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id="toc-General_scalar_quantities" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_scalar_quantities"> <div class="vector-toc-text"> 1.2 General scalar quantities </div> </a> <ul id="toc-General_scalar_quantities-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tensor_resistivity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tensor_resistivity"> <div class="vector-toc-text"> 1.3 Tensor resistivity </div> </a> <ul id="toc-Tensor_resistivity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Conductivity_and_current_carriers" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Conductivity_and_current_carriers"> <div class="vector-toc-text"> 2 Conductivity and current carriers </div> </a> <button aria-controls="toc-Conductivity_and_current_carriers-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Conductivity and current carriers subsection </button> <ul id="toc-Conductivity_and_current_carriers-sublist" class="vector-toc-list"> <li id="toc-Relation_between_current_density_and_electric_current_velocity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relation_between_current_density_and_electric_current_velocity"> <div class="vector-toc-text"> 2.1 Relation between current density and electric current velocity </div> </a> <ul id="toc-Relation_between_current_density_and_electric_current_velocity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Causes_of_conductivity" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Causes_of_conductivity"> <div class="vector-toc-text"> 3 Causes of conductivity </div> </a> <button aria-controls="toc-Causes_of_conductivity-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Causes of conductivity subsection </button> <ul id="toc-Causes_of_conductivity-sublist" class="vector-toc-list"> <li id="toc-Band_theory_simplified" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Band_theory_simplified"> <div class="vector-toc-text"> 3.1 Band theory simplified </div> </a> <ul id="toc-Band_theory_simplified-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_metals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_metals"> <div class="vector-toc-text"> 3.2 In metals </div> </a> <ul id="toc-In_metals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_semiconductors_and_insulators" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_semiconductors_and_insulators"> <div class="vector-toc-text"> 3.3 In semiconductors and insulators </div> </a> <ul id="toc-In_semiconductors_and_insulators-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_ionic_liquids/electrolytes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_ionic_liquids/electrolytes"> <div class="vector-toc-text"> 3.4 In ionic liquids/electrolytes </div> </a> <ul id="toc-In_ionic_liquids/electrolytes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Superconductivity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Superconductivity"> <div class="vector-toc-text"> 3.5 Superconductivity </div> </a> <ul id="toc-Superconductivity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Plasma" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Plasma"> <div class="vector-toc-text"> 3.6 Plasma </div> </a> <ul id="toc-Plasma-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Resistivity_and_conductivity_of_various_materials" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Resistivity_and_conductivity_of_various_materials"> <div class="vector-toc-text"> 4 Resistivity and conductivity of various materials </div> </a> <ul id="toc-Resistivity_and_conductivity_of_various_materials-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Temperature_dependence" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Temperature_dependence"> <div class="vector-toc-text"> 5 Temperature dependence </div> </a> <button aria-controls="toc-Temperature_dependence-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Temperature dependence subsection </button> <ul id="toc-Temperature_dependence-sublist" class="vector-toc-list"> <li id="toc-Linear_approximation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Linear_approximation"> <div class="vector-toc-text"> 5.1 Linear approximation </div> </a> <ul id="toc-Linear_approximation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Metals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Metals"> <div class="vector-toc-text"> 5.2 Metals </div> </a> <ul id="toc-Metals-sublist" class="vector-toc-list"> <li id="toc-Wiedemann–Franz_law" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Wiedemann–Franz_law"> <div class="vector-toc-text"> 5.2.1 Wiedemann–Franz law </div> </a> <ul id="toc-Wiedemann–Franz_law-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Semiconductors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Semiconductors"> <div class="vector-toc-text"> 5.3 Semiconductors </div> </a> <ul id="toc-Semiconductors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kondo_insulators" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kondo_insulators"> <div class="vector-toc-text"> 5.4 Kondo insulators </div> </a> <ul id="toc-Kondo_insulators-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Complex_resistivity_and_conductivity" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Complex_resistivity_and_conductivity"> <div class="vector-toc-text"> 6 Complex resistivity and conductivity </div> </a> <ul id="toc-Complex_resistivity_and_conductivity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Resistance_versus_resistivity_in_complicated_geometries" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Resistance_versus_resistivity_in_complicated_geometries"> <div class="vector-toc-text"> 7 Resistance versus resistivity in complicated geometries </div> </a> <ul id="toc-Resistance_versus_resistivity_in_complicated_geometries-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Resistivity-density_product" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Resistivity-density_product"> <div class="vector-toc-text"> 8 Resistivity-density product </div> </a> <ul id="toc-Resistivity-density_product-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"> 9 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-John_Walsh_and_the_conductivity_of_a_vacuum" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#John_Walsh_and_the_conductivity_of_a_vacuum"> <div class="vector-toc-text"> 9.1 John Walsh and the conductivity of a vacuum </div> </a> <ul id="toc-John_Walsh_and_the_conductivity_of_a_vacuum-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"> 10 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </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"> 11 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"> 12 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"> 13 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"> 14 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">Electrical resistivity and conductivity</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 18 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-18" aria-hidden="true" > 18 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A7%D9%84%D9%85%D9%82%D8%A7%D9%88%D9%85%D9%8A%D8%A9_%D9%88%D8%A7%D9%84%D9%85%D9%88%D8%B5%D9%84%D9%8A%D8%A9_%D8%A7%D9%84%D9%83%D9%87%D8%B1%D8%A8%D8%A7%D8%A6%D9%8A%D8%A9" title="المقاومية والموصلية الكهربائية – Arabic" lang="ar" hreflang="ar" data-title="المقاومية والموصلية الكهربائية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D9%82%D8%A7%D9%88%D9%85%D8%AA_%D9%88%DB%8C%DA%98%D9%87_%D9%88_%D8%B1%D8%B3%D8%A7%D9%86%D9%86%D8%AF%DA%AF%DB%8C_%D8%A7%D9%84%DA%A9%D8%AA%D8%B1%DB%8C%DA%A9%DB%8C" title="مقاومت ویژه و رسانندگی الکتریکی – Persian" lang="fa" hreflang="fa" data-title="مقاومت ویژه و رسانندگی الکتریکی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Elektri%C4%8Dna_otpornost_i_provodnost" title="Električna otpornost i provodnost – Croatian" lang="hr" hreflang="hr" data-title="Električna otpornost i provodnost" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Resistivitas_dan_konduktivitas_listrik" title="Resistivitas dan konduktivitas listrik – Indonesian" lang="id" hreflang="id" data-title="Resistivitas dan konduktivitas listrik" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B5%E0%B2%BF%E0%B2%A6%E0%B3%8D%E0%B2%AF%E0%B3%81%E0%B2%A4%E0%B3%8D_%E0%B2%B0%E0%B3%8B%E0%B2%A7_%E0%B2%AE%E0%B2%A4%E0%B3%8D%E0%B2%A4%E0%B3%81_%E0%B2%B5%E0%B2%BF%E0%B2%A6%E0%B3%8D%E0%B2%AF%E0%B3%81%E0%B2%A4%E0%B3%8D_%E0%B2%B5%E0%B2%BE%E0%B2%B9%E0%B2%95%E0%B2%A4%E0%B3%86" title="ವಿದ್ಯುತ್ ರೋಧ ಮತ್ತು ವಿದ್ಯುತ್ ವಾಹಕತೆ – Kannada" lang="kn" hreflang="kn" data-title="ವಿದ್ಯುತ್ ರೋಧ ಮತ್ತು ವಿದ್ಯುತ್ ವಾಹಕತೆ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B5%E0%B5%88%E0%B4%A6%E0%B5%8D%E0%B4%AF%E0%B5%81%E0%B4%A4%E0%B4%9A%E0%B4%BE%E0%B4%B2%E0%B4%95%E0%B4%A4%E0%B4%AF%E0%B5%81%E0%B4%82_%E0%B4%B5%E0%B5%88%E0%B4%A6%E0%B5%8D%E0%B4%AF%E0%B5%81%E0%B4%A4%E0%B4%AA%E0%B5%8D%E0%B4%B0%E0%B4%A4%E0%B4%BF%E0%B4%B0%E0%B5%8B%E0%B4%A7%E0%B4%B5%E0%B5%81%E0%B4%82" title="വൈദ്യുതചാലകതയും വൈദ്യുതപ്രതിരോധവും – Malayalam" lang="ml" hreflang="ml" data-title="വൈദ്യുതചാലകതയും വൈദ്യുതപ്രതിരോധവും" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Kerintangan_dan_kekonduksian_elektrik" title="Kerintangan dan kekonduksian elektrik – Malay" lang="ms" hreflang="ms" data-title="Kerintangan dan kekonduksian elektrik" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AC%E0%A8%BF%E0%A8%9C%E0%A8%B2%E0%A8%88_%E0%A8%AA%E0%A9%8D%E0%A8%B0%E0%A8%A4%E0%A8%BF%E0%A8%B0%E0%A9%8B%E0%A8%A7%E0%A8%95%E0%A8%A4%E0%A8%BE_%E0%A8%85%E0%A8%A4%E0%A9%87_%E0%A8%A8%E0%A8%BF%E0%A8%B8%E0%A8%BC%E0%A8%9A%E0%A8%BF%E0%A8%A4_%E0%A8%AC%E0%A8%BF%E0%A8%9C%E0%A8%B2%E0%A8%88_%E0%A8%9A%E0%A8%BE%E0%A8%B2%E0%A8%95%E0%A8%A4%E0%A8%BE" title="ਬਿਜਲਈ ਪ੍ਰਤਿਰੋਧਕਤਾ ਅਤੇ ਨਿਸ਼ਚਿਤ ਬਿਜਲਈ ਚਾਲਕਤਾ – Punjabi" lang="pa" hreflang="pa" data-title="ਬਿਜਲਈ ਪ੍ਰਤਿਰੋਧਕਤਾ ਅਤੇ ਨਿਸ਼ਚਿਤ ਬਿਜਲਈ ਚਾਲਕਤਾ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Electrical_resistivity_an_conductivity" title="Electrical resistivity an conductivity – Scots" lang="sco" hreflang="sco" data-title="Electrical resistivity an conductivity" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target">Scots</a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B7%80%E0%B7%92%E0%B6%AF%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%94%E0%B6%AD%E0%B7%8A_%E0%B6%B4%E0%B7%8A%E2%80%8D%E0%B6%BB%E0%B6%AD%E0%B7%92%E0%B6%BB%E0%B7%9D%E0%B6%B0%E0%B6%9A%E0%B6%AD%E0%B7%8F%E0%B7%80_%E0%B7%83%E0%B7%84_%E0%B7%83%E0%B6%B1%E0%B7%8A%E0%B6%B1%E0%B7%8F%E0%B6%BA%E0%B6%9A%E0%B6%AD%E0%B7%8F%E0%B7%80" title="විද්‍යුත් ප්‍රතිරෝධකතාව සහ සන්නායකතාව – Sinhala" lang="si" hreflang="si" data-title="විද්‍යුත් ප්‍රතිරෝධකතාව සහ සන්නායකතාව" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target">සිංහල</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Elektri%C4%8Dna_otpornost_i_provodnost" title="Električna otpornost i provodnost – Serbian" lang="sr" hreflang="sr" data-title="Električna otpornost i provodnost" 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/Specifi%C4%8Dna_elektri%C4%8Dna_vodljivost_i_otpor" title="Specifična električna vodljivost i otpor – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Specifična električna vodljivost i otpor" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AE%BF%E0%AE%A9%E0%AF%8D%E0%AE%95%E0%AE%9F%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AF%81%E0%AE%A4%E0%AE%BF%E0%AE%B1%E0%AE%A9%E0%AF%8D_%E0%AE%AE%E0%AE%B1%E0%AF%8D%E0%AE%B1%E0%AF%81%E0%AE%AE%E0%AF%8D_%E0%AE%AE%E0%AE%BF%E0%AE%A9%E0%AF%8D%E0%AE%A4%E0%AE%9F%E0%AF%88%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AE%BF%E0%AE%B1%E0%AE%A9%E0%AF%8D" title="மின்கடத்துதிறன் மற்றும் மின்தடைத்திறன் – Tamil" lang="ta" hreflang="ta" data-title="மின்கடத்துதிறன் மற்றும் மின்தடைத்திறன்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%AA%E0%B8%A0%E0%B8%B2%E0%B8%9E%E0%B8%95%E0%B9%89%E0%B8%B2%E0%B8%99%E0%B8%97%E0%B8%B2%E0%B8%99%E0%B9%81%E0%B8%A5%E0%B8%B0%E0%B8%AA%E0%B8%A0%E0%B8%B2%E0%B8%9E%E0%B8%99%E0%B8%B3%E0%B9%84%E0%B8%9F%E0%B8%9F%E0%B9%89%E0%B8%B2" title="สภาพต้านทานและสภาพนำไฟฟ้า – Thai" lang="th" hreflang="th" data-title="สภาพต้านทานและสภาพนำไฟฟ้า" 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/Elektrik_direnci_ve_iletkenli%C4%9Fi" title="Elektrik direnci ve iletkenliği – Turkish" lang="tr" hreflang="tr" 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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">This article is about electrical conductivity in general. For other types of conductivity, see <a href="/wiki/Conductivity_(disambiguation)" class="mw-redirect mw-disambig" title="Conductivity (disambiguation)">Conductivity</a>. For specific applications in electrical elements, see <a href="/wiki/Electrical_resistance_and_conductance" title="Electrical resistance and conductance">Electrical resistance and conductance</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above">Resistivity</th></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Common symbols</div></th><td class="infobox-data">ρ</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI unit</a></th><td class="infobox-data">ohm metre (Ω⋅m)</td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Other units</div></th><td class="infobox-data">s (Gaussian/ESU)</td></tr><tr><th scope="row" class="infobox-label">In <a href="/wiki/SI_base_unit" title="SI base unit">SI base units</a></th><td class="infobox-data">kg⋅m3⋅s−3⋅A−2</td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Derivations from other quantities</div></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =R{\frac {A}{\ell }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mi>ℓ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =R{\frac {A}{\ell }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9566636088b5bd9de3551f3a03f63e866065f960" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.644ex; height:5.509ex;" alt="{\displaystyle \rho =R{\frac {A}{\ell }}}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Dimensional_analysis#Formulation" title="Dimensional analysis">Dimension</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {M}}{\mathsf {L}}^{3}{\mathsf {T}}^{-3}{\mathsf {I}}^{-2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {M}}{\mathsf {L}}^{3}{\mathsf {T}}^{-3}{\mathsf {I}}^{-2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c82216dfa6d5931d4a6b411d7b35e89f6046f2b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.243ex; height:2.676ex;" alt="{\displaystyle {\mathsf {M}}{\mathsf {L}}^{3}{\mathsf {T}}^{-3}{\mathsf {I}}^{-2}}"></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546"><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above">Conductivity</th></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Common symbols</div></th><td class="infobox-data">σ, κ, γ</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI unit</a></th><td class="infobox-data">siemens per metre (S/m)</td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Other units</div></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {s} ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {s} ^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a4634ca60f765e9f64d76d09c0097cd8e1f68c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.249ex; height:2.676ex;" alt="{\displaystyle \mathrm {s} ^{-1}}"> (Gaussian/ESU)</td></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Derivations from other quantities</div></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ={\frac {1}{\rho }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ρ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ={\frac {1}{\rho }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f1932f23bbf7c2e9026810a48a9d4722a1c3d75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:6.466ex; height:5.676ex;" alt="{\displaystyle \sigma ={\frac {1}{\rho }}}"></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Dimensional_analysis#Formulation" title="Dimensional analysis">Dimension</a></th><td class="infobox-data"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathsf {M}}^{-1}{\mathsf {L}}^{-3}{\mathsf {T}}^{3}{\mathsf {I}}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">I</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {M}}^{-1}{\mathsf {L}}^{-3}{\mathsf {T}}^{3}{\mathsf {I}}^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a921c259d4e8de62471c86a62aabdb2b2403985e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.297ex; height:2.676ex;" alt="{\displaystyle {\mathsf {M}}^{-1}{\mathsf {L}}^{-3}{\mathsf {T}}^{3}{\mathsf {I}}^{2}}"></td></tr></tbody></table> Electrical resistivity (also called volume resistivity or specific electrical resistance) is a fundamental <a href="/wiki/Specific_property" class="mw-redirect" title="Specific property">specific property</a> of a material that measures its <a href="/wiki/Electrical_resistance" class="mw-redirect" title="Electrical resistance">electrical resistance</a> or how strongly it resists <a href="/wiki/Electric_current" title="Electric current">electric current</a>. A low resistivity indicates a material that readily allows electric current. Resistivity is commonly represented by the <a href="/wiki/Greek_alphabet" title="Greek alphabet">Greek letter</a> ρ (<a href="/wiki/Rho_(letter)" class="mw-redirect" title="Rho (letter)">rho</a>). The <a href="/wiki/SI" class="mw-redirect" title="SI">SI</a> unit of electrical resistivity is the <a href="/wiki/Ohm" title="Ohm">ohm</a>-<a href="/wiki/Metre" title="Metre">metre</a> (Ω⋅m).<a href="#cite_note-1">[1]</a><a href="#cite_note-:0-2">[2]</a><a href="#cite_note-3">[3]</a> For example, if a 1 m3 solid cube of material has sheet contacts on two opposite faces, and the <a href="/wiki/Electrical_resistance" class="mw-redirect" title="Electrical resistance">resistance</a> between these contacts is 1 Ω, then the resistivity of the material is 1 Ω⋅m. Electrical conductivity (or specific conductance) is the reciprocal of electrical resistivity. It represents a material's ability to conduct electric current. It is commonly signified by the Greek letter σ (<a href="/wiki/Sigma_(letter)" class="mw-redirect" title="Sigma (letter)">sigma</a>), but κ (<a href="/wiki/Kappa" title="Kappa">kappa</a>) (especially in electrical engineering)[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] and γ (<a href="/wiki/Gamma" title="Gamma">gamma</a>)[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] are sometimes used. The SI unit of electrical conductivity is <a href="/wiki/Siemens_(unit)" title="Siemens (unit)">siemens</a> per <a href="/wiki/Metre" title="Metre">metre</a> (S/m). Resistivity and conductivity are <a href="/wiki/Intensive_property" class="mw-redirect" title="Intensive property">intensive properties</a> of materials, giving the opposition of a standard cube of material to current. <a href="/wiki/Electrical_resistance_and_conductance" title="Electrical resistance and conductance">Electrical resistance and conductance</a> are corresponding <a href="/wiki/Extensive_property" class="mw-redirect" title="Extensive property">extensive properties</a> that give the opposition of a specific object to electric current. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=1" title="Edit section: Definition">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Ideal_case">Ideal case</h3>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=2" title="Edit section: Ideal case">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Resistivity_geometry.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Resistivity_geometry.png/220px-Resistivity_geometry.png" decoding="async" width="220" height="198" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Resistivity_geometry.png/330px-Resistivity_geometry.