%2Fen%2Fpublications%2Fsi-brochure%2Ftable9.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-KLang-18">^ <a href="#cite_ref-KLang_18-0">a</a> <a href="#cite_ref-KLang_18-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLang2006" class="citation book cs1">Lang, Kenneth R. (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aUjkKuaVIloC&pg=PA176">A Companion to Astronomy and Astrophysics</a>. Springer. p. 176. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387333670" title="Special:BookSources/9780387333670"><bdi>9780387333670</bdi></a>. Retrieved 19 April 2018.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Companion+to+Astronomy+and+Astrophysics&rft.pages=176&rft.pub=Springer&rft.date=2006&rft.isbn=9780387333670&rft.aulast=Lang&rft.aufirst=Kenneth+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DaUjkKuaVIloC%26pg%3DPA176&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://physics.nist.gov/cuu/Units/units.html">"International system of units (SI)"</a>. NIST reference on constants, units, and uncertainty. National Institute of Standards and Technology. 12 April 2010. Retrieved 9 May 2012.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=NIST+reference+on+constants%2C+units%2C+and+uncertainty&rft.atitle=International+system+of+units+%28SI%29&rft.date=2010-04-12&rft_id=http%3A%2F%2Fphysics.nist.gov%2Fcuu%2FUnits%2Funits.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.nasa.gov/pdf/168808main_gp-b_pfar_cvr-pref-execsum.pdf">"Gravity Probe B Executive Summary"</a> (PDF). pp. 10, 21. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://www.nasa.gov/pdf/168808main_gp-b_pfar_cvr-pref-execsum.pdf">Archived</a> (PDF) from the original on 9 October 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Gravity+Probe+B+Executive+Summary&rft.pages=10%2C+21&rft_id=http%3A%2F%2Fwww.nasa.gov%2Fpdf%2F168808main_gp-b_pfar_cvr-pref-execsum.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrown1962" class="citation book cs1">Brown, William Fuller (1962). Magnetostatic Principles in Ferromagnetism. North Holland publishing company. p. 12. <a href="/wiki/ASIN_(identifier)" class="mw-redirect" title="ASIN (identifier)">ASIN</a> <a rel="nofollow" class="external text" href="https://www.amazon.com/dp/B0006AY7F8">B0006AY7F8</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Magnetostatic+Principles+in+Ferromagnetism&rft.pages=12&rft.pub=North+Holland+publishing+company&rft.date=1962&rft_id=https%3A%2F%2Fwww.amazon.com%2Fdp%2FB0006AY7F8%23id-name%3DASIN&rft.aulast=Brown&rft.aufirst=William+Fuller&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-Graham-24"><a href="#cite_ref-Graham_24-0">^</a> See <a href="/wiki/Magnetic_moment#Magnetic_dipoles" title="Magnetic moment">magnetic moment</a>[<a href="/wiki/MOS:BROKENSECTIONLINKS" class="mw-redirect" title="MOS:BROKENSECTIONLINKS">broken anchor</a>] and <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFB._D._CullityC._D._Graham2008" class="citation book cs1">B. D. Cullity; C. D. Graham (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ixAe4qIGEmwC&pg=PA103">Introduction to Magnetic Materials</a> (2 ed.). Wiley-IEEE. p. 103. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-47741-9" title="Special:BookSources/978-0-471-47741-9"><bdi>978-0-471-47741-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Magnetic+Materials&rft.pages=103&rft.edition=2&rft.pub=Wiley-IEEE&rft.date=2008&rft.isbn=978-0-471-47741-9&rft.au=B.+D.+Cullity&rft.au=C.+D.+Graham&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DixAe4qIGEmwC%26pg%3DPA103&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-Trigg-26"><a href="#cite_ref-Trigg_26-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFE._Richard_CohenDavid_R._LideGeorge_L._Trigg2003" class="citation book cs1">E. Richard Cohen; David R. Lide; George L. Trigg (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=JStYf6WlXpgC&pg=PA381">AIP physics desk reference</a> (3 ed.). Birkhäuser. p. 381. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-98973-0" title="Special:BookSources/978-0-387-98973-0"><bdi>978-0-387-98973-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=AIP+physics+desk+reference&rft.pages=381&rft.edition=3&rft.pub=Birkh%C3%A4user&rft.date=2003&rft.isbn=978-0-387-98973-0&rft.au=E.+Richard+Cohen&rft.au=David+R.+Lide&rft.au=George+L.+Trigg&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJStYf6WlXpgC%26pg%3DPA381&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> <a href="#CITEREFGriffiths1999">Griffiths 1999</a>, p. 438 </li> <li id="cite_note-Griffiths4ed-28">^ <a href="#cite_ref-Griffiths4ed_28-0">a</a> <a href="#cite_ref-Griffiths4ed_28-1">b</a> <a href="#cite_ref-Griffiths4ed_28-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGriffiths2017" class="citation book cs1"><a href="/wiki/David_J._Griffiths" title="David J. Griffiths">Griffiths, David J.