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/68/Resistivity_geometry.png/440px-Resistivity_geometry.png 2x" data-file-width="471" data-file-height="424" /></a><figcaption>A piece of resistive material with electrical contacts on both ends</figcaption></figure> In an ideal case, cross-section and physical composition of the examined material are uniform across the sample, and the electric field and current density are both parallel and constant everywhere. Many <a href="/wiki/Resistor" title="Resistor">resistors</a> and <a href="/wiki/Electrical_conductor" title="Electrical conductor">conductors</a> do in fact have a uniform cross section with a uniform flow of electric current, and are made of a single material, so that this is a good model. (See the adjacent diagram.) When this is the case, the resistance of the conductor is directly proportional to its length and inversely proportional to its cross-sectional area, where the electrical resistivity ρ (Greek: <a href="/wiki/Rho_(letter)" class="mw-redirect" title="Rho (letter)">rho</a>) is the constant of proportionality. This is written as: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\propto {\frac {\ell }{A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>∝</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ℓ</mi> <mi>A</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\propto {\frac {\ell }{A}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2dff3133d967b1f57d260260b054c8a47b56316" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.442ex; height:5.509ex;" alt="{\displaystyle R\propto {\frac {\ell }{A}}}"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}R&=\rho {\frac {\ell }{A}}\\[3pt]{}\Leftrightarrow \rho &=R{\frac {A}{\ell }},\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="0.6em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>R</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ℓ</mi> <mi>A</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo stretchy="false">⇔</mo> <mi>ρ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mi>ℓ</mi> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}R&=\rho {\frac {\ell }{A}}\\[3pt]{}\Leftrightarrow \rho &=R{\frac {A}{\ell }},\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc1dfffd808c6e963fb0ca4cc306360d7e1f7e24" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:13.656ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}R&=\rho {\frac {\ell }{A}}\\[3pt]{}\Leftrightarrow \rho &=R{\frac {A}{\ell }},\end{aligned}}}"> where <style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><div class="plainlist"> <ul><li><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}"> is the <a href="/wiki/Electrical_resistance" class="mw-redirect" title="Electrical resistance">electrical resistance</a> of a uniform specimen of the material</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"> is the <a href="/wiki/Length" title="Length">length</a> of the specimen</li> <li><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/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is the <a href="/wiki/Cross_section_(geometry)" title="Cross section (geometry)">cross-sectional area</a> of the specimen</li></ul> </div> The resistivity can be expressed using the <a href="/wiki/SI" class="mw-redirect" title="SI">SI</a> unit <a href="/wiki/Ohm" title="Ohm">ohm</a> <a href="/wiki/Metre" title="Metre">metre</a> (Ω⋅m) — i.e. ohms multiplied by square metres (for the cross-sectional area) then divided by metres (for the length). Both resistance and resistivity describe how difficult it is to make electrical current flow through a material, but unlike resistance, resistivity is an <a href="/wiki/Intrinsic_property" class="mw-redirect" title="Intrinsic property">intrinsic property</a> and does not depend on geometric properties of a material. This means that all pure copper (Cu) wires (which have not been subjected to distortion of their crystalline structure etc.), irrespective of their shape and size, have the same resistivity, but a long, thin copper wire has a much larger resistance than a thick, short copper wire. Every material has its own characteristic resistivity. For example, rubber has a far larger resistivity than copper. In a <a href="/wiki/Hydraulic_analogy" title="Hydraulic analogy">hydraulic analogy</a>, passing current through a high-resistivity material is like pushing water through a pipe full of sand - while passing current through a low-resistivity material is like pushing water through an empty pipe. If the pipes are the same size and shape, the pipe full of sand has higher resistance to flow. Resistance, however, is not solely determined by the presence or absence of sand. It also depends on the length and width of the pipe: short or wide pipes have lower resistance than narrow or long pipes. The above equation can be transposed to get Pouillet's law (named after <a href="/wiki/Claude_Pouillet" title="Claude Pouillet">Claude Pouillet</a>): <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=\rho {\frac {\ell }{A}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ℓ</mi> <mi>A</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=\rho {\frac {\ell }{A}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/346697b860882ec044bf2c1596874eaa8337859d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.29ex; height:5.509ex;" alt="{\displaystyle R=\rho {\frac {\ell }{A}}.}">The resistance of a given element is proportional to the length, but inversely proportional to the cross-sectional area. For example, if A = 1 m2, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"> = 1 m (forming a cube with perfectly conductive contacts on opposite faces), then the resistance of this element in ohms is numerically equal to the resistivity of the material it is made of in Ω⋅m. Conductivity, σ, is the inverse of resistivity: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ={\frac {1}{\rho }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ρ</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ={\frac {1}{\rho }}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ac2b3b3d5bc95f7aad71c06096165b510d7ae02" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:7.113ex; height:5.676ex;" alt="{\displaystyle \sigma ={\frac {1}{\rho }}.}"> Conductivity has SI units of <a href="/wiki/Siemens_(unit)" title="Siemens (unit)">siemens</a> per metre (S/m). <div class="mw-heading mw-heading3"><h3 id="General_scalar_quantities">General scalar quantities</h3>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=3" title="Edit section: General scalar quantities">edit</a>]</div> If the geometry is more complicated, or if the resistivity varies from point to point within the material, the current and electric field will be functions of position. Then it is necessary to use a more general expression in which the resistivity at a particular point is defined as the ratio of the <a href="/wiki/Electric_field" title="Electric field">electric field</a> to the <a href="/wiki/Current_density" title="Current density">density</a> of the current it creates at that point: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (x)={\frac {E(x)}{J(x)}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>J</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (x)={\frac {E(x)}{J(x)}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea3ffe5d8c907513bd3b198b6a165f78747b5c1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.837ex; height:6.509ex;" alt="{\displaystyle \rho (x)={\frac {E(x)}{J(x)}},}"> where <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"> <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f32a26d8795457b2f5c2bdc078758dcbbc71b30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.341ex; height:2.843ex;" alt="{\displaystyle \rho (x)}"> is the resistivity of the conductor material at the point <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}">,</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2411ec03b08babee0aa5cbf7a90ba80690a3a0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.915ex; height:2.843ex;" alt="{\displaystyle E(x)}"> is the electric field at the point <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}">,</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40fcaf0063846d508ceeb126a7a2bc1b0229a0ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.61ex; height:2.843ex;" alt="{\displaystyle J(x)}"> is the <a href="/wiki/Current_density" title="Current density">current density</a> at the point <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}">.</li></ul> </div> The current density is parallel to the electric field by necessity. Conductivity is the inverse (reciprocal) of resistivity. Here, it is given by: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (x)={\frac {1}{\rho (x)}}={\frac {J(x)}{E(x)}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>J</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>E</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (x)={\frac {1}{\rho (x)}}={\frac {J(x)}{E(x)}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c160b25d6f7f93035865c1a455df73be70139c74" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:22.24ex; height:6.509ex;" alt="{\displaystyle \sigma (x)={\frac {1}{\rho (x)}}={\frac {J(x)}{E(x)}}.}"> For example, rubber is a material with large ρ and small σ — because even a very large electric field in rubber makes almost no current flow through it. On the other hand, copper is a material with small ρ and large σ — because even a small electric field pulls a lot of current through it. This expression simplifies to the formula given above under "ideal case" when the resistivity is constant in the material and the geometry has a uniform cross-section. In this case, the electric field and current density are constant and parallel. <dl><dd><table class="toccolours collapsible collapsed" width="80%" style="text-align:left;"> <tbody><tr> <th>Derivation of the constant case from the general case </th></tr> <tr> <td>We will combine three equations. Assume the geometry has a uniform cross-section and the resistivity is constant in the material. Then the electric field and current density are constant and parallel, and by the general definition of resistivity, we obtain <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ={\frac {E}{J}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <mi>J</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ={\frac {E}{J}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8335241d0dfd1f62c465af9a491dfe07429723b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.559ex; height:5.343ex;" alt="{\displaystyle \rho ={\frac {E}{J}},}"> Since the electric field is constant, it is given by the total voltage V across the conductor divided by the length ℓ of the conductor: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E={\frac {V}{\ell }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>V</mi> <mi>ℓ</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E={\frac {V}{\ell }}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2923740e50798e7812b3dfdcd464fbe0b95ad0f5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.144ex; height:5.343ex;" alt="{\displaystyle E={\frac {V}{\ell }}.}"> Since the current density is constant, it is equal to the total current divided by the cross sectional area: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J={\frac {I}{A}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>I</mi> <mi>A</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J={\frac {I}{A}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52ff2cbb00b59daa40de6d82ca0639b9470e07e7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.796ex; height:5.343ex;" alt="{\displaystyle J={\frac {I}{A}}.}"> Plugging in the values of E and J into the first expression, we obtain: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho ={\frac {VA}{I\ell }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>V</mi> <mi>A</mi> </mrow> <mrow> <mi>I</mi> <mi>ℓ</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho ={\frac {VA}{I\ell }}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fdc62cb9e5409a84611eaa4cdf9c522367aa1c9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.314ex; height:5.509ex;" alt="{\displaystyle \rho ={\frac {VA}{I\ell }}.}"> Finally, we apply Ohm's law, V/I = R: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =R{\frac {A}{\ell }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>A</mi> <mi>ℓ</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =R{\frac {A}{\ell }}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5339fb6af466a11023df5dddf93289cf0445755e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.29ex; height:5.509ex;" alt="{\displaystyle \rho =R{\frac {A}{\ell }}.}"> </td></tr></tbody></table></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Tensor_resistivity">Tensor resistivity</h3>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=4" title="Edit section: Tensor resistivity">edit</a>]</div> When the resistivity of a material has a directional component, the most general definition of resistivity must be used. It starts from the tensor-vector form of <a href="/wiki/Ohm%27s_law" title="Ohm's law">Ohm's law</a>, which relates the electric field inside a material to the electric current flow. This equation is completely general, meaning it is valid in all cases, including those mentioned above. However, this definition is the most complicated, so it is only directly used in <a href="/wiki/Anisotropy" title="Anisotropy">anisotropic</a> cases, where the more simple definitions cannot be applied. If the material is not anisotropic, it is safe to ignore the tensor-vector definition, and use a simpler expression instead. Here, <a href="/wiki/Anisotropy" title="Anisotropy">anisotropic</a> means that the material has different properties in different directions. For example, a crystal of <a href="/wiki/Graphite" title="Graphite">graphite</a> consists microscopically of a stack of sheets, and current flows very easily through each sheet, but much less easily from one sheet to the adjacent one.<a href="#cite_note-Pierson-4">[4]</a> In such cases, the current does not flow in exactly the same direction as the electric field. Thus, the appropriate equations are generalized to the three-dimensional tensor form:<a href="#cite_note-5">[5]</a><a href="#cite_note-6">[6]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {J} ={\boldsymbol {\sigma }}\mathbf {E} \,\,\rightleftharpoons \,\,\mathbf {E} ={\boldsymbol {\rho }}\mathbf {J} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mo class="MJX-variant" stretchy="false">⇌</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ρ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J} ={\boldsymbol {\sigma }}\mathbf {E} \,\,\rightleftharpoons \,\,\mathbf {E} ={\boldsymbol {\rho }}\mathbf {J} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98f6bd1bea74252f5865a28b1476f1a8f038344c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.299ex; height:2.509ex;" alt="{\displaystyle \mathbf {J} ={\boldsymbol {\sigma }}\mathbf {E} \,\,\rightleftharpoons \,\,\mathbf {E} ={\boldsymbol {\rho }}\mathbf {J} ,}"> where the conductivity σ and resistivity ρ are rank-2 <a href="/wiki/Tensor" title="Tensor">tensors</a>, and electric field E and current density J are vectors. These tensors can be represented by 3×3 matrices, the vectors with 3×1 matrices, with <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a> used on the right side of these equations. In matrix form, the resistivity relation is given by: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}E_{x}\\E_{y}\\E_{z}\end{bmatrix}}={\begin{bmatrix}\rho _{xx}&\rho _{xy}&\rho _{xz}\\\rho _{yx}&\rho _{yy}&\rho _{yz}\\\rho _{zx}&\rho _{zy}&\rho _{zz}\end{bmatrix}}{\begin{bmatrix}J_{x}\\J_{y}\\J_{z}\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}E_{x}\\E_{y}\\E_{z}\end{bmatrix}}={\begin{bmatrix}\rho _{xx}&\rho _{xy}&\rho _{xz}\\\rho _{yx}&\rho _{yy}&\rho _{yz}\\\rho _{zx}&\rho _{zy}&\rho _{zz}\end{bmatrix}}{\begin{bmatrix}J_{x}\\J_{y}\\J_{z}\end{bmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c086504e9bb39d6234ba9b139750707b1d2cab7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:34.947ex; height:9.843ex;" alt="{\displaystyle {\begin{bmatrix}E_{x}\\E_{y}\\E_{z}\end{bmatrix}}={\begin{bmatrix}\rho _{xx}&\rho _{xy}&\rho _{xz}\\\rho _{yx}&\rho _{yy}&\rho _{yz}\\\rho _{zx}&\rho _{zy}&\rho _{zz}\end{bmatrix}}{\begin{bmatrix}J_{x}\\J_{y}\\J_{z}\end{bmatrix}},}"> where <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><div class="plainlist"> <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7f22b39d51f780fc02859059c1757c606b9de2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.757ex; height:2.176ex;" alt="{\displaystyle \mathbf {E} }"> is the electric field vector, with components (Ex, Ey, Ez);</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\rho }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ρ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\rho }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa514c733a0add8e7c3af8bf4f930fa918d16ff8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.423ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {\rho }}}"> is the resistivity tensor, in general a three by three matrix;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {J} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7686846b1a6b756cb514954000004ab5e7b2a5ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.381ex; height:2.176ex;" alt="{\displaystyle \mathbf {J} }"> is the electric current density vector, with components (Jx, Jy, Jz).</li></ul> </div> Equivalently, resistivity can be given in the more compact <a href="/wiki/Einstein_notation" title="Einstein notation">Einstein notation</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} _{i}={\boldsymbol {\rho }}_{ij}\mathbf {J} _{j}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ρ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} _{i}={\boldsymbol {\rho }}_{ij}\mathbf {J} _{j}~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/690672da5da1387dd180a156dc7c99b82c528cc5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.073ex; height:3.009ex;" alt="{\displaystyle \mathbf {E} _{i}={\boldsymbol {\rho }}_{ij}\mathbf {J} _{j}~.}"> In either case, the resulting expression for each electric field component is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}E_{x}&=\rho _{xx}J_{x}+\rho _{xy}J_{y}+\rho _{xz}J_{z},\\E_{y}&=\rho _{yx}J_{x}+\rho _{yy}J_{y}+\rho _{yz}J_{z},\\E_{z}&=\rho _{zx}J_{x}+\rho _{zy}J_{y}+\rho _{zz}J_{z}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> </msub> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>y</mi> </mrow> </msub> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>z</mi> </mrow> </msub> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}E_{x}&=\rho _{xx}J_{x}+\rho _{xy}J_{y}+\rho _{xz}J_{z},\\E_{y}&=\rho _{yx}J_{x}+\rho _{yy}J_{y}+\rho _{yz}J_{z},\\E_{z}&=\rho _{zx}J_{x}+\rho _{zy}J_{y}+\rho _{zz}J_{z}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e42012c7277af693c1358c42e53d0f059df4ce6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:29.809ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}E_{x}&=\rho _{xx}J_{x}+\rho _{xy}J_{y}+\rho _{xz}J_{z},\\E_{y}&=\rho _{yx}J_{x}+\rho _{yy}J_{y}+\rho _{yz}J_{z},\\E_{z}&=\rho _{zx}J_{x}+\rho _{zy}J_{y}+\rho _{zz}J_{z}.\end{aligned}}}"> Since the choice of the coordinate system is free, the usual convention is to simplify the expression by choosing an x-axis parallel to the current direction, so Jy = Jz = 0. This leaves: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{xx}={\frac {E_{x}}{J_{x}}},\quad \rho _{yx}={\frac {E_{y}}{J_{x}}},{\text{ and }}\rho _{zx}={\frac {E_{z}}{J_{x}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{xx}={\frac {E_{x}}{J_{x}}},\quad \rho _{yx}={\frac {E_{y}}{J_{x}}},{\text{ and }}\rho _{zx}={\frac {E_{z}}{J_{x}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94818e617a0d0513dcfc4c61f0b120e2a9979fda" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:39.769ex; height:6.009ex;" alt="{\displaystyle \rho _{xx}={\frac {E_{x}}{J_{x}}},\quad \rho _{yx}={\frac {E_{y}}{J_{x}}},{\text{ and }}\rho _{zx}={\frac {E_{z}}{J_{x}}}.}"> Conductivity is defined similarly:<a href="#cite_note-7">[7]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}J_{x}\\J_{y}\\J_{z}\end{bmatrix}}={\begin{bmatrix}\sigma _{xx}&\sigma _{xy}&\sigma _{xz}\\\sigma _{yx}&\sigma _{yy}&\sigma _{yz}\\\sigma _{zx}&\sigma _{zy}&\sigma _{zz}\end{bmatrix}}{\begin{bmatrix}E_{x}\\E_{y}\\E_{z}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}J_{x}\\J_{y}\\J_{z}\end{bmatrix}}={\begin{bmatrix}\sigma _{xx}&\sigma _{xy}&\sigma _{xz}\\\sigma _{yx}&\sigma _{yy}&\sigma _{yz}\\\sigma _{zx}&\sigma _{zy}&\sigma _{zz}\end{bmatrix}}{\begin{bmatrix}E_{x}\\E_{y}\\E_{z}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58f4bb548949d6794df2852920064b82b1154ad4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:34.677ex; height:9.843ex;" alt="{\displaystyle {\begin{bmatrix}J_{x}\\J_{y}\\J_{z}\end{bmatrix}}={\begin{bmatrix}\sigma _{xx}&\sigma _{xy}&\sigma _{xz}\\\sigma _{yx}&\sigma _{yy}&\sigma _{yz}\\\sigma _{zx}&\sigma _{zy}&\sigma _{zz}\end{bmatrix}}{\begin{bmatrix}E_{x}\\E_{y}\\E_{z}\end{bmatrix}}}"> or <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {J} _{i}={\boldsymbol {\sigma }}_{ij}\mathbf {E} _{j},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J} _{i}={\boldsymbol {\sigma }}_{ij}\mathbf {E} _{j},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/123dda673da289c871754a895b90c9f8e3b84655" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.664ex; height:2.843ex;" alt="{\displaystyle \mathbf {J} _{i}={\boldsymbol {\sigma }}_{ij}\mathbf {E} _{j},}"> both resulting in: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}J_{x}&=\sigma _{xx}E_{x}+\sigma _{xy}E_{y}+\sigma _{xz}E_{z}\\J_{y}&=\sigma _{yx}E_{x}+\sigma _{yy}E_{y}+\sigma _{yz}E_{z}\\J_{z}&=\sigma _{zx}E_{x}+\sigma _{zy}E_{y}+\sigma _{zz}E_{z}\end{aligned}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>z</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>x</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>y</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> <mi>z</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}J_{x}&=\sigma _{xx}E_{x}+\sigma _{xy}E_{y}+\sigma _{xz}E_{z}\\J_{y}&=\sigma _{yx}E_{x}+\sigma _{yy}E_{y}+\sigma _{yz}E_{z}\\J_{z}&=\sigma _{zx}E_{x}+\sigma _{zy}E_{y}+\sigma _{zz}E_{z}\end{aligned}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b0866241e2d9824e7732cbb944d7a3ddc99d372" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:31.036ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}J_{x}&=\sigma _{xx}E_{x}+\sigma _{xy}E_{y}+\sigma _{xz}E_{z}\\J_{y}&=\sigma _{yx}E_{x}+\sigma _{yy}E_{y}+\sigma _{yz}E_{z}\\J_{z}&=\sigma _{zx}E_{x}+\sigma _{zy}E_{y}+\sigma _{zz}E_{z}\end{aligned}}.}"> Looking at the two expressions, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\rho }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ρ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\rho }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa514c733a0add8e7c3af8bf4f930fa918d16ff8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.423ex; height:2.009ex;" alt="{\displaystyle {\boldsymbol {\rho }}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\sigma }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">σ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\sigma }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e45fe1b9d8dcbc3103fc7805d69798bfe5ca5b16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.594ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\sigma }}}"> are the <a href="/wiki/Invertible_matrix" title="Invertible matrix">matrix inverse</a> of each other. However, in the most general case, the individual matrix elements are not necessarily reciprocals of one another; for example, σxx may not be equal to 1/ρxx. This can be seen in the <a href="/wiki/Hall_effect" title="Hall effect">Hall effect</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{xy}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{xy}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e18e4804b339fd6e032bf29ef36b8fcf7a4bf6a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.191ex; height:2.343ex;" alt="{\displaystyle \rho _{xy}}"> is nonzero. In the Hall effect, due to rotational invariance about the z-axis, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{yy}=\rho _{xx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{yy}=\rho _{xx}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bf3a2b7aa80432abcca0fff7c3f958a07a1d2b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.481ex; height:2.343ex;" alt="{\displaystyle \rho _{yy}=\rho _{xx}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{yx}=-\rho _{xy}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{yx}=-\rho _{xy}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3143edf93368ddac19330ef2b05c87523f9ea7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.29ex; height:2.676ex;" alt="{\displaystyle \rho _{yx}=-\rho _{xy}}">, so the relation between resistivity and conductivity simplifies to:<a href="#cite_note-8">[8]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{xx}={\frac {\rho _{xx}}{\rho _{xx}^{2}+\rho _{xy}^{2}}},\quad \sigma _{xy}={\frac {-\rho _{xy}}{\rho _{xx}^{2}+\rho _{xy}^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mrow> <msubsup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mrow> <mrow> <msubsup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma _{xx}={\frac {\rho _{xx}}{\rho _{xx}^{2}+\rho _{xy}^{2}}},\quad \sigma _{xy}={\frac {-\rho _{xy}}{\rho _{xx}^{2}+\rho _{xy}^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43fa5327951e98b32522358570016d7f67ceeb34" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:37.323ex; height:6.343ex;" alt="{\displaystyle \sigma _{xx}={\frac {\rho _{xx}}{\rho _{xx}^{2}+\rho _{xy}^{2}}},\quad \sigma _{xy}={\frac {-\rho _{xy}}{\rho _{xx}^{2}+\rho _{xy}^{2}}}.}"> If the electric field is parallel to the applied current, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{xy}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{xy}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e18e4804b339fd6e032bf29ef36b8fcf7a4bf6a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.191ex; height:2.343ex;" alt="{\displaystyle \rho _{xy}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{xz}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>z</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{xz}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fe91269766520f7b1e7d591c0f36f86a5a8ddf7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.144ex; height:2.176ex;" alt="{\displaystyle \rho _{xz}}"> are zero. When they are zero, one number, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{xx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{xx}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/906f269fe33416459c066d1e0182c8fef80e8aa8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.315ex; height:2.176ex;" alt="{\displaystyle \rho _{xx}}">, is enough to describe the electrical resistivity. It is then written as simply <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }">, and this reduces to the simpler expression. <div class="mw-heading mw-heading2"><h2 id="Conductivity_and_current_carriers">Conductivity and current carriers</h2>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=5" title="Edit section: Conductivity and current carriers">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Relation_between_current_density_and_electric_current_velocity">Relation between current density and electric current velocity</h3>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=6" title="Edit section: Relation between current density and electric current velocity">edit</a>]</div> Electric current is the ordered movement of <a href="/wiki/Electric_charge" title="Electric charge">electric charges</a>.<a href="#cite_note-:0-2">[2]</a> <div class="mw-heading mw-heading2"><h2 id="Causes_of_conductivity">Causes of conductivity</h2>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=7" title="Edit section: Causes of conductivity">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Band_theory_simplified">Band theory simplified</h3>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=8" title="Edit section: Band theory simplified">edit</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/Band_theory" class="mw-redirect" title="Band theory">Band theory</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Band_filling_diagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Band_filling_diagram.svg/400px-Band_filling_diagram.svg.png" decoding="async" width="400" height="195" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Band_filling_diagram.svg/600px-Band_filling_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9d/Band_filling_diagram.svg/800px-Band_filling_diagram.svg.png 2x" data-file-width="351" data-file-height="171" /></a><figcaption>Filling of the electronic states in various types of materials at <a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">equilibrium</a>. Here, height is energy while width is the <a href="/wiki/Density_of_states" title="Density of states">density of available states</a> for a certain energy in the material listed. The shade follows the <a href="/wiki/Fermi%E2%80%93Dirac_statistics" title="Fermi–Dirac statistics">Fermi–Dirac distribution</a> (black: all states filled, white: no state filled). In <a href="/wiki/Metal" title="Metal">metals</a> and <a href="/wiki/Semimetal" title="Semimetal">semimetals</a> the <a href="/wiki/Fermi_level" title="Fermi level">Fermi level</a> EF lies inside at least one band.<div class="paragraphbreak" style="margin-top:0.5em"></div> In <a href="/wiki/Insulator_(electricity)" title="Insulator (electricity)">insulators</a> and <a href="/wiki/Semiconductor" title="Semiconductor">semiconductors</a> the Fermi level is inside a <a href="/wiki/Band_gap" title="Band gap">band gap</a>; however, in semiconductors the bands are near enough to the Fermi level to be <a href="/wiki/Fermi%E2%80%93Dirac_statistics" title="Fermi–Dirac statistics">thermally populated</a> with electrons or <a href="/wiki/Electron_hole" title="Electron hole">holes</a>. "intrin." indicates <a href="/wiki/Intrinsic_semiconductor" title="Intrinsic semiconductor">intrinsic semiconductors</a>. <div class="editsection noprint editlink plainlinks" align="right" style="float: right; margin-left: 5px;"><a class="external text" href="https://en.wikipedia.org/w/index.php?title=Template:Band_structure_filling_diagram&action=edit">edit</a></div></figcaption></figure> According to elementary <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, an electron in an atom or crystal can only have certain precise energy levels; energies between these levels are impossible. When a large number of such allowed levels have close-spaced energy values – i.e. have energies that differ only minutely – those close energy levels in combination are called an "energy band". There can be many such energy bands in a material, depending on the atomic number of the constituent atoms<a href="#cite_note-9">[a]</a> and their distribution within the crystal.<a href="#cite_note-10">[b]</a> The material's electrons seek to minimize the total energy in the material by settling into low energy states; however, the <a href="/wiki/Pauli_exclusion_principle" title="Pauli exclusion principle">Pauli exclusion principle</a> means that only one can exist in each such state. So the electrons "fill up" the band structure starting from the bottom. The characteristic energy level up to which the electrons have filled is called the <a href="/wiki/Fermi_level" title="Fermi level">Fermi level</a>. The position of the Fermi level with respect to the band structure is very important for electrical conduction: Only electrons in energy levels near or above the <a href="/wiki/Fermi_level" title="Fermi level">Fermi level</a> are free to move within the broader material structure, since the electrons can easily jump among the partially occupied states in that region. In contrast, the low energy states are completely filled with a fixed limit on the number of electrons at all times, and the high energy states are empty of electrons at all times. Electric current consists of a flow of electrons. In metals there are many electron energy levels near the Fermi level, so there are many electrons available to move. This is what causes the high electronic conductivity of metals. An important part of band theory is that there may be forbidden bands of energy: energy intervals that contain no energy levels. In insulators and semiconductors, the number of electrons is just the right amount to fill a certain integer number of low energy bands, exactly to the boundary. In this case, the Fermi level falls within a band gap. Since there are no available states near the Fermi level, and the electrons are not freely movable, the electronic conductivity is very low. <div class="mw-heading mw-heading3"><h3 id="In_metals">In metals</h3>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=9" title="Edit section: In metals">edit</a>]</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/Free_electron_model" title="Free electron model">Free electron model</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Newtons_cradle_animation.ogv/240px--Newtons_cradle_animation.ogv.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="240" height="135" data-durationhint="26" data-mwtitle="Newtons_cradle_animation.ogv" data-mwprovider="wikimediacommons" resource="/wiki/File:Newtons_cradle_animation.ogv"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/13/Newtons_cradle_animation.ogv/Newtons_cradle_animation.ogv.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="854" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/13/Newtons_cradle_animation.ogv/Newtons_cradle_animation.ogv.720p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="720p.vp9.webm" data-width="1280" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/1/13/Newtons_cradle_animation.ogv" type="video/ogg; codecs="theora"" data-width="1280" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/13/Newtons_cradle_animation.ogv/Newtons_cradle_animation.ogv.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="256" data-height="144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/13/Newtons_cradle_animation.ogv/Newtons_cradle_animation.ogv.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="426" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/13/Newtons_cradle_animation.ogv/Newtons_cradle_animation.ogv.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="640" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/1/13/Newtons_cradle_animation.ogv/Newtons_cradle_animation.ogv.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="640" data-height="360" /></video><figcaption>An animation of <a href="/wiki/Newton%27s_cradle" title="Newton's cradle">Newton's cradle</a>. Like the balls in Newton's cradle, electrons in a metal quickly transfer energy from one terminal to another, despite their own negligible movement.</figcaption></figure> A <a href="/wiki/Metal" title="Metal">metal</a> consists of a <a href="/wiki/Crystal_lattice" class="mw-redirect" title="Crystal lattice">lattice</a> of <a href="/wiki/Atom" title="Atom">atoms</a>, each with an outer shell of electrons that freely dissociate from their parent atoms and travel through the lattice. This is also known as a positive ionic lattice.<a href="#cite_note-11">[9]</a> This 'sea' of dissociable electrons allows the metal to conduct electric current. When an electrical potential difference (a <a href="/wiki/Voltage" title="Voltage">voltage</a>) is applied across the metal, the resulting electric field causes electrons to drift towards the positive terminal. The actual <a href="/wiki/Drift_velocity" title="Drift velocity">drift velocity</a> of electrons is typically small, on the order of magnitude of metres per hour. However, due to the sheer number of moving electrons, even a slow drift velocity results in a large <a href="/wiki/Current_density" title="Current density">current density</a>.<a href="#cite_note-classroom-12">[10]</a> The mechanism is similar to transfer of momentum of balls in a <a href="/wiki/Newton%27s_cradle" title="Newton's cradle">Newton's cradle</a><a href="#cite_note-13">[11]</a> but the rapid propagation of an electric energy along a wire is not due to the mechanical forces, but the propagation of an energy-carrying electromagnetic field guided by the wire. Most metals have electrical resistance. In simpler models (non quantum mechanical models) this can be explained by replacing electrons and the crystal lattice by a wave-like structure. When the electron wave travels through the lattice, the waves <a href="/wiki/Wave_interference" title="Wave interference">interfere</a>, which causes resistance. The more regular the lattice is, the less disturbance happens and thus the less resistance. The amount of resistance is thus mainly caused by two factors. First, it is caused by the temperature and thus amount of vibration of the crystal lattice. Higher temperatures cause bigger vibrations, which act as irregularities in the lattice. Second, the purity of the metal is relevant as a mixture of different ions is also an irregularity.<a href="#cite_note-14">[12]</a><a href="#cite_note-15">[13]</a> The small decrease in conductivity on melting of pure metals is due to the loss of long range crystalline order. The short range order remains and strong correlation between positions of ions results in coherence between waves diffracted by adjacent ions.<a href="#cite_note-16">[14]</a> <div class="mw-heading mw-heading3"><h3 id="In_semiconductors_and_insulators">In semiconductors and insulators</h3>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=10" title="Edit section: In semiconductors and insulators">edit</a>]</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/Semiconductor" title="Semiconductor">Semiconductor</a>, <a href="/wiki/Insulator_(electricity)" title="Insulator (electricity)">Insulator (electricity)</a>, and <a href="/wiki/Charge_carrier_density" title="Charge carrier density">Charge carrier density</a></div> In metals, the <a href="/wiki/Fermi_level" title="Fermi level">Fermi level</a> lies in the <a href="/wiki/Conduction_band" class="mw-redirect" title="Conduction band">conduction band</a> (see Band Theory, above) giving rise to free conduction electrons. However, in <a href="/wiki/Semiconductors" class="mw-redirect" title="Semiconductors">semiconductors</a> the position of the Fermi level is within the band gap, about halfway between the conduction band minimum (the bottom of the first band of unfilled electron energy levels) and the valence band maximum (the top of the band below the conduction band, of filled electron energy levels). That applies for intrinsic (undoped) semiconductors. This means that at absolute zero temperature, there would be no free conduction electrons, and the resistance is infinite. However, the resistance decreases as the <a href="/wiki/Charge_carrier_density" title="Charge carrier density">charge carrier density</a> (i.e., without introducing further complications, the density of electrons) in the conduction band increases. In extrinsic (doped) semiconductors, <a href="/wiki/Dopant" title="Dopant">dopant</a> atoms increase the majority charge carrier concentration by donating electrons to the conduction band or producing holes in the valence band. (A "hole" is a position where an electron is missing; such holes can behave in a similar way to electrons.) For both types of donor or acceptor atoms, increasing dopant density reduces resistance. Hence, highly doped semiconductors behave metallically. At very high temperatures, the contribution of thermally generated carriers dominates over the contribution from dopant atoms, and the resistance decreases exponentially with temperature. <div class="mw-heading mw-heading3"><h3 id="In_ionic_liquids/electrolytes">In ionic liquids/electrolytes</h3>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=11" title="Edit section: In ionic liquids/electrolytes">edit</a>]</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/Conductivity_(electrolytic)" title="Conductivity (electrolytic)">Conductivity (electrolytic)</a></div> In <a href="/wiki/Electrolyte" title="Electrolyte">electrolytes</a>, electrical conduction happens not by band electrons or holes, but by full atomic species (<a href="/wiki/Ion" title="Ion">ions</a>) traveling, each carrying an electrical charge. The resistivity of ionic solutions (electrolytes) varies tremendously with concentration – while distilled water is almost an insulator, <a href="/wiki/Saline_water" title="Saline water">salt water</a> is a reasonable electrical conductor. Conduction in <a href="/wiki/Ionic_liquid" title="Ionic liquid">ionic liquids</a> is also controlled by the movement of ions, but here we are talking about molten salts rather than solvated ions. In <a href="/wiki/Cell_membrane" title="Cell membrane">biological membranes</a>, currents are carried by ionic salts. Small holes in cell membranes, called <a href="/wiki/Ion_channel" title="Ion channel">ion channels</a>, are selective to specific ions and determine the membrane resistance. The concentration of ions in a liquid (e.g., in an aqueous solution) depends on the degree of dissociation of the dissolved substance, characterized by a dissociation coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }">, which is the ratio of the concentration of ions <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}"> to the concentration of molecules of the dissolved substance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b6328fbe0cded37216c90735c89ee188be26a30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.92ex; height:2.509ex;" alt="{\displaystyle N_{0}}">: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=\alpha N_{0}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mi>α</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=\alpha N_{0}~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/482687777c37f6a8c84f1fafb4827a872e4b98b1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.798ex; height:2.509ex;" alt="{\displaystyle N=\alpha N_{0}~.