</a> (2017). <a href="/wiki/Introduction_to_Electrodynamics" title="Introduction to Electrodynamics">Introduction to Electrodynamics</a> (4th ed.). Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781108357142" title="Special:BookSources/9781108357142"><bdi>9781108357142</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Electrodynamics&rft.edition=4th&rft.pub=Cambridge+University+Press&rft.date=2017&rft.isbn=9781108357142&rft.aulast=Griffiths&rft.aufirst=David+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> <a href="#CITEREFGriffiths1999">Griffiths 1999</a>, pp. 222–225 </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://puhep1.princeton.edu/~kirkmcd/examples/disk.pdf">"K. McDonald's Physics Examples - Disk"</a> (PDF). puhep1.princeton.edu. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/https://puhep1.princeton.edu/~kirkmcd/examples/disk.pdf">Archived</a> (PDF) from the original on 9 October 2022. Retrieved 13 February 2021.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=puhep1.princeton.edu&rft.atitle=K.+McDonald%27s+Physics+Examples+-+Disk&rft_id=https%3A%2F%2Fpuhep1.princeton.edu%2F~kirkmcd%2Fexamples%2Fdisk.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://physics.princeton.edu//~mcdonald/examples/railgun.pdf">"K. 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(2008). <a rel="nofollow" class="external text" href="http://academic.csuohio.edu/deissler/PhysRevE_77_036609.pdf">"Dipole in a magnetic field, work, and quantum spin"</a> (PDF). <a href="/wiki/Physical_Review_E" title="Physical Review E">Physical Review E</a>. 77 (3, pt 2): 036609. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008PhRvE..77c6609D">2008PhRvE..77c6609D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevE.77.036609">10.1103/PhysRevE.77.036609</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/18517545">18517545</a>. <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/http://academic.csuohio.edu/deissler/PhysRevE_77_036609.pdf">Archived</a> (PDF) from the original on 9 October 2022.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+E&rft.atitle=Dipole+in+a+magnetic+field%2C+work%2C+and+quantum+spin&rft.volume=77&rft.issue=3%2C+pt+2&rft.pages=036609&rft.date=2008&rft_id=info%3Apmid%2F18517545&rft_id=info%3Adoi%2F10.1103%2FPhysRevE.77.036609&rft_id=info%3Abibcode%2F2008PhRvE..77c6609D&rft.aulast=Deissler&rft.aufirst=R.J.&rft_id=http%3A%2F%2Facademic.csuohio.edu%2Fdeissler%2FPhysRevE_77_036609.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-35"><a href="#cite_ref-35">^</a> <a href="#CITEREFGriffiths1999">Griffiths 1999</a>, pp. 266–268 </li> <li id="cite_note-Slater-36"><a href="#cite_ref-Slater_36-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_Clarke_SlaterNathaniel_Herman_Frank1969" class="citation book cs1">John Clarke Slater; Nathaniel Herman Frank (1969). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GYsphnFwUuUC&pg=PA69">Electromagnetism</a> (first published in 1947 ed.). Courier Dover Publications. p. 69. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-62263-7" title="Special:BookSources/978-0-486-62263-7"><bdi>978-0-486-62263-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electromagnetism&rft.pages=69&rft.edition=first+published+in+1947&rft.pub=Courier+Dover+Publications&rft.date=1969&rft.isbn=978-0-486-62263-7&rft.au=John+Clarke+Slater&rft.au=Nathaniel+Herman+Frank&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DGYsphnFwUuUC%26pg%3DPA69&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> <a href="#CITEREFGriffiths1999">Griffiths 1999</a>, p. 332 </li> <li id="cite_note-Tilley-39">^ <a href="#cite_ref-Tilley_39-0">a</a> <a href="#cite_ref-Tilley_39-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRJD_Tilley2004" class="citation book cs1">RJD Tilley (2004). <a rel="nofollow" class="external text" href="https://archive.org/details/understandingsol0000till">Understanding Solids</a>. 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Cambridge University Press. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2Fcbo9781139005043">10.1017/cbo9781139005043</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-107-01360-5" title="Special:BookSources/978-1-107-01360-5"><bdi>978-1-107-01360-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electricity+and+Magnetism&rft.pub=Cambridge+University+Press&rft.date=2011-09-22&rft_id=info%3Adoi%2F10.1017%2Fcbo9781139005043&rft.isbn=978-1-107-01360-5&rft.aulast=Purcell&rft.aufirst=Edward&rft_id=http%3A%2F%2Fdx.doi.org%2F10.1017%2Fcbo9781139005043&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-51"><a href="#cite_ref-51">^</a> C. Doran and A. Lasenby (2003) Geometric Algebra for Physicists, Cambridge University Press, p. 233. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0521715954" title="Special:BookSources/0521715954">0521715954</a>. </li> <li id="cite_note-52"><a href="#cite_ref-52">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFE._J._Konopinski1978" class="citation journal cs1"><a href="/wiki/Emil_Konopinski" title="Emil Konopinski">E. J. Konopinski</a> (1978). "What the electromagnetic vector potential describes". Am. J. Phys. 46 (5): 499–502. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1978AmJPh..46..499K">1978AmJPh..46..499K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.11298">10.1119/1.11298</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Am.+J.+Phys.&rft.atitle=What+the+electromagnetic+vector+potential+describes&rft.volume=46&rft.issue=5&rft.pages=499-502&rft.date=1978&rft_id=info%3Adoi%2F10.1119%2F1.11298&rft_id=info%3Abibcode%2F1978AmJPh..46..499K&rft.au=E.+J.+Konopinski&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-53"><a href="#cite_ref-53">^</a> <a href="#CITEREFGriffiths1999">Griffiths 1999</a>, p. 422 </li> <li id="cite_note-54"><a href="#cite_ref-54">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNaber,_Gregory_L.2012" class="citation book cs1">Naber, Gregory L. (2012). <a rel="nofollow" class="external text" href="http://worldcat.org/oclc/804823303">The Geometry of Minkowski spacetime : an introduction to the mathematics of the special theory of relativity</a>. Springer. pp. 4–5. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4419-7837-0" title="Special:BookSources/978-1-4419-7837-0"><bdi>978-1-4419-7837-0</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/804823303">804823303</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Geometry+of+Minkowski+spacetime+%3A+an+introduction+to+the+mathematics+of+the+special+theory+of+relativity&rft.pages=4-5&rft.pub=Springer&rft.date=2012&rft_id=info%3Aoclcnum%2F804823303&rft.isbn=978-1-4419-7837-0&rft.au=Naber%2C+Gregory+L.&rft_id=http%3A%2F%2Fworldcat.org%2Foclc%2F804823303&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-55"><a href="#cite_ref-55">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRosser1968" class="citation book cs1">Rosser, W. G. V. (1968). <a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007/978-1-4899-6559-2">Classical Electromagnetism via Relativity</a>. Boston, MA: Springer. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4899-6559-2">10.1007/978-1-4899-6559-2</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4899-6258-4" title="Special:BookSources/978-1-4899-6258-4"><bdi>978-1-4899-6258-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Classical+Electromagnetism+via+Relativity&rft.place=Boston%2C+MA&rft.pub=Springer&rft.date=1968&rft_id=info%3Adoi%2F10.1007%2F978-1-4899-6559-2&rft.isbn=978-1-4899-6258-4&rft.aulast=Rosser&rft.aufirst=W.+G.+V.&rft_id=https%3A%2F%2Flink.springer.com%2Fbook%2F10.1007%2F978-1-4899-6559-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-56"><a href="#cite_ref-56">^</a> For a good qualitative introduction see: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRichard_Feynman2006" class="citation book cs1"><a href="/wiki/Richard_Feynman" title="Richard Feynman">Richard Feynman</a> (2006). <a href="/wiki/QED_(book)" class="mw-redirect" title="QED (book)">QED: the strange theory of light and matter</a>. <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-12575-6" title="Special:BookSources/978-0-691-12575-6"><bdi>978-0-691-12575-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=QED%3A+the+strange+theory+of+light+and+matter&rft.pub=Princeton+University+Press&rft.date=2006&rft.isbn=978-0-691-12575-6&rft.au=Richard+Feynman&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-Weiss-57"><a href="#cite_ref-Weiss_57-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeiss2002" class="citation journal cs1">Weiss, Nigel (2002). <a rel="nofollow" class="external text" href="https://doi.org/10.1046%2Fj.1468-4004.2002.43309.x">"Dynamos in planets, stars and galaxies"</a>. Astronomy and Geophysics. 43 (3): 3.09–3.15. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2002A&G....43c...9W">2002A&G....43c...9W</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1046%2Fj.1468-4004.2002.43309.x">10.1046/j.1468-4004.2002.43309.x</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Astronomy+and+Geophysics&rft.atitle=Dynamos+in+planets%2C+stars+and+galaxies&rft.volume=43&rft.issue=3&rft.pages=3.09-3.15&rft.date=2002&rft_id=info%3Adoi%2F10.1046%2Fj.1468-4004.2002.43309.x&rft_id=info%3Abibcode%2F2002A%26G....43c...9W&rft.aulast=Weiss&rft.aufirst=Nigel&rft_id=https%3A%2F%2Fdoi.org%2F10.1046%252Fj.1468-4004.2002.43309.x&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-58"><a href="#cite_ref-58">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.ngdc.noaa.gov/geomag/faqgeom.shtml#What_is_the_Earths_magnetic_field">"What is the Earth's magnetic field?"</a>. Geomagnetism Frequently Asked Questions. National Centers for Environmental Information, National Oceanic and Atmospheric Administration. Retrieved 19 April 2018.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Geomagnetism+Frequently+Asked+Questions&rft.atitle=What+is+the+Earth%27s+magnetic+field%3F&rft_id=https%3A%2F%2Fwww.ngdc.noaa.gov%2Fgeomag%2Ffaqgeom.shtml%23What_is_the_Earths_magnetic_field&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-59"><a href="#cite_ref-59">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRaymond_A._SerwayChris_VuilleJerry_S._Faughn2009" class="citation book cs1">Raymond A. Serway; Chris Vuille; Jerry S. Faughn (2009). <a rel="nofollow" class="external text" href="https://archive.org/details/collegephysics00serw_139">College physics</a> (8th ed.). Belmont, CA: Brooks/Cole, Cengage Learning. p. <a rel="nofollow" class="external text" href="https://archive.org/details/collegephysics00serw_139/page/n659">628</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-495-38693-3" title="Special:BookSources/978-0-495-38693-3"><bdi>978-0-495-38693-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=College+physics&rft.place=Belmont%2C+CA&rft.pages=628&rft.edition=8th&rft.pub=Brooks%2FCole%2C+Cengage+Learning&rft.date=2009&rft.isbn=978-0-495-38693-3&rft.au=Raymond+A.+Serway&rft.au=Chris+Vuille&rft.au=Jerry+S.+Faughn&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcollegephysics00serw_139&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-60"><a href="#cite_ref-60">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMerrillMcElhinnyMcFadden1996" class="citation book cs1">Merrill, Ronald T.; McElhinny, Michael W.; McFadden, Phillip L. (1996). "2. The present geomagnetic field: analysis and description from historical observations". The magnetic field of the earth: paleomagnetism, the core, and the deep mantle. <a href="/wiki/Academic_Press" title="Academic Press">Academic Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-491246-5" title="Special:BookSources/978-0-12-491246-5"><bdi>978-0-12-491246-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=2.+The+present+geomagnetic+field%3A+analysis+and+description+from+historical+observations&rft.btitle=The+magnetic+field+of+the+earth%3A+paleomagnetism%2C+the+core%2C+and+the+deep+mantle&rft.pub=Academic+Press&rft.date=1996&rft.isbn=978-0-12-491246-5&rft.aulast=Merrill&rft.aufirst=Ronald+T.&rft.au=McElhinny%2C+Michael+W.&rft.au=McFadden%2C+Phillip+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-61"><a href="#cite_ref-61">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPhillips2003" class="citation news cs1">Phillips, Tony (29 December 2003). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20221101165248/https://science.nasa.gov/science-news/science-at-nasa/2003/29dec_magneticfield/">"Earth's Inconstant Magnetic Field"</a>. Science@Nasa. Archived from <a rel="nofollow" class="external text" href="https://science.nasa.gov/science-news/science-at-nasa/2003/29dec_magneticfield/">the original</a> on 1 November 2022. Retrieved 27 December 2009.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science%40Nasa&rft.atitle=Earth%27s+Inconstant+Magnetic+Field&rft.date=2003-12-29&rft.aulast=Phillips&rft.aufirst=Tony&rft_id=https%3A%2F%2Fscience.nasa.gov%2Fscience-news%2Fscience-at-nasa%2F2003%2F29dec_magneticfield%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-62"><a href="#cite_ref-62">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoykoBykovDolotenkoKolokolchikov1999" class="citation book cs1">Boyko, B.A.; Bykov, A.I.; Dolotenko, M.I.; Kolokolchikov, N.P.; Markevtsev, I.M.; Tatsenko, O.M.; Shuvalov, K. (1999). "With record magnetic fields to the 21st Century". Digest of Technical Papers. 12th IEEE International Pulsed Power Conference. (Cat. No.99CH36358). Vol. 2. pp. 746–749. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FPPC.1999.823621">10.1109/PPC.1999.823621</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7803-5498-2" title="Special:BookSources/0-7803-5498-2"><bdi>0-7803-5498-2</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:42588549">42588549</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=With+record+magnetic+fields+to+the+21st+Century&rft.btitle=Digest+of+Technical+Papers.