}"> The specific electrical conductivity (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" 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 \sigma }">) of a solution is equal to: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =q\left(b^{+}+b^{-}\right)\alpha N_{0}~,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <mi>q</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>α</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =q\left(b^{+}+b^{-}\right)\alpha N_{0}~,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8306656c252e1d9787b37f7cf11b1ec8c2f53f60" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.894ex; height:3.176ex;" alt="{\displaystyle \sigma =q\left(b^{+}+b^{-}\right)\alpha N_{0}~,}"> 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}">: module of the ion charge, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{+}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f74596292a463a86aaf05d61249084ef137bc746" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.508ex; height:2.509ex;" alt="{\displaystyle b^{+}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{-}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3081f709261a0e3f93991f77e2b36b8813a45614" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.508ex; height:2.509ex;" alt="{\displaystyle b^{-}}">: mobility of positively and negatively charged ions, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b6328fbe0cded37216c90735c89ee188be26a30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.92ex; height:2.509ex;" alt="{\displaystyle N_{0}}">: concentration of molecules of the dissolved substance, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }">: the coefficient of dissociation. <div class="mw-heading mw-heading3"><h3 id="Superconductivity">Superconductivity</h3>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=12" title="Edit section: Superconductivity">edit</a>]</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/Superconductivity" title="Superconductivity">Superconductivity</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Superconductivity_1911.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Superconductivity_1911.gif/220px-Superconductivity_1911.gif" decoding="async" width="220" height="291" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Superconductivity_1911.gif/330px-Superconductivity_1911.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Superconductivity_1911.gif/440px-Superconductivity_1911.gif 2x" data-file-width="648" data-file-height="856" /></a><figcaption>Original data from the 1911 experiment by <a href="/wiki/Heike_Kamerlingh_Onnes" title="Heike Kamerlingh Onnes">Heike Kamerlingh Onnes</a> showing the resistance of a mercury wire as a function of temperature. The abrupt drop in resistance is the superconducting transition.</figcaption></figure> The electrical resistivity of a metallic conductor decreases gradually as temperature is lowered. In normal (that is, non-superconducting) conductors, such as <a href="/wiki/Copper" title="Copper">copper</a> or <a href="/wiki/Silver" title="Silver">silver</a>, this decrease is limited by impurities and other defects. Even near <a href="/wiki/Absolute_zero" title="Absolute zero">absolute zero</a>, a real sample of a normal conductor shows some resistance. In a superconductor, the resistance drops abruptly to zero when the material is cooled below its critical temperature. In a normal conductor, the current is driven by a voltage gradient, whereas in a superconductor, there is no voltage gradient and the current is instead related to the phase gradient of the superconducting order parameter.<a href="#cite_note-17">[15]</a> A consequence of this is that an electric current flowing in a loop of <a href="/wiki/Superconducting_wire" title="Superconducting wire">superconducting wire</a> can persist indefinitely with no power source.<a href="#cite_note-Gallop-18">[16]</a> In a class of superconductors known as <a href="/wiki/Type_II_superconductor" class="mw-redirect" title="Type II superconductor">type II superconductors</a>, including all known <a href="/wiki/High-temperature_superconductor" class="mw-redirect" title="High-temperature superconductor">high-temperature superconductors</a>, an extremely low but nonzero resistivity appears at temperatures not too far below the nominal superconducting transition when an electric current is applied in conjunction with a strong magnetic field, which may be caused by the electric current. This is due to the motion of <a href="/wiki/Abrikosov_vortex" title="Abrikosov vortex">magnetic vortices</a> in the electronic superfluid, which dissipates some of the energy carried by the current. The resistance due to this effect is tiny compared with that of non-superconducting materials, but must be taken into account in sensitive experiments. However, as the temperature decreases far enough below the nominal superconducting transition, these vortices can become frozen so that the resistance of the material becomes truly zero. <div class="mw-heading mw-heading3"><h3 id="Plasma">Plasma</h3>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=13" title="Edit section: Plasma">edit</a>]</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/Plasma_(physics)" title="Plasma (physics)">Plasma (physics)</a></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Lightning_over_Oradea_Romania_3.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Lightning_over_Oradea_Romania_3.jpg/220px-Lightning_over_Oradea_Romania_3.jpg" decoding="async" width="220" height="340" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Lightning_over_Oradea_Romania_3.jpg/330px-Lightning_over_Oradea_Romania_3.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/13/Lightning_over_Oradea_Romania_3.jpg/440px-Lightning_over_Oradea_Romania_3.jpg 2x" data-file-width="1935" data-file-height="2989" /></a><figcaption><a href="/wiki/Lightning" title="Lightning">Lightning</a> is an example of plasma present at Earth's surface. Typically, lightning discharges 30,000 amperes at up to 100 million volts, and emits light, radio waves, and X-rays.<a href="#cite_note-19">[17]</a> Plasma temperatures in lightning might approach 30,000 kelvin (29,727 °C) (53,540 °F), and electron densities may exceed 1024 m−3.</figcaption></figure> Plasmas are very good conductors and electric potentials play an important role. The potential as it exists on average in the space between charged particles, independent of the question of how it can be measured, is called the plasma potential, or space potential. If an electrode is inserted into a plasma, its potential generally lies considerably below the plasma potential, due to what is termed a <a href="/wiki/Debye_sheath" title="Debye sheath">Debye sheath</a>. The good electrical conductivity of plasmas makes their electric fields very small. This results in the important concept of quasineutrality, which says the density of negative charges is approximately equal to the density of positive charges over large volumes of the plasma (ne = ⟨Z⟩ > ni), but on the scale of the <a href="/wiki/Debye_length" title="Debye length">Debye length</a> there can be charge imbalance. In the special case that <a href="/wiki/Double_layer_(plasma)" class="mw-redirect" title="Double layer (plasma)">double layers</a> are formed, the charge separation can extend some tens of Debye lengths. The magnitude of the potentials and electric fields must be determined by means other than simply finding the net <a href="/wiki/Charge_density" title="Charge density">charge density</a>. A common example is to assume that the electrons satisfy the <a href="/wiki/Boltzmann_relation" title="Boltzmann relation">Boltzmann relation</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{\text{e}}\propto \exp \left(e\Phi /k_{\text{B}}T_{\text{e}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> <mo>∝</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>e</mi> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{\text{e}}\propto \exp \left(e\Phi /k_{\text{B}}T_{\text{e}}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e7705413ef93a3de6484043f0a221002d8bad33" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.315ex; height:2.843ex;" alt="{\displaystyle n_{\text{e}}\propto \exp \left(e\Phi /k_{\text{B}}T_{\text{e}}\right).}"> Differentiating this relation provides a means to calculate the electric field from the density: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} =-{\frac {k_{\text{B}}T_{\text{e}}}{e}}{\frac {\nabla n_{\text{e}}}{n_{\text{e}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> </mrow> <mi>e</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∇</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> </mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} =-{\frac {k_{\text{B}}T_{\text{e}}}{e}}{\frac {\nabla n_{\text{e}}}{n_{\text{e}}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a27bb13360c187bc4f1055e380cf5dcadd0ef3f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:18.202ex; height:5.676ex;" alt="{\displaystyle \mathbf {E} =-{\frac {k_{\text{B}}T_{\text{e}}}{e}}{\frac {\nabla n_{\text{e}}}{n_{\text{e}}}}.}"> (∇ is the vector gradient operator; see <a href="/wiki/Nabla_symbol" title="Nabla symbol">nabla symbol</a> and <a href="/wiki/Gradient" title="Gradient">gradient</a> for more information.) It is possible to produce a plasma that is not quasineutral. An electron beam, for example, has only negative charges. The density of a non-neutral plasma must generally be very low, or it must be very small. Otherwise, the repulsive <a href="/wiki/Electrostatic_force" class="mw-redirect" title="Electrostatic force">electrostatic force</a> dissipates it. In <a href="/wiki/Astrophysical" class="mw-redirect" title="Astrophysical">astrophysical</a> plasmas, <a href="/wiki/Electric_field_screening" class="mw-redirect" title="Electric field screening">Debye screening</a> prevents electric fields from directly affecting the plasma over large distances, i.e., greater than the <a href="/wiki/Debye_length" title="Debye length">Debye length</a>. However, the existence of charged particles causes the plasma to generate, and be affected by, <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic fields</a>. This can and does cause extremely complex behavior, such as the generation of plasma double layers, an object that separates charge over a few tens of <a href="/wiki/Debye_length" title="Debye length">Debye lengths</a>. The dynamics of plasmas interacting with external and self-generated magnetic fields are studied in the academic discipline of <a href="/wiki/Magnetohydrodynamics" title="Magnetohydrodynamics">magnetohydrodynamics</a>. Plasma is often called the fourth <a href="/wiki/State_of_matter" title="State of matter">state of matter</a> after solid, liquids and gases.<a href="#cite_note-20">[18]</a><a href="#cite_note-21">[19]</a> It is distinct from these and other lower-energy <a href="/wiki/States_of_matter" class="mw-redirect" title="States of matter">states of matter</a>. Although it is closely related to the gas phase in that it also has no definite form or volume, it differs in a number of ways, including the following: <table class="wikitable"> <tbody><tr> <th>Property</th> <th>Gas</th> <th>Plasma </th></tr> <tr> <th>Electrical conductivity </th> <td>Very low: air is an excellent insulator until it breaks down into plasma at electric field strengths above 30 kilovolts per centimetre.<a href="#cite_note-22">[20]</a> </td> <td>Usually very high: for many purposes, the conductivity of a plasma may be treated as infinite. </td></tr> <tr> <th>Independently acting species </th> <td>One: all gas particles behave in a similar way, influenced by <a href="/wiki/Gravity" title="Gravity">gravity</a> and by <a href="/wiki/Collision" title="Collision">collisions</a> with one another. </td> <td>Two or three: <a href="/wiki/Electron" title="Electron">electrons</a>, <a href="/wiki/Ion" title="Ion">ions</a>, <a href="/wiki/Proton" title="Proton">protons</a> and <a href="/wiki/Neutron" title="Neutron">neutrons</a> can be distinguished by the sign and value of their <a href="/wiki/Electric_charge" title="Electric charge">charge</a> so that they behave independently in many circumstances, with different bulk velocities and temperatures, allowing phenomena such as new types of <a href="/wiki/Waves_in_plasma" class="mw-redirect" title="Waves in plasma">waves</a> and <a href="/wiki/Instability" title="Instability">instabilities</a>. </td></tr> <tr> <th>Velocity distribution </th> <td><a href="/wiki/Maxwell%E2%80%93Boltzmann_distribution" title="Maxwell–Boltzmann distribution">Maxwellian</a>: collisions usually lead to a Maxwellian velocity distribution of all gas particles, with very few relatively fast particles. </td> <td>Often non-Maxwellian: collisional interactions are often weak in hot plasmas and external forcing can drive the plasma far from local equilibrium and lead to a significant population of unusually fast particles. </td></tr> <tr> <th>Interactions </th> <td>Binary: two-particle collisions are the rule, three-body collisions extremely rare. </td> <td>Collective: waves, or organized motion of plasma, are very important because the particles can interact at long ranges through the electric and magnetic forces. </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Resistivity_and_conductivity_of_various_materials">Resistivity and conductivity of various materials</h2>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=14" title="Edit section: Resistivity and conductivity of various materials">edit</a>]</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/Electrical_resistivities_of_the_elements_(data_page)" title="Electrical resistivities of the elements (data page)">Electrical resistivities of the elements (data page)</a></div> <ul><li>A conductor such as a metal has high conductivity and a low resistivity.</li> <li>An <a href="/wiki/Electrical_insulation" class="mw-redirect" title="Electrical insulation">insulator</a> such as <a href="/wiki/Glass" title="Glass">glass</a> has low conductivity and a high resistivity.</li> <li>The conductivity of a <a href="/wiki/Semiconductor" title="Semiconductor">semiconductor</a> is generally intermediate, but varies widely under different conditions, such as exposure of the material to electric fields or specific frequencies of <a href="/wiki/Light" title="Light">light</a>, and, most important, with <a href="/wiki/Temperature" title="Temperature">temperature</a> and composition of the semiconductor material.</li></ul> The degree of <a href="/wiki/Doping_(semiconductor)" title="Doping (semiconductor)">semiconductors doping</a> makes a large difference in conductivity. To a point, more doping leads to higher conductivity. The conductivity of a <a href="/wiki/Water_(molecule)" class="mw-redirect" title="Water (molecule)">water</a>/<a href="/wiki/Aqueous" class="mw-redirect" title="Aqueous">aqueous</a> <a href="/wiki/Solution_(chemistry)" title="Solution (chemistry)">solution</a> is highly dependent on its <a href="/wiki/Concentration" title="Concentration">concentration</a> of dissolved <a href="/wiki/Salts" class="mw-redirect" title="Salts">salts</a>, and other chemical species that <a href="/wiki/Ionization" title="Ionization">ionize</a> in the solution. Electrical conductivity of water samples is used as an indicator of how salt-free, ion-free, or impurity-free the sample is; the purer the water, the lower the conductivity (the higher the resistivity). Conductivity measurements in water are often reported as specific conductance, relative to the conductivity of pure water at 25 °C. An <a href="/wiki/EC_meter" class="mw-redirect" title="EC meter">EC meter</a> is normally used to measure conductivity in a solution. A rough summary is as follows: <table class="wikitable plainrowheaders"> <caption>Resistivity of classes of materials </caption> <tbody><tr> <th scope="col">Material </th> <th scope="col">Resistivity, ρ (Ω·m) </th></tr> <tr> <th scope="row"><a href="/wiki/Superconductors" class="mw-redirect" title="Superconductors">Superconductors</a> </th> <td>0 </td></tr> <tr> <th scope="row"><a href="/wiki/Metal" title="Metal">Metals</a> </th> <td>10−8 </td></tr> <tr> <th scope="row"><a href="/wiki/Semiconductor" title="Semiconductor">Semiconductors</a> </th> <td>Variable </td></tr> <tr> <th scope="row"><a href="/wiki/Electrolyte" title="Electrolyte">Electrolytes</a> </th> <td>Variable </td></tr> <tr> <th scope="row"><a href="/wiki/Electrical_insulation" class="mw-redirect" title="Electrical insulation">Insulators</a> </th> <td>1016 </td></tr> <tr> <th scope="row"><a href="/wiki/Superinsulator" title="Superinsulator">Superinsulators</a> </th> <td>∞ </td></tr></tbody></table> This table shows the resistivity (ρ), conductivity and <a href="/wiki/Temperature_coefficient" title="Temperature coefficient">temperature coefficient</a> of various materials at 20 °C (68 °F; 293 K). <table class="wikitable sortable plainrowheaders"> <caption>Resistivity, conductivity, and temperature coefficient for several materials </caption> <tbody><tr> <th scope="col">Material </th> <th scope="col" data-sort-type="number">Resistivity, ρ, at 20 °C (Ω·m) </th> <th scope="col" data-sort-type="number">Conductivity, σ, at 20 °C (S/m) </th> <th scope="col" data-sort-type="number">Temperature coefficient<a href="#cite_note-23">[c]</a> (K−1) </th> <th scope="col" class="unsortable">Reference </th></tr> <tr> <th scope="row"><a href="/wiki/Silver" title="Silver">Silver</a><a href="#cite_note-24">[d]</a> </th> <td data-sort-value="0.0000000159">1.59×10−8</td> <td data-sort-value="6.30E7">63.0×106</td> <td data-sort-value="3.80E-3">3.80×10−3</td> <td><a href="#cite_note-serway-25">[21]</a><a href="#cite_note-Griffiths-26">[22]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Copper" title="Copper">Copper</a><a href="#cite_note-27">[e]</a> </th> <td data-sort-value="0.0000000168">1.68×10−8</td> <td data-sort-value="5.96E7">59.6×106</td> <td data-sort-value="4.04E-3">4.04×10−3</td> <td><a href="#cite_note-28">[23]</a><a href="#cite_note-Giancoli-29">[24]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Annealing_(metallurgy)" class="mw-redirect" title="Annealing (metallurgy)">Annealed</a> <a href="/wiki/Copper" title="Copper">copper</a><a href="#cite_note-30">[f]</a> </th> <td data-sort-value="0.0000000172">1.72×10−8</td> <td data-sort-value="5.80E7">58.0×106</td> <td data-sort-value="3.93E-3">3.93×10−3</td> <td><a href="#cite_note-31">[25]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Gold" title="Gold">Gold</a><a href="#cite_note-32">[g]</a> </th> <td data-sort-value="2.44E-8">2.44×10−8</td> <td data-sort-value="4.11E7">41.1×106</td> <td data-sort-value="3.40E-3">3.40×10−3</td> <td><a href="#cite_note-serway-25">[21]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Aluminium" title="Aluminium">Aluminium</a><a href="#cite_note-33">[h]</a> </th> <td data-sort-value="0.0000000265">2.65×10−8</td> <td data-sort-value="3.8E7">37.7×106</td> <td data-sort-value="3.90E-3">3.90×10−3</td> <td><a href="#cite_note-serway-25">[21]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Brass" title="Brass">Brass (5% Zn)</a> </th> <td data-sort-value="0.00000003">3.00×10−8</td> <td data-sort-value="3.34E7">33.4×106</td> <td></td> <td><a href="#cite_note-34">[26]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Calcium" title="Calcium">Calcium</a> </th> <td data-sort-value="0.0000000336">3.36×10−8</td> <td data-sort-value="2.98E7">29.8×106</td> <td data-sort-value="4.10E-3">4.10×10−3</td> <td> </td></tr> <tr> <th scope="row"><a href="/wiki/Rhodium" title="Rhodium">Rhodium</a> </th> <td data-sort-value="0.0000000433">4.33×10−8</td> <td data-sort-value="2.31E7">23.1×106</td> <td></td> <td> </td></tr> <tr> <th scope="row"><a href="/wiki/Tungsten" title="Tungsten">Tungsten</a> </th> <td data-sort-value="0.0000000560">5.60×10−8</td> <td data-sort-value="1.79E7">17.9×106</td> <td data-sort-value="4.50E-3">4.50×10−3</td> <td><a href="#cite_note-serway-25">[21]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Zinc" title="Zinc">Zinc</a> </th> <td data-sort-value="0.0000000590">5.90×10−8</td> <td data-sort-value="1.69E7">16.9×106</td> <td data-sort-value="3.70E-3">3.70×10−3</td> <td><a href="#cite_note-35">[27]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Brass" title="Brass">Brass (30% Zn)</a> </th> <td data-sort-value="0.0000000599">5.99×10−8</td> <td data-sort-value="1.67E7">16.7×106</td> <td></td> <td><a href="#cite_note-36">[28]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Cobalt" title="Cobalt">Cobalt</a><a href="#cite_note-ic-38">[i]</a> </th> <td data-sort-value="0.0000000624">6.24×10−8</td> <td data-sort-value="1.60E7">16.0×106</td> <td data-sort-value="7.00E-3">7.00×10−3<a href="#cite_note-39">[30]</a> [<a href="/wiki/Wikipedia:Reliable_sources" title="Wikipedia:Reliable sources">unreliable source?</a>]</td> <td> </td></tr> <tr> <th scope="row"><a href="/wiki/Nickel" title="Nickel">Nickel</a> </th> <td data-sort-value="0.0000000699">6.99×10−8</td> <td data-sort-value="1.43E7">14.3×106</td> <td data-sort-value="6.00E-3">6.00×10−3</td> <td> </td></tr> <tr> <th scope="row"><a href="/wiki/Ruthenium" title="Ruthenium">Ruthenium</a><a href="#cite_note-ic-38">[i]</a> </th> <td data-sort-value="0.0000000710">7.10×10−8</td> <td data-sort-value="1.41E7">14.1×106</td> <td></td> <td> </td></tr> <tr> <th scope="row"><a href="/wiki/Lithium" title="Lithium">Lithium</a> </th> <td data-sort-value="0.0000000928">9.28×10−8</td> <td data-sort-value="1.08E7">10.8×106</td> <td data-sort-value="6.00E-3">6.00×10−3</td> <td> </td></tr> <tr> <th scope="row"><a href="/wiki/Iron" title="Iron">Iron</a> </th> <td data-sort-value="0.000000097">9.70×10−8</td> <td data-sort-value="1.03E7">10.3×106</td> <td data-sort-value="5.00E-3">5.00×10−3</td> <td><a href="#cite_note-serway-25">[21]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Platinum" title="Platinum">Platinum</a> </th> <td data-sort-value="0.000000106">10.6×10−8</td> <td data-sort-value="9.43E6">9.43×106</td> <td data-sort-value="3.92E-3">3.92×10−3</td> <td><a href="#cite_note-serway-25">[21]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Tin" title="Tin">Tin</a> </th> <td data-sort-value="0.000000109">10.9×10−8</td> <td data-sort-value="9.17E6">9.17×106</td> <td data-sort-value="4.50E-3">4.50×10−3</td> <td> </td></tr> <tr> <th scope="row"><a href="/wiki/Phosphor_bronze" title="Phosphor bronze">Phosphor Bronze (0.2% P / 5% Sn)</a> </th> <td data-sort-value="0.000000112">11.2×10−8</td> <td data-sort-value="8.94E6">8.94×106</td> <td></td> <td><a href="#cite_note-40">[31]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Gallium" title="Gallium">Gallium</a> </th> <td data-sort-value="0.000000140">14.0×10−8</td> <td data-sort-value="7.10E6">7.10×106</td> <td data-sort-value="4.00E-3">4.00×10−3</td> <td> </td></tr> <tr> <th scope="row"><a href="/wiki/Niobium" title="Niobium">Niobium</a> </th> <td data-sort-value="0.000000140">14.0×10−8</td> <td data-sort-value="7.00E6">7.00×106</td> <td></td> <td><a href="#cite_note-41">[32]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Carbon_steel" title="Carbon steel">Carbon steel</a> (1010) </th> <td data-sort-value="0.000000143">14.3×10−8</td> <td data-sort-value="6.99E6">6.99×106</td> <td></td> <td><a href="#cite_note-42">[33]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Lead" title="Lead">Lead</a> </th> <td data-sort-value="0.000000220">22.0×10−8</td> <td data-sort-value="4.55E6">4.55×106</td> <td data-sort-value="3.90E-3">3.90×10−3</td> <td><a href="#cite_note-serway-25">[21]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Galinstan" title="Galinstan">Galinstan</a> </th> <td data-sort-value="28.9E-8">28.9×10−8</td> <td data-sort-value="3.46E6">3.46×106</td> <td><a href="#cite_note-43">[34]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Titanium" title="Titanium">Titanium</a> </th> <td data-sort-value="0.000000420">42.0×10−8</td> <td data-sort-value="2.38E6">2.38×106</td> <td>3.80×10−3</td> <td> </td></tr> <tr> <th scope="row">Grain oriented <a href="/wiki/Electrical_steel" title="Electrical steel">electrical steel</a> </th> <td data-sort-value="0.000000460">46.0×10−8</td> <td data-sort-value="2.17E6">2.17×106</td> <td></td> <td><a href="#cite_note-44">[35]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Manganin" title="Manganin">Manganin</a> </th> <td data-sort-value="0.000000482">48.2×10−8</td> <td data-sort-value="2.07E6">2.07×106</td> <td data-sort-value="0.002E-3">0.002×10−3</td> <td><a href="#cite_note-giancoli-45">[36]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Constantan" title="Constantan">Constantan</a> </th> <td data-sort-value="0.000000490">49.0×10−8</td> <td data-sort-value="2.04E6">2.04×106</td> <td data-sort-value="0.008E-3">0.008×10−3</td> <td><a href="#cite_note-46">[37]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Stainless_steel" title="Stainless steel">Stainless steel</a><a href="#cite_note-47">[j]</a> </th> <td data-sort-value="0.000000690">69.0×10−8</td> <td data-sort-value="1.45E6">1.45×106</td> <td data-sort-value="0.94E-3">0.94×10−3</td> <td><a href="#cite_note-48">[38]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Mercury_(element)" title="Mercury (element)">Mercury</a> </th> <td data-sort-value="0.000000980">98.0×10−8</td> <td data-sort-value="1.02E6">1.02×106</td> <td data-sort-value="0.90E-3">0.90×10−3</td> <td><a href="#cite_note-giancoli-45">[36]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Bismuth" title="Bismuth">Bismuth</a> </th> <td data-sort-value="0.00000129">129×10−8</td> <td data-sort-value="7.75E5">7.75×105</td> <td></td> <td> </td></tr> <tr> <th scope="row"><a href="/wiki/Manganese" title="Manganese">Manganese</a> </th> <td data-sort-value="0.00000144">144×10−8</td> <td data-sort-value="6.94E5">6.94×105</td> <td></td> <td> </td></tr> <tr> <th scope="row"><a href="/wiki/Plutonium" title="Plutonium">Plutonium</a><a href="#cite_note-49">[39]</a> (0 °C) </th> <td data-sort-value="0.00000146">146×10−8</td> <td data-sort-value="6.85E5">6.85×105</td> <td></td> <td> </td></tr> <tr> <th scope="row"><a href="/wiki/Nichrome" title="Nichrome">Nichrome</a><a href="#cite_note-50">[k]</a> </th> <td data-sort-value="0.0000011">110×10−8</td> <td data-sort-value="6.70E5">6.70×105 [<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>]</td> <td data-sort-value="0.40E-3">0.40×10−3</td> <td><a href="#cite_note-serway-25">[21]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Graphite" title="Graphite">Carbon (graphite)</a> parallel to <a href="/wiki/Basal_plane" class="mw-redirect" title="Basal plane">basal plane</a><a href="#cite_note-51">[l]</a> </th> <td data-sort-value="2.5E-6">250×10−8 to 500×10−8</td> <td data-sort-value="2E5">2×105 to 3×105 [<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>]</td> <td></td> <td><a href="#cite_note-Pierson-4">[4]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Amorphous_carbon" title="Amorphous carbon">Carbon (amorphous)</a> </th> <td data-sort-value="5E-4">0.5×10−3 to 0.8×10−3</td> <td data-sort-value="1.25E3">1.25×103 to 2.00×103</td> <td data-sort-value="-0.50E-3">−0.50×10−3</td> <td><a href="#cite_note-serway-25">[21]</a><a href="#cite_note-52">[40]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Graphite" title="Graphite">Carbon (graphite)</a> perpendicular to basal plane </th> <td data-sort-value="3E-3">3.0×10−3</td> <td data-sort-value="3.3E2">3.3×102</td> <td></td> <td><a href="#cite_note-Pierson-4">[4]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/GaAs" class="mw-redirect" title="GaAs">GaAs</a> </th> <td data-sort-value="0.001">10−3 to 108 [<a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify">clarification needed</a>]</td> <td data-sort-value="1E-8">10−8 to 103 [<a href="/wiki/Wikipedia:Accuracy_dispute#Disputed_statement" title="Wikipedia:Accuracy dispute">dubious</a> – <a href="/wiki/Talk:Electrical_resistivity_and_conductivity#Dubious" title="Talk:Electrical resistivity and conductivity">discuss</a>]</td> <td></td> <td><a href="#cite_note-Ohring-53">[41]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Germanium" title="Germanium">Germanium</a><a href="#cite_note-semi-54">[m]</a> </th> <td data-sort-value="4.6E-1">4.6×10−1</td> <td>2.17</td> <td data-sort-value="-48.0E-3">−48.0×10−3</td> <td><a href="#cite_note-serway-25">[21]</a><a href="#cite_note-Griffiths-26">[22]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Sea_water" class="mw-redirect" title="Sea water">Sea water</a><a href="#cite_note-55">[n]</a> </th> <td data-sort-value="2E-1">2.1×10−1</td> <td data-sort-value="4.8">4.8</td> <td></td> <td><a href="#cite_note-56">[42]</a> </td></tr> <tr> <th scope="row">Swimming pool water<a href="#cite_note-57">[o]</a> </th> <td data-sort-value="4E-1">3.3×10−1 to 4.0×10−1</td> <td data-sort-value="0.25">0.25 to 0.30</td> <td></td> <td><a href="#cite_note-58">[43]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Drinking_water" title="Drinking water">Drinking water</a><a href="#cite_note-59">[p]</a> </th> <td data-sort-value="2E1">2×101 to 2×103</td> <td data-sort-value="5E-4">5×10−4 to 5×10−2</td> <td></td> <td>[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] </td></tr> <tr> <th scope="row"><a href="/wiki/Bone" title="Bone">Bone</a> </th> <td data-sort-value="1.66E2">1.66×102</td> <td data-sort-value="6E-3">6×10−3</td> <td></td> <td><a href="#cite_note-60">[44]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Silicon" title="Silicon">Silicon</a><a href="#cite_note-semi-54">[m]</a> </th> <td data-sort-value="2.3E3">2.3×103</td> <td data-sort-value="4.35E-4">4.35×10−4</td> <td data-sort-value="-75.0E-3">−75.0×10−3</td> <td><a href="#cite_note-Eranna2014-61">[45]</a><a href="#cite_note-serway-25">[21]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Wood" title="Wood">Wood</a> (damp) </th> <td data-sort-value="1E3">103 to 104</td> <td data-sort-value="1E-4">10−4 to 10−3</td> <td></td> <td><a href="#cite_note-Transmission_Lines_data-62">[46]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Deionized_water" class="mw-redirect" title="Deionized water">Deionized water</a><a href="#cite_note-63">[q]</a> </th> <td data-sort-value="1.8E5">1.8×105</td> <td data-sort-value="4.2E-5">4.2×10−5</td> <td></td> <td><a href="#cite_note-64">[47]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Ultrapure_water#Conductivity/resistivity" title="Ultrapure water">Ultrapure water</a> </th> <td data-sort-value="1.82E9">1.82×109</td> <td data-sort-value="5.49E-10">5.49×10−10</td> <td></td> <td><a href="#cite_note-65">[48]</a><a href="#cite_note-66">[49]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Glass" title="Glass">Glass</a> </th> <td data-sort-value="1E11">1011 to 1015</td> <td data-sort-value="1E-11">10−15 to 10−11</td> <td></td> <td><a href="#cite_note-serway-25">[21]</a><a href="#cite_note-Griffiths-26">[22]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Diamond" title="Diamond">Carbon (diamond)</a> </th> <td data-sort-value="1E12">1012</td> <td data-sort-value="1E-13">~10−13</td> <td></td> <td><a href="#cite_note-67">[50]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Hard_rubber" class="mw-redirect" title="Hard rubber">Hard rubber</a> </th> <td data-sort-value="1E13">1013</td> <td data-sort-value="1E-14">10−14</td> <td></td> <td><a href="#cite_note-serway-25">[21]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Air" class="mw-redirect" title="Air">Air</a> </th> <td data-sort-value="1E9">109 to 1015</td> <td data-sort-value="1E-15">~10−15 to 10−9</td> <td></td> <td><a href="#cite_note-68">[51]</a><a href="#cite_note-69">[52]</a> </td></tr> <tr> <th scope="row">Wood (oven dry) </th> <td data-sort-value="1E14">1014 to 1016</td> <td data-sort-value="1E-16">10−16 to 10−14</td> <td></td> <td><a href="#cite_note-Transmission_Lines_data-62">[46]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Sulfur" title="Sulfur">Sulfur</a> </th> <td data-sort-value="1E15">1015</td> <td data-sort-value="1E-16">10−16</td> <td></td> <td><a href="#cite_note-serway-25">[21]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Fused_quartz" title="Fused quartz">Fused quartz</a> </th> <td data-sort-value="7.5E17">7.5×1017</td> <td data-sort-value="1.3E-18">1.3×10−18</td> <td></td> <td><a href="#cite_note-serway-25">[21]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Polyethylene_terephthalate" title="Polyethylene terephthalate">PET</a> </th> <td data-sort-value="1E21">1021</td> <td data-sort-value="1E-21">10−21</td> <td></td> <td> </td></tr> <tr> <th scope="row"><a href="/wiki/PTFE" class="mw-redirect" title="PTFE">PTFE</a> (teflon) </th> <td data-sort-value="1E23">1023 to 1025</td> <td data-sort-value="1E-25">10−25 to 10−23</td> <td></td> <td> </td></tr></tbody></table> The effective temperature coefficient varies with temperature and purity level of the material. The 20 °C value is only an approximation when used at other temperatures. For example, the coefficient becomes lower at higher temperatures for copper, and the value 0.00427 is commonly specified at 0 °C.<a href="#cite_note-70">[53]</a> The extremely low resistivity (high conductivity) of silver is characteristic of metals. <a href="/wiki/George_Gamow" title="George Gamow">George Gamow</a> tidily summed up the nature of the metals' dealings with electrons in his popular science book One, Two, Three...Infinity (1947): <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"> The metallic substances differ from all other materials by the fact that the outer shells of their atoms are bound rather loosely, and often let one of their electrons go free. Thus the interior of a metal is filled up with a large number of unattached electrons that travel aimlessly around like a crowd of displaced persons. When a metal wire is subjected to electric force applied on its opposite ends, these free electrons rush in the direction of the force, thus forming what we call an electric current.</blockquote> More technically, the <a href="/wiki/Free_electron_model" title="Free electron model">free electron model</a> gives a basic description of electron flow in metals. Wood is widely regarded as an extremely good insulator, but its resistivity is sensitively dependent on moisture content, with damp wood being a factor of at least 1010 worse insulator than oven-dry.<a href="#cite_note-Transmission_Lines_data-62">[46]</a> In any case, a sufficiently high voltage – such as that in lightning strikes or some high-tension power lines – can lead to insulation breakdown and electrocution risk even with apparently dry wood.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <div class="mw-heading mw-heading2"><h2 id="Temperature_dependence">Temperature dependence</h2>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=15" title="Edit section: Temperature dependence">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Linear_approximation">Linear approximation</h3>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=16" title="Edit section: Linear approximation">edit</a>]</div> The electrical resistivity of most materials changes with temperature. If the temperature T does not vary too much, a <a href="/wiki/Linear_approximation" title="Linear approximation">linear approximation</a> is typically used: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (T)=\rho _{0}[1+\alpha (T-T_{0})],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">[</mo> <mn>1</mn> <mo>+</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>−</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (T)=\rho _{0}[1+\alpha (T-T_{0})],}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4124cbfa1149d902f84b2b13f1ccbad6c188fad3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.131ex; height:2.843ex;" alt="{\displaystyle \rho (T)=\rho _{0}[1+\alpha (T-T_{0})],}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"> is called the <a href="/wiki/Temperature_coefficient" title="Temperature coefficient">temperature coefficient</a> of resistivity, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55b9e7d7b96196b5a6a26f4349caa3ac82fd67e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{0}}"> is a fixed reference temperature (usually room temperature), and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9c04a9d26b86af8c6205ba2a6287fd655b6b714" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.256ex; height:2.176ex;" alt="{\displaystyle \rho _{0}}"> is the resistivity at temperature <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55b9e7d7b96196b5a6a26f4349caa3ac82fd67e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{0}}">. The parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"> is an empirical parameter fitted from measurement data, equal to 1/<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }">[<a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify">clarify</a>]. Because the linear approximation is only an approximation, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"> is different for different reference temperatures. For this reason it is usual to specify the temperature that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"> was measured at with a suffix, such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{15}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>15</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{15}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e93f2a04a064beebf70cf0756e0cb1c8fb5a1bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.364ex; height:2.009ex;" alt="{\displaystyle \alpha _{15}}">, and the relationship only holds in a range of temperatures around the reference.<a href="#cite_note-71">[54]</a> When the temperature varies over a large temperature range, the <a href="/wiki/Linear_approximation" title="Linear approximation">linear approximation</a> is inadequate and a more detailed analysis and understanding should be used. <div class="mw-heading mw-heading3"><h3 id="Metals">Metals</h3>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=17" title="Edit section: Metals">edit</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/Bloch%E2%80%93Gr%C3%BCneisen_temperature" title="Bloch–Grüneisen temperature">Bloch–Grüneisen temperature</a> and <a href="/wiki/Free_electron_model#Mean_free_dependence_of_the_resistivity_of_gold,_copper_and_silver." title="Free electron model">Free electron model § Mean free dependence of the resistivity of gold, copper and silver.</a></div> In general, electrical resistivity of metals increases with temperature. Electron–<a href="/wiki/Phonon" title="Phonon">phonon</a> interactions can play a key role. At high temperatures, the resistance of a metal increases linearly with temperature. As the temperature of a metal is reduced, the temperature dependence of resistivity follows a power law function of temperature. Mathematically the temperature dependence of the resistivity ρ of a metal can be approximated through the Bloch–Grüneisen formula:<a href="#cite_note-72">[55]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (T)=\rho (0)+A\left({\frac {T}{\Theta _{R}}}\right)^{n}\int _{0}^{\Theta _{R}/T}{\frac {x^{n}}{(e^{x}-1)(1-e^{-x})}}\,dx,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>A</mi> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>T</mi> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>T</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (T)=\rho (0)+A\left({\frac {T}{\Theta _{R}}}\right)^{n}\int _{0}^{\Theta _{R}/T}{\frac {x^{n}}{(e^{x}-1)(1-e^{-x})}}\,dx,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c76a521c47ed5bba624b65304e097b0d7f8a62d1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:55.002ex; height:6.676ex;" alt="{\displaystyle \rho (T)=\rho (0)+A\left({\frac {T}{\Theta _{R}}}\right)^{n}\int _{0}^{\Theta _{R}/T}{\frac {x^{n}}{(e^{x}-1)(1-e^{-x})}}\,dx,}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd3300b3c092a939f5e977152e2b2b66cd9f7401" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.174ex; height:2.843ex;" alt="{\displaystyle \rho (0)}"> is the residual resistivity due to defect scattering, A is a constant that depends on the velocity of electrons at the <a href="/wiki/Fermi_surface" title="Fermi surface">Fermi surface</a>, the <a href="/wiki/Debye_radius" class="mw-redirect" title="Debye radius">Debye radius</a> and the <a href="/wiki/Number_density" title="Number density">number density</a> of electrons in the metal. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Theta _{R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Theta _{R}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87306cb04677c97ea1e4f7e58596ba585e2184ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.288ex; height:2.509ex;" alt="{\displaystyle \Theta _{R}}"> is the <a href="/wiki/Debye_temperature" class="mw-redirect" title="Debye temperature">Debye temperature</a> as obtained from resistivity measurements and matches very closely with the values of Debye temperature obtained from specific heat measurements. n is an integer that depends upon the nature of interaction: <ul><li>n = 5 implies that the resistance is due to scattering of electrons by phonons (as it is for simple metals)</li> <li>n = 3 implies that the resistance is due to s-d electron scattering (as is the case for transition metals)</li> <li>n = 2 implies that the resistance is due to electron–electron interaction.</li></ul> The Bloch–Grüneisen formula is an approximation obtained assuming that the studied metal has spherical Fermi surface inscribed within the first <a href="/wiki/Brillouin_zone" title="Brillouin zone">Brillouin zone</a> and a <a href="/wiki/Debye_model" title="Debye model">Debye phonon spectrum</a>.<a href="#cite_note-73">[56]</a> If more than one source of scattering is simultaneously present, Matthiessen's rule (first formulated by <a href="/wiki/Augustus_Matthiessen" title="Augustus Matthiessen">Augustus Matthiessen</a> in the 1860s)<a href="#cite_note-74">[57]</a><a href="#cite_note-75">[58]</a> states that the total resistance can be approximated by adding up several different terms, each with the appropriate value of n. As the temperature of the metal is sufficiently reduced (so as to 'freeze' all the phonons), the resistivity usually reaches a constant value, known as the residual resistivity. This value depends not only on the type of metal, but on its purity and thermal history. The value of the residual resistivity of a metal is decided by its impurity concentration. Some materials lose all electrical resistivity at sufficiently low temperatures, due to an effect known as <a href="/wiki/Superconductivity" title="Superconductivity">superconductivity</a>. An investigation of the low-temperature resistivity of metals was the motivation to <a href="/wiki/Heike_Kamerlingh_Onnes" title="Heike Kamerlingh Onnes">Heike Kamerlingh Onnes</a>'s experiments that led in 1911 to discovery of <a href="/wiki/Superconductivity" title="Superconductivity">superconductivity</a>. For details see <a href="/wiki/History_of_superconductivity" title="History of superconductivity">History of superconductivity</a>. <div class="mw-heading mw-heading4"><h4 id="Wiedemann–Franz_law">Wiedemann–Franz law</h4>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=18" title="Edit section: Wiedemann–Franz law">edit</a>]</div> The <a href="/wiki/Wiedemann%E2%80%93Franz_law" title="Wiedemann–Franz law">Wiedemann–Franz law</a> states that for materials where heat and charge transport is dominated by electrons, the ratio of thermal to electrical conductivity is proportional to the temperature: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\kappa \over \sigma }={\pi ^{2} \over 3}\left({\frac {k}{e}}\right)^{2}T,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>κ</mi> <mi>σ</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mn>3</mn> </mfrac> </mrow> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>e</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>T</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\kappa \over \sigma }={\pi ^{2} \over 3}\left({\frac {k}{e}}\right)^{2}T,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2546553f5d76a07c5bc2692d85911e9e92390558" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.304ex; height:6.509ex;" alt="{\displaystyle {\kappa \over \sigma }={\pi ^{2} \over 3}\left({\frac {k}{e}}\right)^{2}T,}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }"> is the <a href="/wiki/Thermal_conductivity" class="mw-redirect" title="Thermal conductivity">thermal conductivity</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> is the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"> is the electron charge, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> is temperature, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" 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 \sigma }"> is the <a href="/wiki/Electric_conductivity" class="mw-redirect" title="Electric conductivity">electric conductivity</a>. The ratio on the rhs is called the Lorenz number. <div class="mw-heading mw-heading3"><h3 id="Semiconductors">Semiconductors</h3>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=19" title="Edit section: Semiconductors">edit</a>]</div> In general, <a href="/wiki/Intrinsic_semiconductor" title="Intrinsic semiconductor">intrinsic semiconductor</a> resistivity decreases with increasing temperature. The electrons are bumped to the <a href="/wiki/Conduction_band" class="mw-redirect" title="Conduction band">conduction energy band</a> by thermal energy, where they flow freely, and in doing so leave behind <a href="/wiki/Electron_hole" title="Electron hole">holes</a> in the <a href="/wiki/Valence_band" class="mw-redirect" title="Valence band">valence band</a>, which also flow freely. The electric resistance of a typical <a href="/wiki/Intrinsic_semiconductor" title="Intrinsic semiconductor">intrinsic</a> (non doped) <a href="/wiki/Semiconductor" title="Semiconductor">semiconductor</a> decreases <a href="/wiki/Exponential_decay" title="Exponential decay">exponentially</a> with temperature following an <a href="/wiki/Arrhenius_equation" title="Arrhenius equation">Arrhenius model</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =\rho _{0}e^{\frac {E_{A}}{k_{B}T}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =\rho _{0}e^{\frac {E_{A}}{k_{B}T}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95145e1a4a2967525d383f4bffa885bb121dcb54" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.136ex; height:5.009ex;" alt="{\displaystyle \rho =\rho _{0}e^{\frac {E_{A}}{k_{B}T}}.}"> An even better approximation of the temperature dependence of the resistivity of a semiconductor is given by the <a href="/wiki/Steinhart%E2%80%93Hart_equation" title="Steinhart–Hart equation">Steinhart–Hart equation</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{T}}=A+B\ln \rho +C(\ln \rho )^{3},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <mo>=</mo> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mi>ln</mi> <mo>⁡</mo> <mi>ρ</mi> <mo>+</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi>ρ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{T}}=A+B\ln \rho +C(\ln \rho )^{3},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/715f2402f00087ae2e9a61385f81a6f01aea3c73" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.479ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{T}}=A+B\ln \rho +C(\ln \rho )^{3},}"> where A, B and C are the so-called Steinhart–Hart coefficients. This equation is used to calibrate <a href="/wiki/Thermistor" title="Thermistor">thermistors</a>. <a href="/wiki/Extrinsic_semiconductor" title="Extrinsic semiconductor">Extrinsic (doped) semiconductors</a> have a far more complicated temperature profile. As temperature increases starting from absolute zero they first decrease steeply in resistance as the carriers leave the donors or acceptors. After most of the donors or acceptors have lost their carriers, the resistance starts to increase again slightly due to the reducing mobility of carriers (much as in a metal). At higher temperatures, they behave like intrinsic semiconductors as the carriers from the donors/acceptors become insignificant compared to the thermally generated carriers.<a href="#cite_note-76">[59]</a> In non-crystalline semiconductors, conduction can occur by charges <a href="/wiki/Quantum_tunnelling" title="Quantum tunnelling">quantum tunnelling</a> from one localised site to another. This is known as <a href="/wiki/Variable_range_hopping" class="mw-redirect" title="Variable range hopping">variable range hopping</a> and has the characteristic form of <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho =A\exp \left(T^{-1/n}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <mi>A</mi> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho =A\exp \left(T^{-1/n}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/713c5cd099ee7be1e00fdcfdfae9f18a4588d5a6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.266ex; height:4.843ex;" alt="{\displaystyle \rho =A\exp \left(T^{-1/n}\right),}"> where n = 2, 3, 4, depending on the dimensionality of the system. <div class="mw-heading mw-heading3"><h3 id="Kondo_insulators">Kondo insulators</h3>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=20" title="Edit section: Kondo insulators">edit</a>]</div> <a href="/wiki/Kondo_insulator" title="Kondo insulator">Kondo insulators</a> are materials where the resistivity follows the formula <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (T)=\rho _{0}+aT^{2}+bT^{5}+c_{m}\ln {\frac {\mu }{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>a</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>μ</mi> <mi>T</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (T)=\rho _{0}+aT^{2}+bT^{5}+c_{m}\ln {\frac {\mu }{T}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bb8f9ff0ead241749498f1f33b8cc248021de06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.167ex; height:4.843ex;" alt="{\displaystyle \rho (T)=\rho _{0}+aT^{2}+bT^{5}+c_{m}\ln {\frac {\mu }{T}}}"></dd></dl> where <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}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{m}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5a92f980a7ccf6827b6925c6d6421984d9c5859" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.682ex; height:2.009ex;" alt="{\displaystyle c_{m}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"> are constant parameters, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9c04a9d26b86af8c6205ba2a6287fd655b6b714" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.256ex; height:2.176ex;" alt="{\displaystyle \rho _{0}}"> the residual resistivity, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f632936d25e5207ab3ad0e6182882ea3c6a9c7ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.774ex; height:2.676ex;" alt="{\displaystyle T^{2}}"> the <a href="/wiki/Fermi_liquid" class="mw-redirect" title="Fermi liquid">Fermi liquid</a> contribution, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T^{5}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5faa08e3daf65c0aee7c5587ed7968076ee0342c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.774ex; height:2.676ex;" alt="{\displaystyle T^{5}}"> a lattice vibrations term and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ln {\frac {1}{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ln</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ln {\frac {1}{T}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11b9e701b351b7f87c725979734d94e3ee7d841f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:4.799ex; height:5.176ex;" alt="{\displaystyle \ln {\frac {1}{T}}}"> the <a href="/wiki/Kondo_effect" title="Kondo effect">Kondo effect</a>. <div class="mw-heading mw-heading2"><h2 id="Complex_resistivity_and_conductivity">Complex resistivity and conductivity</h2>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=21" title="Edit section: Complex resistivity and conductivity">edit</a>]</div> When analyzing the response of materials to alternating electric fields (<a href="/wiki/Dielectric_spectroscopy" title="Dielectric spectroscopy">dielectric spectroscopy</a>),<a href="#cite_note-77">[60]</a> in applications such as <a href="/wiki/Electrical_impedance_tomography" title="Electrical impedance tomography">electrical impedance tomography</a>,<a href="#cite_note-78">[61]</a> it is convenient to replace resistivity with a <a href="/wiki/Complex_number" title="Complex number">complex</a> quantity called impedivity (in analogy to <a href="/wiki/Electrical_impedance" title="Electrical impedance">electrical impedance</a>). Impedivity is the sum of a real component, the resistivity, and an imaginary component, the reactivity (in analogy to <a href="/wiki/Reactance_(electronics)" class="mw-redirect" title="Reactance (electronics)">reactance</a>). The magnitude of impedivity is the square root of sum of squares of magnitudes of resistivity and reactivity. Conversely, in such cases the conductivity must be expressed as a <a href="/wiki/Complex_number" title="Complex number">complex number</a> (or even as a matrix of complex numbers, in the case of <a href="/wiki/Anisotropic" class="mw-redirect" title="Anisotropic">anisotropic</a> materials) called the <a href="/wiki/Admittance" title="Admittance">admittivity</a>. Admittivity is the sum of a real component called the conductivity and an imaginary component called the <a href="/wiki/Susceptance" title="Susceptance">susceptivity</a>. An alternative description of the response to alternating currents uses a real (but frequency-dependent) conductivity, along with a real <a href="/wiki/Permittivity" title="Permittivity">permittivity</a>. The larger the conductivity is, the more quickly the alternating-current signal is absorbed by the material (i.e., the more <a href="/wiki/Opacity_(optics)" class="mw-redirect" title="Opacity (optics)">opaque</a> the material is). For details, see <a href="/wiki/Mathematical_descriptions_of_opacity" title="Mathematical descriptions of opacity">Mathematical descriptions of opacity</a>. <div class="mw-heading mw-heading2"><h2 id="Resistance_versus_resistivity_in_complicated_geometries">Resistance versus resistivity in complicated geometries</h2>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=22" title="Edit section: Resistance versus resistivity in complicated geometries">edit</a>]</div> Even if the material's resistivity is known, calculating the resistance of something made from it may, in some cases, be much more complicated than the formula <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=\rho \ell /A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mi>ρ</mi> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=\rho \ell /A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5beb516669f18b2c32411e2a75e3e1c7bdc3b9db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.94ex; height:2.843ex;" alt="{\displaystyle R=\rho \ell /A}"> above. One example is <a href="/wiki/Spreading_resistance_profiling" title="Spreading resistance profiling">spreading resistance profiling</a>, where the material is inhomogeneous (different resistivity in different places), and the exact paths of current flow are not obvious. In cases like this, the formulas <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=\sigma E\,\,\rightleftharpoons \,\,E=\rho J}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <mi>σ</mi> <mi>E</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mo class="MJX-variant" stretchy="false">⇌</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mi>E</mi> <mo>=</mo> <mi>ρ</mi> <mi>J</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=\sigma E\,\,\rightleftharpoons \,\,E=\rho J}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38669da5c8873e486cbb2e7a2d52af8da1ec3fc3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.385ex; height:2.676ex;" alt="{\displaystyle J=\sigma E\,\,\rightleftharpoons \,\,E=\rho J}"> must be replaced with <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {J} (\mathbf {r} )=\sigma (\mathbf {r} )\mathbf {E} (\mathbf {r} )\,\,\rightleftharpoons \,\,\mathbf {E} (\mathbf {r} )=\rho (\mathbf {r} )\mathbf {J} (\mathbf {r} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</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 stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mo class="MJX-variant" stretchy="false">⇌</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ρ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J} (\mathbf {r} )=\sigma (\mathbf {r} )\mathbf {E} (\mathbf {r} )\,\,\rightleftharpoons \,\,\mathbf {E} (\mathbf {r} )=\rho (\mathbf {r} )\mathbf {J} (\mathbf {r} ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ace87cd6afed7cdb3ae62b212beb5fa70aae974c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.282ex; height:2.843ex;" alt="{\displaystyle \mathbf {J} (\mathbf {r} )=\sigma (\mathbf {r} )\mathbf {E} (\mathbf {r} )\,\,\rightleftharpoons \,\,\mathbf {E} (\mathbf {r} )=\rho (\mathbf {r} )\mathbf {J} (\mathbf {r} ),}"> where E and J are now <a href="/wiki/Vector_field" title="Vector field">vector fields</a>. This equation, along with the <a href="/wiki/Continuity_equation" title="Continuity equation">continuity equation</a> for J and the <a href="/wiki/Poisson%27s_equation" title="Poisson's equation">Poisson's equation</a> for E, form a set of <a href="/wiki/Partial_differential_equation" title="Partial differential equation">partial differential equations</a>. In special cases, an exact or approximate solution to these equations can be worked out by hand, but for very accurate answers in complex cases, computer methods like <a href="/wiki/Finite_element_method" title="Finite element method">finite element analysis</a> may be required. <div class="mw-heading mw-heading2"><h2 id="Resistivity-density_product">Resistivity-density product</h2>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=23" title="Edit section: Resistivity-density product">edit</a>]</div> In some applications where the weight of an item is very important, the product of resistivity and <a href="/wiki/Density" title="Density">density</a> is more important than absolute low resistivity – it is often possible to make the conductor thicker to make up for a higher resistivity; and then a low-resistivity-density-product material (or equivalently a high conductivity-to-density ratio) is desirable. For example, for long-distance <a href="/wiki/Overhead_power_line" title="Overhead power line">overhead power lines</a>, aluminium is frequently used rather than copper (Cu) because it is lighter for the same conductance. Silver, although it is the least resistive metal known, has a high density and performs similarly to copper by this measure, but is much more expensive. Calcium and the alkali metals have the best resistivity-density products, but are rarely used for conductors due to their high reactivity with water and oxygen (and lack of physical strength). Aluminium is far more stable. Toxicity excludes the choice of beryllium.