+12th+IEEE+International+Pulsed+Power+Conference.+%28Cat.+No.99CH36358%29&rft.pages=746-749&rft.date=1999&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A42588549%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1109%2FPPC.1999.823621&rft.isbn=0-7803-5498-2&rft.aulast=Boyko&rft.aufirst=B.A.&rft.au=Bykov%2C+A.I.&rft.au=Dolotenko%2C+M.I.&rft.au=Kolokolchikov%2C+N.P.&rft.au=Markevtsev%2C+I.M.&rft.au=Tatsenko%2C+O.M.&rft.au=Shuvalov%2C+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-smithsonianMagnetRecord-63">^ <a href="#cite_ref-smithsonianMagnetRecord_63-0">a</a> <a href="#cite_ref-smithsonianMagnetRecord_63-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDaley" class="citation web cs1">Daley, Jason. <a rel="nofollow" class="external text" href="https://www.smithsonianmag.com/smart-news/strongest-indoor-magnetic-field-blows-doors-tokyo-lab-180970436/">"Watch the Strongest Indoor Magnetic Field Blast Doors of Tokyo Lab Wide Open"</a>. Smithsonian Magazine. Retrieved 8 September 2020.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Smithsonian+Magazine&rft.atitle=Watch+the+Strongest+Indoor+Magnetic+Field+Blast+Doors+of+Tokyo+Lab+Wide+Open&rft.aulast=Daley&rft.aufirst=Jason&rft_id=https%3A%2F%2Fwww.smithsonianmag.com%2Fsmart-news%2Fstrongest-indoor-magnetic-field-blows-doors-tokyo-lab-180970436%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-64"><a href="#cite_ref-64">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTuchin2013" class="citation journal cs1">Tuchin, Kirill (2013). <a rel="nofollow" class="external text" href="https://doi.org/10.1155%2F2013%2F490495">"Particle production in strong electromagnetic fields in relativistic heavy-ion collisions"</a>. Adv. High Energy Phys. 2013: 490495. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1301.0099">1301.0099</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013arXiv1301.0099T">2013arXiv1301.0099T</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1155%2F2013%2F490495">10.1155/2013/490495</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:4877952">4877952</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Adv.+High+Energy+Phys.&rft.atitle=Particle+production+in+strong+electromagnetic+fields+in+relativistic+heavy-ion+collisions&rft.volume=2013&rft.pages=490495&rft.date=2013&rft_id=info%3Aarxiv%2F1301.0099&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A4877952%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1155%2F2013%2F490495&rft_id=info%3Abibcode%2F2013arXiv1301.0099T&rft.aulast=Tuchin&rft.aufirst=Kirill&rft_id=https%3A%2F%2Fdoi.org%2F10.1155%252F2013%252F490495&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-65"><a href="#cite_ref-65">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBzdakSkokov2012" class="citation journal cs1">Bzdak, Adam; Skokov, Vladimir (29 March 2012). "Event-by-event fluctuations of magnetic and electric fields in heavy ion collisions". Physics Letters B. 710 (1): 171–174. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1111.1949">1111.1949</a>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2012PhLB..710..171B">2012PhLB..710..171B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.physletb.2012.02.065">10.1016/j.physletb.2012.02.065</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118462584">118462584</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+Letters+B&rft.atitle=Event-by-event+fluctuations+of+magnetic+and+electric+fields+in+heavy+ion+collisions&rft.volume=710&rft.issue=1&rft.pages=171-174&rft.date=2012-03-29&rft_id=info%3Aarxiv%2F1111.1949&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118462584%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1016%2Fj.physletb.2012.02.065&rft_id=info%3Abibcode%2F2012PhLB..710..171B&rft.aulast=Bzdak&rft.aufirst=Adam&rft.au=Skokov%2C+Vladimir&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-66"><a href="#cite_ref-66">^</a> Kouveliotou, C.; Duncan, R. C.; Thompson, C. (February 2003). "<a rel="nofollow" class="external text" href="http://solomon.as.utexas.edu/~duncan/sciam.pdf">Magnetars</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20070611144829/http://solomon.as.utexas.edu/~duncan/sciam.pdf">Archived</a> 11 June 2007 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>". <a href="/wiki/Scientific_American" title="Scientific American">Scientific American</a>; Page 36. </li> <li id="cite_note-67"><a href="#cite_ref-67">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChapman2007" class="citation book cs1">Chapman, Allan (2007). "Peregrinus, Petrus (Flourished 1269)". Encyclopedia of Geomagnetism and Paleomagnetism. Dordrecht: Springer. pp. 808–809. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4020-4423-6_261">10.1007/978-1-4020-4423-6_261</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4020-3992-8" title="Special:BookSources/978-1-4020-3992-8"><bdi>978-1-4020-3992-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Peregrinus%2C+Petrus+%28Flourished+1269%29&rft.btitle=Encyclopedia+of+Geomagnetism+and+Paleomagnetism&rft.place=Dordrecht&rft.pages=808-809&rft.pub=Springer&rft.date=2007&rft_id=info%3Adoi%2F10.1007%2F978-1-4020-4423-6_261&rft.isbn=978-1-4020-3992-8&rft.aulast=Chapman&rft.aufirst=Allan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-Whittaker1910-69">^ <a href="#cite_ref-Whittaker1910_69-0">a</a> <a href="#cite_ref-Whittaker1910_69-1">b</a> <a href="#cite_ref-Whittaker1910_69-2">c</a> <a href="#cite_ref-Whittaker1910_69-3">d</a> <a href="#cite_ref-Whittaker1910_69-4">e</a> <a href="#cite_ref-Whittaker1910_69-5">f</a> <a href="#cite_ref-Whittaker1910_69-6">g</a> <a href="#cite_ref-Whittaker1910_69-7">h</a> <a href="#cite_ref-Whittaker1910_69-8">i</a> <a href="#cite_ref-Whittaker1910_69-9">j</a> <a href="#cite_ref-Whittaker1910_69-10">k</a> <a href="#cite_ref-Whittaker1910_69-11">l</a> <a href="#cite_ref-Whittaker1910_69-12">m</a> <a href="#cite_ref-Whittaker1910_69-13">n</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhittaker1910" class="citation book cs1"><a href="/wiki/E._T._Whittaker" title="E. T. Whittaker">Whittaker, E. T.</a> (1910). <a href="/wiki/A_History_of_the_Theories_of_Aether_and_Electricity" title="A History of the Theories of Aether and Electricity">A History of the Theories of Aether and Electricity</a>. <a href="/wiki/Dover_Publications" title="Dover Publications">Dover Publications</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-26126-3" title="Special:BookSources/978-0-486-26126-3"><bdi>978-0-486-26126-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+the+Theories+of+Aether+and+Electricity&rft.pub=Dover+Publications&rft.date=1910&rft.isbn=978-0-486-26126-3&rft.aulast=Whittaker&rft.aufirst=E.+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-71"><a href="#cite_ref-71">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilliams1974" class="citation encyclopaedia cs1">Williams, L. Pearce (1974). <a rel="nofollow" class="external text" href="https://archive.org/details/dictionaryofscie10gill/page/184">"Oersted, Hans Christian"</a>. In Gillespie, C. C. (ed.). Dictionary of Scientific Biography. New York: Charles Scribner's Sons. p. 185.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Oersted%2C+Hans+Christian&rft.btitle=Dictionary+of+Scientific+Biography&rft.place=New+York&rft.pages=185&rft.pub=Charles+Scribner%27s+Sons&rft.date=1974&rft.aulast=Williams&rft.aufirst=L.+Pearce&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdictionaryofscie10gill%2Fpage%2F184&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-72"><a href="#cite_ref-72">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlundell2012" class="citation book cs1">Blundell, Stephen J. (2012). Magnetism: A Very Short Introduction. OUP Oxford. p. 31. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780191633720" title="Special:BookSources/9780191633720"><bdi>9780191633720</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Magnetism%3A+A+Very+Short+Introduction&rft.pages=31&rft.pub=OUP+Oxford&rft.date=2012&rft.isbn=9780191633720&rft.aulast=Blundell&rft.aufirst=Stephen+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-Tricker23-73">^ <a href="#cite_ref-Tricker23_73-0">a</a> <a href="#cite_ref-Tricker23_73-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTricker1965" class="citation book cs1">Tricker, R. A. R. (1965). <a rel="nofollow" class="external text" href="https://archive.org/details/earlyelectrodyna0000tric">Early electrodynamics</a>. Oxford: Pergamon. p. <a rel="nofollow" class="external text" href="https://archive.org/details/earlyelectrodyna0000tric/page/23">23</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Early+electrodynamics&rft.place=Oxford&rft.pages=23&rft.pub=Pergamon&rft.date=1965&rft.aulast=Tricker&rft.aufirst=R.+A.+R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fearlyelectrodyna0000tric&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-74"><a href="#cite_ref-74">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFErlichson1998" class="citation journal cs1">Erlichson, Herman (1998). <a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.18878">"The experiments of Biot and Savart concerning the force exerted by a current on a magnetic needle"</a>. American Journal of Physics. 66 (5): 389. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1998AmJPh..66..385E">1998AmJPh..66..385E</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.18878">10.1119/1.18878</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Physics&rft.atitle=The+experiments+of+Biot+and+Savart+concerning+the+force+exerted+by+a+current+on+a+magnetic+needle&rft.volume=66&rft.issue=5&rft.pages=389&rft.date=1998&rft_id=info%3Adoi%2F10.1119%2F1.18878&rft_id=info%3Abibcode%2F1998AmJPh..66..385E&rft.aulast=Erlichson&rft.aufirst=Herman&rft_id=https%3A%2F%2Fdoi.org%2F10.1119%252F1.18878&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-75"><a href="#cite_ref-75">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrankel1972" class="citation book cs1">Frankel, Eugene (1972). Jean-Baptiste Biot: The career of a physicist in nineteenth-century France. Princeton University: Doctoral dissertation. p. 334.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Jean-Baptiste+Biot%3A+The+career+of+a+physicist+in+nineteenth-century+France&rft.place=Princeton+University&rft.pages=334&rft.pub=Doctoral+dissertation&rft.date=1972&rft.aulast=Frankel&rft.aufirst=Eugene&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-77"><a href="#cite_ref-77">^</a> <a rel="nofollow" class="external text" href="http://www.physik.uni-augsburg.de/lehrstuehle/did/Physik/Schulservice/Physikerkalender/06_26_Lord-Kelvin-of-Largs.pdf">Lord Kelvin of Largs</a>. physik.uni-augsburg.de. 26 June 1824 </li> <li id="cite_note-h202-78"><a href="#cite_ref-h202_78-0">^</a> Huurdeman, Anton A. (2003) The Worldwide History of Telecommunications. Wiley. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0471205052" title="Special:BookSources/0471205052">0471205052</a>. p. 202 </li> <li id="cite_note-79"><a href="#cite_ref-79">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.hhi.fraunhofer.de/fraunhofer-hhi-the-institute/about-us/history-of-hhi/the-most-important-experiments.html">"The most important Experiments – The most important Experiments and their Publication between 1886 and 1889"</a>. Fraunhofer Heinrich Hertz Institute. Retrieved 19 February 2016.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+most+important+Experiments+%E2%80%93+The+most+important+Experiments+and+their+Publication+between+1886+and+1889&rft.pub=Fraunhofer+Heinrich+Hertz+Institute&rft_id=http%3A%2F%2Fwww.hhi.fraunhofer.de%2Ffraunhofer-hhi-the-institute%2Fabout-us%2Fhistory-of-hhi%2Fthe-most-important-experiments.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-80"><a href="#cite_ref-80">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=g07Q9M4agp4C&pg=PA117">Networks of Power: Electrification in Western Society, 1880–1930</a>. JHU Press. March 1993. p. 117. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780801846144" title="Special:BookSources/9780801846144"><bdi>9780801846144</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Networks+of+Power%3A+Electrification+in+Western+Society%2C+1880%E2%80%931930&rft.pages=117&rft.pub=JHU+Press&rft.date=1993-03&rft.isbn=9780801846144&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dg07Q9M4agp4C%26pg%3DPA117&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-81"><a href="#cite_ref-81">^</a> Thomas Parke Hughes, Networks of Power: Electrification in Western Society, 1880–1930, pp. 115–118 </li> <li id="cite_note-82"><a href="#cite_ref-82">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLtdSmithsonian_Institution1998" class="citation book cs1">Ltd, Nmsi Trading; Smithsonian Institution (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1AsFdUxOwu8C&pg=PA204">Robert Bud, Instruments of Science: An Historical Encyclopedia</a>. Taylor & Francis. p. 204. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780815315612" title="Special:BookSources/9780815315612"><bdi>9780815315612</bdi></a>. Retrieved 18 March 2013.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Robert+Bud%2C+Instruments+of+Science%3A+An+Historical+Encyclopedia&rft.pages=204&rft.pub=Taylor+%26+Francis&rft.date=1998&rft.isbn=9780815315612&rft.aulast=Ltd&rft.aufirst=Nmsi+Trading&rft.au=Smithsonian+Institution&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D1AsFdUxOwu8C%26pg%3DPA204&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-83"><a href="#cite_ref-83">^</a> <a rel="nofollow" class="external text" href="https://patents.google.com/patent/US381968">U.S. patent 381,968</a> </li> <li id="cite_note-84"><a href="#cite_ref-84">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPorterProut1924" class="citation journal cs1">Porter, H. F. J.; Prout, Henry G. (January 1924). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.2307/1838546">"A Life of George Westinghouse"</a>. The American Historical Review. 