<a href="#cite_note-79">[62]</a> (Pure beryllium is also brittle.) Thus, aluminium is usually the metal of choice when the weight or cost of a conductor is the driving consideration. <table class="wikitable sortable plainrowheaders"> <caption>Resistivity, density, and resistivity-density products of selected materials </caption> <tbody><tr> <th rowspan="2">Material </th> <th rowspan="2" data-sort-type="number">Resistivity (<abbr title="nanoohm metres">nΩ·m</abbr>) </th> <th rowspan="2" data-sort-type="number"><a href="/wiki/Density" title="Density">Density</a> (<abbr title="gram per cubic centimetre">g/cm3</abbr>) </th> <th colspan="2">Resistivity × density </th> <th style="width:10em;" rowspan="2">Resistivity relative to <abbr title="copper">Cu</abbr>, i.e. cross-sectional area required to give same conductance </th> <th colspan="2">Approx. price, at 9 December 2018 [<a href="/wiki/Wikipedia:Accuracy_dispute#Disputed_statement" title="Wikipedia:Accuracy dispute">dubious</a> – <a href="/wiki/Talk:Electrical_resistivity_and_conductivity#Dubious" title="Talk:Electrical resistivity and conductivity">discuss</a>] </th></tr> <tr> <th data-sort-type="number">(<abbr title="gram milliohm per square metre">g·mΩ/m2</abbr>) </th> <th data-sort-type="number">Relative to <abbr title="copper">Cu</abbr> </th> <th>(USD per kg) </th> <th>Relative to <abbr title="copper">Cu</abbr> </th></tr> <tr> <th scope="row"><a href="/wiki/Sodium" title="Sodium">Sodium</a> </th> <td>47.7 </td> <td>0.97 </td> <td>46 </td> <td>31% </td> <td>2.843 </td> <td> </td> <td> </td></tr> <tr> <th scope="row"><a href="/wiki/Lithium" title="Lithium">Lithium</a> </th> <td>92.8 </td> <td>0.53 </td> <td>49 </td> <td>33% </td> <td>5.531 </td> <td> </td> <td> </td></tr> <tr> <th scope="row"><a href="/wiki/Calcium" title="Calcium">Calcium</a> </th> <td>33.6 </td> <td>1.55 </td> <td>52 </td> <td>35% </td> <td>2.002 </td> <td> </td> <td> </td></tr> <tr> <th scope="row"><a href="/wiki/Potassium" title="Potassium">Potassium</a> </th> <td>72.0 </td> <td>0.89 </td> <td>64 </td> <td>43% </td> <td>4.291 </td> <td> </td> <td> </td></tr> <tr> <th scope="row"><a href="/wiki/Beryllium" title="Beryllium">Beryllium</a> </th> <td>35.6 </td> <td>1.85 </td> <td>66 </td> <td>44% </td> <td>2.122 </td> <td> </td> <td> </td></tr> <tr> <th scope="row"><a href="/wiki/Aluminium" title="Aluminium">Aluminium</a> </th> <td>26.50 </td> <td>2.70 </td> <td>72 </td> <td>48% </td> <td>1.579 </td> <td>2.0 </td> <td>0.16 </td></tr> <tr> <th scope="row"><a href="/wiki/Magnesium" title="Magnesium">Magnesium</a> </th> <td>43.90 </td> <td>1.74 </td> <td>76 </td> <td>51% </td> <td>2.616 </td> <td> </td> <td> </td></tr> <tr> <th scope="row"><a href="/wiki/Copper" title="Copper">Copper</a> </th> <td>16.78 </td> <td>8.96 </td> <td>150 </td> <td style="background:#b87333;">100% </td> <td style="background:#b87333;">1 </td> <td>6.0 </td> <td style="background:#b87333;">1 </td></tr> <tr> <th scope="row"><a href="/wiki/Silver" title="Silver">Silver</a> </th> <td>15.87 </td> <td>10.49 </td> <td>166 </td> <td>111% </td> <td>0.946 </td> <td>456 </td> <td>84 </td></tr> <tr> <th scope="row"><a href="/wiki/Gold" title="Gold">Gold</a> </th> <td>22.14 </td> <td>19.30 </td> <td>427 </td> <td>285% </td> <td>1.319 </td> <td>39,000 </td> <td>19,000 </td></tr> <tr> <th scope="row"><a href="/wiki/Iron" title="Iron">Iron</a> </th> <td>96.1 </td> <td>7.874 </td> <td>757 </td> <td>505% </td> <td>5.727 </td> <td> </td> <td> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=24" title="Edit section: History">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="John_Walsh_and_the_conductivity_of_a_vacuum">John Walsh and the conductivity of a vacuum</h3>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=25" title="Edit section: John Walsh and the conductivity of a vacuum">edit</a>]</div> In a 1774 letter to Dutch-born British scientist <a href="/wiki/Jan_Ingenhousz" title="Jan Ingenhousz">Jan Ingenhousz</a>, <a href="/wiki/Benjamin_Franklin" title="Benjamin Franklin">Benjamin Franklin</a> relates an experiment by another British scientist, <a href="/wiki/John_Walsh_(scientist)" title="John Walsh (scientist)">John Walsh</a>, that purportedly showed this astonishing fact: Although rarified air conducts electricity better than common air, a vacuum does not conduct electricity at all.<a href="#cite_note-:1-80">[63]</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote">Mr. Walsh ... has just made a curious Discovery in Electricity. You know we find that in rarify’d Air it would pass more freely, and leap thro’ greater Spaces than in dense Air; and thence it was concluded that in a perfect Vacuum it would pass any distance without the least Obstruction. But having made a perfect Vacuum by means of boil’d Mercury in a long Torricellian bent Tube, its Ends immers’d in Cups full of Mercury, he finds that the Vacuum will not conduct at all, but resists the Passage of the Electric Fluid absolutely.</blockquote> However, to this statement a note (based on modern knowledge) was added by the editors—at the American Philosophical Society and Yale University—of the webpage hosting the letter:<a href="#cite_note-:1-80">[63]</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1244412712"><blockquote class="templatequote">We can only assume that something was wrong with Walsh’s findings. ... Although the conductivity of a gas, as it approaches a vacuum, increases up to a point and then decreases, that point is far beyond what the technique described might have been expected to reach. Boiling replaced the air with mercury vapor, which as it cooled created a vacuum that could scarcely have been complete enough to decrease, let alone eliminate, the vapor’s conductivity.</blockquote> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=26" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col"> <ul><li><a href="/wiki/Charge_transport_mechanisms" title="Charge transport mechanisms">Charge transport mechanisms</a></li> <li><a href="/wiki/Chemiresistor" title="Chemiresistor">Chemiresistor</a></li> <li><a href="/wiki/Permittivity#Classification_of_materials" title="Permittivity">Classification of materials based on permittivity</a></li> <li><a href="/wiki/Conductivity_near_the_percolation_threshold" title="Conductivity near the percolation threshold">Conductivity near the percolation threshold</a></li> <li><a href="/wiki/Contact_resistance" title="Contact resistance">Contact resistance</a></li> <li><a href="/wiki/Electrical_resistivities_of_the_elements_(data_page)" title="Electrical resistivities of the elements (data page)">Electrical resistivities of the elements (data page)</a></li> <li><a href="/wiki/Electrical_resistivity_tomography" title="Electrical resistivity tomography">Electrical resistivity tomography</a></li> <li><a href="/wiki/Sheet_resistance" title="Sheet resistance">Sheet resistance</a></li> <li><a href="/wiki/SI_electromagnetism_units" class="mw-redirect" title="SI electromagnetism units">SI electromagnetism units</a></li> <li><a href="/wiki/Skin_effect" title="Skin effect">Skin effect</a></li> <li><a href="/wiki/Spitzer_resistivity" title="Spitzer resistivity">Spitzer resistivity</a></li> <li><a href="/wiki/Dielectric_strength" title="Dielectric strength">Dielectric strength</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=27" title="Edit section: Notes">edit</a>]</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-lower-alpha"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-9"><a href="#cite_ref-9">^</a> The atomic number is the count of electrons in an atom that is electrically neutral – has no net electric charge. </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> Other relevant factors that are specifically not considered are the size of the whole crystal and external factors of the surrounding environment that modify the energy bands, such as imposed electric or magnetic fields. </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> The numbers in this column increase or decrease the <a href="/wiki/Significand" title="Significand">significand</a> portion of the resistivity. For example, at 30 °C (303 K), the resistivity of silver is 1.65×10−8. This is calculated as Δρ = α ΔT ρ0 where ρ0 is the resistivity at 20 °C (in this case) and α is the temperature coefficient. </li> <li id="cite_note-24"><a href="#cite_ref-24">^</a> The conductivity of metallic silver is not significantly better than metallic copper for most practical purposes – the difference between the two can be easily compensated for by thickening the copper wire by only 3%. However silver is preferred for exposed electrical contact points because corroded silver is a tolerable conductor, but corroded copper is a fairly good insulator, like most corroded metals. </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> Copper is widely used in electrical equipment, building wiring, and telecommunication cables. </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> Referred to as 100% IACS or International Annealed Copper Standard. The unit for expressing the conductivity of nonmagnetic materials by testing using the <a href="/wiki/Eddy_current" title="Eddy current">eddy current</a> method. Generally used for temper and alloy verification of aluminium. </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> Despite being less conductive than copper, gold is commonly used in <a href="/wiki/Electrical_contacts" class="mw-redirect" title="Electrical contacts">electrical contacts</a> because it does not easily corrode. </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> Commonly used for <a href="/wiki/Overhead_power_line" title="Overhead power line">overhead power line</a> with steel reinforced <a href="/wiki/Aluminium_conductor_steel-reinforced_cable" class="mw-redirect" title="Aluminium conductor steel-reinforced cable">(ACSR)</a> </li> <li id="cite_note-ic-38">^ <a href="#cite_ref-ic_38-0">a</a> <a href="#cite_ref-ic_38-1">b</a> <a href="/wiki/Cobalt" title="Cobalt">Cobalt</a> and <a href="/wiki/Ruthenium" title="Ruthenium">ruthenium</a> are considered to replace <a href="/wiki/Copper" title="Copper">copper</a> in <a href="/wiki/Integrated_circuits" class="mw-redirect" title="Integrated circuits">integrated circuits</a> fabricated in advanced nodes<a href="#cite_note-37">[29]</a> </li> <li id="cite_note-47"><a href="#cite_ref-47">^</a> 18% chromium and 8% nickel austenitic stainless steel </li> <li id="cite_note-50"><a href="#cite_ref-50">^</a> Nickel-iron-chromium alloy commonly used in heating elements. </li> <li id="cite_note-51"><a href="#cite_ref-51">^</a> Graphite is strongly anisotropic. </li> <li id="cite_note-semi-54">^ <a href="#cite_ref-semi_54-0">a</a> <a href="#cite_ref-semi_54-1">b</a> The resistivity of <a href="/wiki/Semiconductor" title="Semiconductor">semiconductors</a> depends strongly on the presence of <a href="/wiki/Impurity" class="mw-redirect" title="Impurity">impurities</a> in the material. </li> <li id="cite_note-55"><a href="#cite_ref-55">^</a> Corresponds to an average salinity of 35 g/kg at 20 °C. </li> <li id="cite_note-57"><a href="#cite_ref-57">^</a> The pH should be around 8.4 and the conductivity in the range of 2.5–3 mS/cm. The lower value is appropriate for freshly prepared water. The conductivity is used for the determination of TDS (total dissolved particles). </li> <li id="cite_note-59"><a href="#cite_ref-59">^</a> This value range is typical of high quality drinking water and not an indicator of water quality </li> <li id="cite_note-63"><a href="#cite_ref-63">^</a> Conductivity is lowest with monatomic gases present; changes to 12×10−5 upon complete de-gassing, or to 7.5×10−5 upon equilibration to the atmosphere due to dissolved CO2 </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=28" title="Edit section: References">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <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="CITEREFLowrie2007" class="citation book cs1">Lowrie, William (2007). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=h2-NjUg4RtEC&pg=PA254">Fundamentals of Geophysics</a>. Cambridge University Press. pp. 254–55. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-05-2185-902-8" title="Special:BookSources/978-05-2185-902-8"><bdi>978-05-2185-902-8</bdi></a>. Retrieved March 24, 2019.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Geophysics&rft.pages=254-55&rft.pub=Cambridge+University+Press&rft.date=2007&rft.isbn=978-05-2185-902-8&rft.aulast=Lowrie&rft.aufirst=William&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dh2-NjUg4RtEC%26pg%3DPA254&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrical+resistivity+and+conductivity" class="Z3988"> </li> <li id="cite_note-:0-2">^ <a href="#cite_ref-:0_2-0">a</a> <a href="#cite_ref-:0_2-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKumar2003" class="citation book cs1">Kumar, Narinder (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=IryMtwHHngIC&pg=PA282">Comprehensive Physics for Class XII</a>. New Delhi: Laxmi Publications. pp. 280–84. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-81-7008-592-8" title="Special:BookSources/978-81-7008-592-8"><bdi>978-81-7008-592-8</bdi></a>. 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Prentice Hall Professional. p. 114. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-066946-9" title="Special:BookSources/978-0-13-066946-9"><bdi>978-0-13-066946-9</bdi></a>. Retrieved March 24, 2019.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Signal+Integrity%3A+Simplified&rft.pages=114&rft.pub=Prentice+Hall+Professional&rft.date=2004&rft.isbn=978-0-13-066946-9&rft.aulast=Bogatin&rft.aufirst=Eric&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_IiONSphoB4C%26q%3DSignal%2520integrity%26pg%3DPA114&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrical+resistivity+and+conductivity" class="Z3988"> </li> <li id="cite_note-Pierson-4">^ <a href="#cite_ref-Pierson_4-0">a</a> <a href="#cite_ref-Pierson_4-1">b</a> <a href="#cite_ref-Pierson_4-2">c</a> Hugh O. Pierson, Handbook of carbon, graphite, diamond, and fullerenes: properties, processing, and applications, p. 61, William Andrew, 1993 <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/0-8155-1339-9" title="Special:BookSources/0-8155-1339-9">0-8155-1339-9</a>. </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> J.R. Tyldesley (1975) An introduction to Tensor Analysis: For Engineers and Applied Scientists, Longman, <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/0-582-44355-5" title="Special:BookSources/0-582-44355-5">0-582-44355-5</a> </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> G. 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Geophysical Journal International. 128 (3): 505–521. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1365-246X.1997.tb05314.x">10.1111/j.1365-246X.1997.tb05314.x</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Geophysical+Journal+International&rft.atitle=Finite-difference+modelling+of+magnetotelluric+fields+in+two-dimensional+anisotropic+media&rft.volume=128&rft.issue=3&rft.pages=505-521&rft.date=2007-04-03&rft_id=info%3Adoi%2F10.1111%2Fj.1365-246X.1997.tb05314.x&rft.au=Josef+Pek%2C+Tomas+Verner&rft_id=https%3A%2F%2Fdoi.org%2F10.1111%252Fj.1365-246X.1997.tb05314.x&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrical+resistivity+and+conductivity" 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="CITEREFDavid_Tong2016" class="citation web cs1">David Tong (Jan 2016). <a rel="nofollow" class="external text" href="http://www.damtp.cam.ac.uk/user/tong/qhe/qhe.pdf">"The Quantum Hall Effect: TIFR Infosys Lectures"</a> (PDF). 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John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-14704-7" title="Special:BookSources/978-0-470-14704-7"><bdi>978-0-470-14704-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electronics+All-in-One+For+Dummies&rft.pub=John+Wiley+%26+Sons&rft.date=2012&rft.isbn=978-0-470-14704-7&rft.aulast=Lowe&rft.aufirst=Doug&rft_id=http%3A%2F%2Fwww.dummies.com%2Fhow-to%2Fcontent%2Felectronics-basics-direct-and-alternating-current.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrical+resistivity+and+conductivity" class="Z3988"> </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKeith_Welch" class="citation web cs1">Keith Welch. <a rel="nofollow" class="external text" href="http://education.jlab.org/qa/current_02.html">"Questions & Answers – How do you explain electrical resistance?"</a>. <a href="/wiki/Thomas_Jefferson_National_Accelerator_Facility" title="Thomas Jefferson National Accelerator Facility">Thomas Jefferson National Accelerator Facility</a>. Retrieved 28 April 2017.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Questions+%26+Answers+%E2%80%93+How+do+you+explain+electrical+resistance%3F&rft.pub=Thomas+Jefferson+National+Accelerator+Facility&rft.au=Keith+Welch&rft_id=http%3A%2F%2Feducation.jlab.org%2Fqa%2Fcurrent_02.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrical+resistivity+and+conductivity" class="Z3988"> </li> <li id="cite_note-15"><a href="#cite_ref-15">^</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.csl.mete.metu.edu.tr/Electromigration/emig.htm">"Electromigration : What is electromigration?"</a>. Middle East Technical University. Retrieved 31 July 2017. <q>When electrons are conducted through a metal, they interact with imperfections in the lattice and scatter. […] Thermal energy produces scattering by causing atoms to vibrate. This is the source of resistance of metals.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Electromigration+%3A+What+is+electromigration%3F&rft.pub=Middle+East+Technical+University&rft_id=http%3A%2F%2Fwww.csl.mete.metu.edu.tr%2FElectromigration%2Femig.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrical+resistivity+and+conductivity" class="Z3988"> </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFaber1972" class="citation book cs1">Faber, T.E. (1972). <a rel="nofollow" class="external text" href="https://www.cambridge.org/us/academic/subjects/physics/condensed-matter-physics-nanoscience-and-mesoscopic-physics/introduction-theory-liquid-metals?format=PB">Introduction to the Theory of Liquid Metals</a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521154499" title="Special:BookSources/9780521154499"><bdi>9780521154499</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+the+Theory+of+Liquid+Metals&rft.pub=Cambridge+University+Press&rft.date=1972&rft.isbn=9780521154499&rft.aulast=Faber&rft.aufirst=T.E.&rft_id=https%3A%2F%2Fwww.cambridge.org%2Fus%2Facademic%2Fsubjects%2Fphysics%2Fcondensed-matter-physics-nanoscience-and-mesoscopic-physics%2Fintroduction-theory-liquid-metals%3Fformat%3DPB&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrical+resistivity+and+conductivity" class="Z3988"> </li> <li id="cite_note-17"><a href="#cite_ref-17">^</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://feynmanlectures.caltech.edu/III_21.html">"The Feynman Lectures in Physics, Vol. III, Chapter 21: The Schrödinger Equation in a Classical Context: A Seminar on Superconductivity"</a>. Retrieved 26 December 2021.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Feynman+Lectures+in+Physics%2C+Vol.+III%2C+Chapter+21%3A+The+Schr%C3%B6dinger+Equation+in+a+Classical+Context%3A+A+Seminar+on+Superconductivity&rft_id=https%3A%2F%2Ffeynmanlectures.caltech.edu%2FIII_21.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrical+resistivity+and+conductivity" class="Z3988"> </li> <li id="cite_note-Gallop-18"><a href="#cite_ref-Gallop_18-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_C._Gallop1990" class="citation book cs1">John C. Gallop (1990). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ad8_JsfCdKQC">SQUIDS, the Josephson Effects and Superconducting Electronics</a>. <a href="/wiki/CRC_Press" title="CRC Press">CRC Press</a>. pp. 3, 20. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7503-0051-3" title="Special:BookSources/978-0-7503-0051-3"><bdi>978-0-7503-0051-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=SQUIDS%2C+the+Josephson+Effects+and+Superconducting+Electronics&rft.pages=3%2C+20&rft.pub=CRC+Press&rft.date=1990&rft.isbn=978-0-7503-0051-3&rft.au=John+C.+Gallop&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dad8_JsfCdKQC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrical+resistivity+and+conductivity" class="Z3988"> </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> See <a rel="nofollow" class="external text" href="https://www.nasa.gov/vision/universe/solarsystem/rhessi_tgf.html">Flashes in the Sky: Earth's Gamma-Ray Bursts Triggered by Lightning</a> </li> <li id="cite_note-20"><a href="#cite_ref-20">^</a> Yaffa Eliezer, Shalom Eliezer, The Fourth State of Matter: An Introduction to the Physics of Plasma, Publisher: Adam Hilger, 1989, <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/978-0-85274-164-1" title="Special:BookSources/978-0-85274-164-1">978-0-85274-164-1</a>, 226 pages, page 5 </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBittencourt,_J.A.2004" class="citation book cs1">Bittencourt, J.A. (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=qCA64ys-5bUC&pg=PA1">Fundamentals of Plasma Physics</a>. Springer. p. 1. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387209753" title="Special:BookSources/9780387209753"><bdi>9780387209753</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Plasma+Physics&rft.pages=1&rft.pub=Springer&rft.date=2004&rft.isbn=9780387209753&rft.au=Bittencourt%2C+J.A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DqCA64ys-5bUC%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrical+resistivity+and+conductivity" class="Z3988"> </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHong2000" class="citation web cs1">Hong, Alice (2000). <a rel="nofollow" class="external text" href="http://hypertextbook.com/facts/2000/AliceHong.shtml">"Dielectric Strength of Air"</a>. The Physics Factbook.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=The+Physics+Factbook&rft.atitle=Dielectric+Strength+of+Air&rft.date=2000&rft.aulast=Hong&rft.aufirst=Alice&rft_id=http%3A%2F%2Fhypertextbook.com%2Ffacts%2F2000%2FAliceHong.shtml&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrical+resistivity+and+conductivity" class="Z3988"> </li> <li id="cite_note-serway-25">^ <a href="#cite_ref-serway_25-0">a</a> <a href="#cite_ref-serway_25-1">b</a> <a href="#cite_ref-serway_25-2">c</a> <a href="#cite_ref-serway_25-3">d</a> <a href="#cite_ref-serway_25-4">e</a> <a href="#cite_ref-serway_25-5">f</a> <a href="#cite_ref-serway_25-6">g</a> <a href="#cite_ref-serway_25-7">h</a> <a href="#cite_ref-serway_25-8">i</a> <a href="#cite_ref-serway_25-9">j</a> <a href="#cite_ref-serway_25-10">k</a> <a href="#cite_ref-serway_25-11">l</a> <a href="#cite_ref-serway_25-12">m</a> <a href="#cite_ref-serway_25-13">n</a> <a href="#cite_ref-serway_25-14">o</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRaymond_A._Serway1998" class="citation book cs1">Raymond A. Serway (1998). <a rel="nofollow" class="external text" href="https://archive.org/details/principlesofphys00serw/page/602">Principles of Physics</a> (2nd ed.). Fort Worth, Texas; London: Saunders College Pub. p. <a rel="nofollow" class="external text" href="https://archive.org/details/principlesofphys00serw/page/602">602</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-03-020457-9" title="Special:BookSources/978-0-03-020457-9"><bdi>978-0-03-020457-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Principles+of+Physics&rft.place=Fort+Worth%2C+Texas%3B+London&rft.pages=602&rft.edition=2nd&rft.pub=Saunders+College+Pub&rft.date=1998&rft.isbn=978-0-03-020457-9&rft.au=Raymond+A.+Serway&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fprinciplesofphys00serw%2Fpage%2F602&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrical+resistivity+and+conductivity" class="Z3988"> </li> <li id="cite_note-Griffiths-26">^ <a href="#cite_ref-Griffiths_26-0">a</a> <a href="#cite_ref-Griffiths_26-1">b</a> <a href="#cite_ref-Griffiths_26-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDavid_Griffiths1999" class="citation book cs1"><a href="/wiki/David_Griffiths_(physicist)" class="mw-redirect" title="David Griffiths (physicist)">David Griffiths</a> (1999) [1981]. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoel00grif_0">"7 Electrodynamics"</a>. In Alison Reeves (ed.). Introduction to Electrodynamics (3rd ed.). Upper Saddle River, New Jersey: <a href="/wiki/Prentice_Hall" title="Prentice Hall">Prentice Hall</a>. p. <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoel00grif_0/page/286">286</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-805326-0" title="Special:BookSources/978-0-13-805326-0"><bdi>978-0-13-805326-0</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/40251748">40251748</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=7+Electrodynamics&rft.btitle=Introduction+to+Electrodynamics&rft.place=Upper+Saddle+River%2C+New+Jersey&rft.pages=286&rft.edition=3rd&rft.pub=Prentice+Hall&rft.date=1999&rft_id=info%3Aoclcnum%2F40251748&rft.isbn=978-0-13-805326-0&rft.au=David+Griffiths&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoel00grif_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrical+resistivity+and+conductivity" class="Z3988"> </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMatula1979" class="citation journal cs1">Matula, R.A. (1979). <a rel="nofollow" class="external text" href="https://srd.nist.gov/JPCRD/jpcrd155.pdf">"Electrical resistivity of copper, gold, palladium, and silver"</a> (PDF). Journal of Physical and Chemical Reference Data. 8 (4): 1147. <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/1979JPCRD...8.1147M">1979JPCRD...8.1147M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.555614">10.1063/1.555614</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:95005999">95005999</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Physical+and+Chemical+Reference+Data&rft.atitle=Electrical+resistivity+of+copper%2C+gold%2C+palladium%2C+and+silver&rft.volume=8&rft.issue=4&rft.pages=1147&rft.date=1979&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A95005999%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1063%2F1.555614&rft_id=info%3Abibcode%2F1979JPCRD...8.1147M&rft.aulast=Matula&rft.aufirst=R.A.&rft_id=https%3A%2F%2Fsrd.nist.gov%2FJPCRD%2Fjpcrd155.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrical+resistivity+and+conductivity" class="Z3988"> </li> <li id="cite_note-Giancoli-29"><a href="#cite_ref-Giancoli_29-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDouglas_Giancoli2009" class="citation book cs1">Douglas Giancoli (2009) [1984]. "25 Electric Currents and Resistance". In Jocelyn Phillips (ed.). Physics for Scientists and Engineers with Modern Physics (4th ed.). Upper Saddle River, New Jersey: <a href="/wiki/Prentice_Hall" title="Prentice Hall">Prentice Hall</a>. p. 658. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-149508-1" title="Special:BookSources/978-0-13-149508-1"><bdi>978-0-13-149508-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=25+Electric+Currents+and+Resistance&rft.btitle=Physics+for+Scientists+and+Engineers+with+Modern+Physics&rft.place=Upper+Saddle+River%2C+New+Jersey&rft.pages=658&rft.edition=4th&rft.pub=Prentice+Hall&rft.date=2009&rft.isbn=978-0-13-149508-1&rft.au=Douglas+Giancoli&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrical+resistivity+and+conductivity" class="Z3988"> </li> <li id="cite_note-31"><a href="#cite_ref-31">^</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://archive.org/details/copperwiretables31unituoft">"Copper wire tables"</a>. United States National Bureau of Standards. Retrieved 3 February 2014 – via Internet Archive - archive.org (archived 2001-03-10).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Copper+wire+tables&rft.pub=United+States+National+Bureau+of+Standards&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcopperwiretables31unituoft&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrical+resistivity+and+conductivity" class="Z3988"> </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> <a rel="nofollow" class="external autonumber" href="https://www.copper.org/applications/industrial/DesignGuide/selection/conductbrass02.html">[1]</a>. (Calculated as "56% conductivity of pure copper" (5.96E-7)). Retrieved on 2023-1-12. </li> <li id="cite_note-35"><a href="#cite_ref-35">^</a> <a rel="nofollow" class="external text" href="http://physics.mipt.ru/S_III/t">Physical constants</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20111123121944/http://physics.mipt.ru/S_III/t">Archived</a> 2011-11-23 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. (PDF format; see page 2, table in the right lower corner). Retrieved on 2011-12-17. </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> <a rel="nofollow" class="external autonumber" href="https://www.copper.org/applications/industrial/DesignGuide/selection/conductbrass02.html">[2]</a>. (Calculated as "28% conductivity of pure copper" (5.96E-7)). Retrieved on 2023-1-12. </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> <a rel="nofollow" class="external text" href="https://semiwiki.com/semiconductor-services/ic-knowledge/7569-iitc-imec-presents-copper-cobalt-and-ruthenium-interconnect-results/">IITC – Imec Presents Copper, Cobalt and Ruthenium Interconnect Results</a> </li> <li id="cite_note-39"><a href="#cite_ref-39">^</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.electronics-notes.com/articles/basic_concepts/resistance/resistance-resistivity-temperature-coefficient.php">"Temperature Coefficient of Resistance | Electronics Notes"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Temperature+Coefficient+of+Resistance+%26%23124%3B+Electronics+Notes&rft_id=https%3A%2F%2Fwww.electronics-notes.com%2Farticles%2Fbasic_concepts%2Fresistance%2Fresistance-resistivity-temperature-coefficient.php&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrical+resistivity+and+conductivity" class="Z3988"> </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> <a rel="nofollow" class="external autonumber" href="https://www.copper.org/applications/industrial/DesignGuide/selection/conductbronze02.html">[3]</a>. (Calculated as "15% conductivity of pure copper" (5.96E-7)). Retrieved on 2023-1-12. </li> <li id="cite_note-41"><a href="#cite_ref-41">^</a> <a rel="nofollow" class="external text" href="https://www.plansee.com/en/materials/niobium.html">Material properties of niobium.</a> </li> <li id="cite_note-42"><a href="#cite_ref-42">^</a> <a rel="nofollow" class="external text" href="http://www.matweb.com/search/DataSheet.aspx?MatGUID=025d4a04c2c640c9b0eaaef28318d761">AISI 1010 Steel, cold drawn</a>. Matweb </li> <li id="cite_note-43"><a href="#cite_ref-43">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKarcherKocourek2007" class="citation journal cs1">Karcher, Ch.; Kocourek, V. (December 2007). <a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fpamm.200700645">"Free-surface instabilities during electromagnetic shaping of liquid metals"</a>. Proceedings in Applied Mathematics and Mechanics. 7 (1): 4140009–4140010. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fpamm.200700645">10.1002/pamm.200700645</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/1617-7061">1617-7061</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+in+Applied+Mathematics+and+Mechanics&rft.atitle=Free-surface+instabilities+during+electromagnetic+shaping+of+liquid+metals&rft.volume=7&rft.issue=1&rft.pages=4140009-4140010&rft.date=2007-12&rft_id=info%3Adoi%2F10.1002%2Fpamm.200700645&rft.issn=1617-7061&rft.aulast=Karcher&rft.aufirst=Ch.&rft.au=Kocourek%2C+V.&rft_id=https%3A%2F%2Fdoi.org%2F10.1002%252Fpamm.200700645&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrical+resistivity+and+conductivity" class="Z3988"> </li> <li id="cite_note-44"><a href="#cite_ref-44">^</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.jfe-steel.co.jp/en/products/electrical/catalog/f1e-001.pdf">"JFE steel"</a> (PDF). Retrieved 2012-10-20.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=JFE+steel&rft_id=http%3A%2F%2Fwww.jfe-steel.co.jp%2Fen%2Fproducts%2Felectrical%2Fcatalog%2Ff1e-001.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrical+resistivity+and+conductivity" class="Z3988"> </li> <li id="cite_note-giancoli-45">^ <a href="#cite_ref-giancoli_45-0">a</a> <a href="#cite_ref-giancoli_45-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDouglas_C._Giancoli1995" class="citation book cs1">Douglas C. Giancoli (1995). <a rel="nofollow" class="external text" href="https://archive.org/details/physicsprinciple00gian_0">Physics: Principles with Applications</a> (4th ed.). London: Prentice Hall. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-102153-2" title="Special:BookSources/978-0-13-102153-2"><bdi>978-0-13-102153-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Physics%3A+Principles+with+Applications&rft.place=London&rft.edition=4th&rft.pub=Prentice+Hall&rft.date=1995&rft.isbn=978-0-13-102153-2&rft.au=Douglas+C.+Giancoli&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fphysicsprinciple00gian_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrical+resistivity+and+conductivity" class="Z3988"> (see also <a rel="nofollow" class="external text" href="http://hyperphysics.phy-astr.gsu.edu/hbase/Tables/rstiv.html">Table of Resistivity</a>. hyperphysics.phy-astr.gsu.edu) </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> John O'Malley (1992) Schaum's outline of theory and problems of basic circuit analysis, p. 19, McGraw-Hill Professional, <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/0-07-047824-4" title="Special:BookSources/0-07-047824-4">0-07-047824-4</a> </li> <li id="cite_note-48"><a href="#cite_ref-48">^</a> Glenn Elert (ed.), <a rel="nofollow" class="external text" href="http://hypertextbook.com/facts/2006/UmranUgur.shtml">"Resistivity of steel"</a>, The Physics Factbook, retrieved and <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110606042043/http://hypertextbook.com/facts/2006/UmranUgur.shtml">archived</a> 16 June 2011. </li> <li id="cite_note-49"><a href="#cite_ref-49">^</a> Probably, the metal with highest value of electrical resistivity. </li> <li id="cite_note-52"><a href="#cite_ref-52">^</a> Y. Pauleau, Péter B. Barna, P. B. Barna (1997) Protective coatings and thin films: synthesis, characterization, and applications, p. 215, Springer, <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/0-7923-4380-8" title="Special:BookSources/0-7923-4380-8">0-7923-4380-8</a>. </li> <li id="cite_note-Ohring-53"><a href="#cite_ref-Ohring_53-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMilton_Ohring1995" class="citation book cs1">Milton Ohring (1995). Engineering materials science, Volume 1 (3rd ed.). Academic Press. p. 561. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0125249959" title="Special:BookSources/978-0125249959"><bdi>978-0125249959</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Engineering+materials+science%2C+Volume+1&rft.pages=561&rft.edition=3rd&rft.pub=Academic+Press&rft.date=1995&rft.isbn=978-0125249959&rft.au=Milton+Ohring&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrical+resistivity+and+conductivity" class="Z3988"> </li> <li id="cite_note-56"><a href="#cite_ref-56">^</a> <a rel="nofollow" class="external text" href="http://www.kayelaby.npl.co.uk/general_physics/2_7/2_7_9.html">Physical properties of sea water</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180118072121/http://www.kayelaby.npl.co.uk/general_physics/2_7/2_7_9.html">Archived</a> 2018-01-18 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. 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Matthiessen, Rep. Brit. Ass. 32, 144 (1862) </li> <li id="cite_note-75"><a href="#cite_ref-75">^</a> A. Matthiessen, Progg. Anallen, 122, 47 (1864) </li> <li id="cite_note-76"><a href="#cite_ref-76">^</a> J. Seymour (1972) Physical Electronics, chapter 2, Pitman </li> <li id="cite_note-77"><a href="#cite_ref-77">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStephensonHubler2015" class="citation journal cs1">Stephenson, C.; Hubler, A. (2015). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4604515">"Stability and conductivity of self-assembled wires in a transverse electric field"</a>. Sci. Rep. 5: 15044. <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/2015NatSR...515044S">2015NatSR...515044S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fsrep15044">10.1038/srep15044</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4604515">4604515</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/26463476">26463476</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Sci.+Rep.&rft.atitle=Stability+and+conductivity+of+self-assembled+wires+in+a+transverse+electric+field&rft.volume=5&rft.pages=15044&rft.date=2015&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4604515%23id-name%3DPMC&rft_id=info%3Apmid%2F26463476&rft_id=info%3Adoi%2F10.1038%2Fsrep15044&rft_id=info%3Abibcode%2F2015NatSR...515044S&rft.aulast=Stephenson&rft.aufirst=C.&rft.au=Hubler%2C+A.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4604515&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrical+resistivity+and+conductivity" class="Z3988"> </li> <li id="cite_note-78"><a href="#cite_ref-78">^</a> Otto H. Schmitt, University of Minnesota <a rel="nofollow" class="external text" href="https://web.archive.org/web/20101013215436/http://www.otto-schmitt.org/OttoPagesFinalForm/Sounds/Speeches/MutualImpedivity.htm">Mutual Impedivity Spectrometry and the Feasibility of its Incorporation into Tissue-Diagnostic Anatomical Reconstruction and Multivariate Time-Coherent Physiological Measurements</a>. otto-schmitt.org. Retrieved on 2011-12-17. </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="https://www.lenntech.com/periodic/elements/be.htm">"Berryllium (Be) - Chemical properties, Health and Environmental effects"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Berryllium+%28Be%29+-+Chemical+properties%2C+Health+and+Environmental+effects&rft_id=https%3A%2F%2Fwww.lenntech.com%2Fperiodic%2Felements%2Fbe.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrical+resistivity+and+conductivity" class="Z3988"> </li> <li id="cite_note-:1-80">^ <a href="#cite_ref-:1_80-0">a</a> <a href="#cite_ref-:1_80-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFranklin1978" class="citation book cs1 cs1-prop-long-vol">Franklin, Benjamin (1978) [1774]. <a rel="nofollow" class="external text" href="https://founders.archives.gov/documents/Franklin/01-21-02-0062">"From Benjamin Franklin to Jan Ingenhousz, 18 March 1774"</a>. In Willcox, William B. (ed.). The Papers of Benjamin Franklin. Vol. 21, January 1, 1774, through March 22, 1775. Yale University Press. pp. 147–149 – via Founders Online, National Archives.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=From+Benjamin+Franklin+to+Jan+Ingenhousz%2C+18+March+1774&rft.btitle=The+Papers+of+Benjamin+Franklin&rft.pages=147-149&rft.pub=Yale+University+Press&rft.date=1978&rft.aulast=Franklin&rft.aufirst=Benjamin&rft_id=https%3A%2F%2Ffounders.archives.gov%2Fdocuments%2FFranklin%2F01-21-02-0062&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrical+resistivity+and+conductivity" class="Z3988"> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=29" title="Edit section: Further reading">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPaul_Tipler2004" class="citation book cs1">Paul Tipler (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>.</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&rft.edition=5th&rft.pub=W.+H.+Freeman&rft.date=2004&rft.isbn=978-0-7167-0810-0&rft.au=Paul+Tipler&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrical+resistivity+and+conductivity" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="https://www.academia.edu/29112469/Electrical_Conductivity_and_Resistivity">Measuring Electrical Resistivity and Conductivity</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Electrical_resistivity_and_conductivity&action=edit&section=30" title="Edit section: External links">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/60px-Wikibooks-logo-en-noslogan.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/80px-Wikibooks-logo-en-noslogan.svg.png 2x" data-file-width="400" data-file-height="400" /></div> <div class="side-box-text plainlist">Wikibooks has a book on the topic of: <a href="https://en.wikibooks.org/wiki/A-level_Physics_(Advancing_Physics)/Resistivity_and_Conductivity" class="extiw" title="wikibooks:A-level Physics (Advancing Physics)/Resistivity and Conductivity">A-level Physics (Advancing Physics)/Resistivity and Conductivity</a></div></div> </div> <ul><li><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.sixtysymbols.com/videos/conductivity.htm">"Electrical Conductivity"</a>. Sixty Symbols. <a href="/wiki/Brady_Haran" title="Brady Haran">Brady Haran</a> for the <a href="/wiki/University_of_Nottingham" title="University of Nottingham">University of Nottingham</a>. 2010.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Sixty+Symbols&rft.atitle=Electrical+Conductivity&rft.date=2010&rft_id=http%3A%2F%2Fwww.sixtysymbols.com%2Fvideos%2Fconductivity.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrical+resistivity+and+conductivity" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="http://www.wolframalpha.com/input/?i=conductivity+sulfur%2C+silicon%2C+copper&lk=3">Comparison of the electrical conductivity of various elements in WolframAlpha</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPartial_and_total_conductivity" class="citation web cs1">Partial and total conductivity. <a rel="nofollow" class="external text" href="https://www.uio.no/studier/emner/matnat/kjemi/KJM5120/v09/undervisningsmateriale/Defects-and-transport-2009-Ch6-Electrical-conductivity.pdf">"Electrical conductivity"</a> (PDF).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Electrical+conductivity&rft.au=Partial+and+total+conductivity&rft_id=https%3A%2F%2Fwww.uio.no%2Fstudier%2Femner%2Fmatnat%2Fkjemi%2FKJM5120%2Fv09%2Fundervisningsmateriale%2FDefects-and-transport-2009-Ch6-Electrical-conductivity.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AElectrical+resistivity+and+conductivity" class="Z3988"></li></ul> <ul><li><a rel="nofollow" class="external free" href="https://edu-physics.com/2021/01/07/resistivity-of-the-material-of-a-wire-physics-practical/">https://edu-physics.com/2021/01/07/resistivity-of-the-material-of-a-wire-physics-practical/</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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