29 (2): 129. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1838546">10.2307/1838546</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2027%2Fcoo1.ark%3A%2F13960%2Ft15m6rz0r">2027/coo1.ark:/13960/t15m6rz0r</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-8762">0002-8762</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1838546">1838546</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Historical+Review&rft.atitle=A+Life+of+George+Westinghouse&rft.volume=29&rft.issue=2&rft.pages=129&rft.date=1924-01&rft_id=info%3Ahdl%2F2027%2Fcoo1.ark%3A%2F13960%2Ft15m6rz0r&rft.issn=0002-8762&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1838546%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F1838546&rft.aulast=Porter&rft.aufirst=H.+F.+J.&rft.au=Prout%2C+Henry+G.&rft_id=http%3A%2F%2Fdx.doi.org%2F10.2307%2F1838546&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> <li id="cite_note-85"><a href="#cite_ref-85">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20210709182151/http://ieeemilestones.ethw.org/images/a/a0/1-Galileo_Ferraris_Rotating-field--Rotazioni_Elettrodinamiche.pdf">"Galileo Ferraris (March 1888) Rotazioni elettrodinamiche prodotte per mezzo di correnti alternate (Electrodynamic rotations by means of alternating currents), memory read at Accademia delle Scienze, Torino, in Opere di Galileo Ferraris, Hoepli, Milano,1902 vol I pages 333 to 348"</a> (PDF). Archived from <a rel="nofollow" class="external text" href="http://ieeemilestones.ethw.org/images/a/a0/1-Galileo_Ferraris_Rotating-field--Rotazioni_Elettrodinamiche.pdf">the original</a> (PDF) on 9 July 2021. Retrieved 2 July 2021.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Galileo+Ferraris+%28March+1888%29+Rotazioni+elettrodinamiche+prodotte+per+mezzo+di+correnti+alternate+%28Electrodynamic+rotations+by+means+of+alternating+currents%29%2C+memory+read+at+Accademia+delle+Scienze%2C+Torino%2C+in+Opere+di+Galileo+Ferraris%2C+Hoepli%2C+Milano%2C1902+vol+I+pages+333+to+348&rft_id=http%3A%2F%2Fieeemilestones.ethw.org%2Fimages%2Fa%2Fa0%2F1-Galileo_Ferraris_Rotating-field--Rotazioni_Elettrodinamiche.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJiles1994" class="citation book cs1">Jiles, David (1994). Introduction to Electronic Properties of Materials (1st ed.). Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-412-49580-9" title="Special:BookSources/978-0-412-49580-9"><bdi>978-0-412-49580-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Electronic+Properties+of+Materials&rft.edition=1st&rft.pub=Springer&rft.date=1994&rft.isbn=978-0-412-49580-9&rft.aulast=Jiles&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTipler2004" class="citation book cs1">Tipler, Paul (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.). W. H. Freeman. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7167-0810-0" title="Special:BookSources/978-0-7167-0810-0"><bdi>978-0-7167-0810-0</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/51095685">51095685</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics+for+Scientists+and+Engineers%3A+Electricity%2C+Magnetism%2C+Light%2C+and+Elementary+Modern+Physics+%285th+ed.%29&rft.pub=W.+H.+Freeman&rft.date=2004&rft_id=info%3Aoclcnum%2F51095685&rft.isbn=978-0-7167-0810-0&rft.aulast=Tipler&rft.aufirst=Paul&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMagnetic+field" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div> <ul><li><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Magnetic_fields" class="extiw" title="commons:Category:Magnetic fields">Magnetic fields</a> at Wikimedia Commons</li> <li>Crowell, B., "<a rel="nofollow" class="external text" href="http://www.lightandmatter.com/html_books/0sn/ch11/ch11.html">Electromagnetism</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20100430152732/http://www.lightandmatter.com/html_books/0sn/ch11/ch11.html">Archived</a> 30 April 2010 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>".</li> <li>Nave, R., "<a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magfie.html">Magnetic Field</a>". HyperPhysics.</li> <li>"Magnetism", <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060709195126/http://theory.uwinnipeg.ca/physics/mag/node2.html#SECTION00110000000000000000">The Magnetic Field</a> (archived 9 July 2006). theory.uwinnipeg.ca.</li> <li>Hoadley, Rick, "<a rel="nofollow" class="external text" href="http://my.execpc.com/~rhoadley/magfield.htm">What do magnetic fields look like</a>?" 17 July 2005.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.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 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