{"id":10271,"date":"2020-05-24T14:38:27","date_gmt":"2020-05-24T17:38:27","guid":{"rendered":"https:\/\/www.antonioguilherme.web.br.com\/blog\/?page_id=10271"},"modified":"2025-09-23T15:45:59","modified_gmt":"2025-09-23T18:45:59","slug":"condutividade-e-resistividade-ii","status":"publish","type":"page","link":"https:\/\/www.antonioguilherme.web.br.com\/blog\/materiais-eletricos-e-magneticos\/propriedades-eletricas\/condutividade-e-resistividade-ii\/","title":{"rendered":"Condutividade El\u00e9trica CA"},"content":{"rendered":"<p><span style=\"font-family: verdana, geneva, sans-serif;\">O <a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/video-supercondutividade\/\">Cap\u00edtulo anterior<\/a> analisou a Condutividade e a Resistividade el\u00e9trica em corrente cont\u00ednua porque a eletricidade evoluiu a partir da corrente cont\u00ednua.\u00a0<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Em 1880 Thomas Edson e George Westinghouse, com o suporte t\u00e9cnico de Nicolas Tesla, travaram uma disputa t\u00e9cnico-comercial nos EUA entre a utiliza\u00e7\u00e3o da corrente cont\u00ednua e a corrente alternada para a gera\u00e7\u00e3o, transmiss\u00e3o e distribui\u00e7\u00e3o de energia el\u00e9trica. Essa disputa resultou em duas empresas gigantes; a General Electric e a Westinghouse, e no filme a <a href=\"http:\/\/kue18AxK1tU\">Guerra das Correntes<\/a>, cujo trailer se encontra no link acima.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Portanto, a descoberta da corrente alternada ocorreu antes da teoria de Drude apresenta no cap\u00edtulo anterior, e esse cap\u00edtulo analisa os impactos da corrente alterna na condutividade e resistividade el\u00e9trica.<\/span><\/p>\n<h2 id='condutividade-el\u00e9trica-em-corrente-alternada'  id=\"boomdevs_1\">Condutividade El\u00e9trica em Corrente Alternada<\/h2>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Conforme visto em <a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/materiais-eletricos-e-magneticos\/propriedades-eletricas\/resistividade-eletrica\/\">Condutividade e Resistividade I<\/a>, a Mec\u00e2nica Cl\u00e1ssica associa for\u00e7a a varia\u00e7\u00f5es de momento da seguinte maneira:<\/span><\/p>\n<figure id=\"attachment_10252\" aria-describedby=\"caption-attachment-10252\" style=\"width: 211px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/05\/momentoxforca.png\"><img decoding=\"async\" class=\"wp-image-10252 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/05\/momentoxforca.png\" alt=\"\" width=\"211\" height=\"49\" \/><\/a><figcaption id=\"caption-attachment-10252\" class=\"wp-caption-text\">Equa\u00e7\u00e3o 1. Rela\u00e7\u00e3o entre varia\u00e7\u00e3o de momento e for\u00e7a<\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Onde:<\/span><\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">\u0394p \u00e9 a varia\u00e7\u00e3o do momento linear;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">F \u00e9 a for\u00e7a;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">\u0394t \u00e9 o intervalo de tempo.<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Pode-se reescrever a Equa\u00e7\u00e3o 1 aplicada ao El\u00e9tron \u00a0submetido a campo el\u00e9trico E da seguinte maneira:<\/span><\/p>\n<figure id=\"attachment_10277\" aria-describedby=\"caption-attachment-10277\" style=\"width: 441px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/05\/torque_drude_1.png\"><img decoding=\"async\" class=\"wp-image-10277 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/05\/torque_drude_1.png\" alt=\"\" width=\"441\" height=\"90\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/05\/torque_drude_1.png 441w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/05\/torque_drude_1-300x61.png 300w\" sizes=\"(max-width: 441px) 100vw, 441px\" \/><\/a><figcaption id=\"caption-attachment-10277\" class=\"wp-caption-text\">Equa\u00e7\u00e3o 2. Varia\u00e7\u00e3o do momento de um El\u00e9tron<\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Onde:<\/span><\/p>\n<ul>\n<li style=\"list-style-type: none;\">\n<ul>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">p \u00e9 o momento linear;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">\u03c4 \u00e9 tempo de relaxa\u00e7\u00e3o;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\"><em>e<\/em> \u00e9 a carga elementar;<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">E(t) \u00e9 o campo el\u00e9trico em fun\u00e7\u00e3o do tempo.<\/span><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Aplicando a Transforma de Laplace na Equa\u00e7\u00e3o 2<span id='easy-footnote-1-10271' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/materiais-eletricos-e-magneticos\/propriedades-eletricas\/condutividade-e-resistividade-ii\/#easy-footnote-bottom-1-10271' title=' e considerando as condi\u00e7\u00f5es iniciais nulas'><sup>1<\/sup><\/a><\/span>, obtemos a seguinte express\u00e3o:<\/span><\/p>\n<figure id=\"attachment_22749\" aria-describedby=\"caption-attachment-22749\" style=\"width: 477px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/10\/drude_ac_1.png\"><img decoding=\"async\" class=\"size-full wp-image-22749\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/10\/drude_ac_1.png\" alt=\"\" width=\"477\" height=\"188\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/10\/drude_ac_1.png 477w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/10\/drude_ac_1-300x118.png 300w\" sizes=\"(max-width: 477px) 100vw, 477px\" \/><\/a><figcaption id=\"caption-attachment-22749\" class=\"wp-caption-text\">Equa\u00e7\u00e3o 3.<\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Substituindo a velocidade pelo momento linear na Equa\u00e7\u00e3o 4 da se\u00e7\u00e3o <a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/materiais-eletricos-e-magneticos\/propriedades-eletricas\/resistividade-eletrica\/\">Condutividade e Resistividade I<\/a>, obtem-se a seguinte express\u00e3o para a densidade de corrente:<\/span><\/p>\n<figure id=\"attachment_22752\" aria-describedby=\"caption-attachment-22752\" style=\"width: 346px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/10\/drude_ac_2.png\"><img decoding=\"async\" class=\"size-full wp-image-22752\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/10\/drude_ac_2.png\" alt=\"\" width=\"346\" height=\"78\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/10\/drude_ac_2.png 346w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/10\/drude_ac_2-300x68.png 300w\" sizes=\"(max-width: 346px) 100vw, 346px\" \/><\/a><figcaption id=\"caption-attachment-22752\" class=\"wp-caption-text\">Equa\u00e7\u00e3o 4.<\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Substituindo a Equa\u00e7\u00e3o 3 na Equa\u00e7\u00e3o 4, obtem-se o seguinte resultado:<\/span><\/p>\n<figure id=\"attachment_22754\" aria-describedby=\"caption-attachment-22754\" style=\"width: 368px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/10\/drude_ac_3.png\"><img decoding=\"async\" class=\"size-full wp-image-22754\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/10\/drude_ac_3.png\" alt=\"\" width=\"368\" height=\"110\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/10\/drude_ac_3.png 368w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/10\/drude_ac_3-300x90.png 300w\" sizes=\"(max-width: 368px) 100vw, 368px\" \/><\/a><figcaption id=\"caption-attachment-22754\" class=\"wp-caption-text\">Equa\u00e7\u00e3o 5.<\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">A partir da Equa\u00e7\u00e3o 5, determina-se a condutividade da seguinte maneira:<\/span><\/p>\n<figure id=\"attachment_22756\" aria-describedby=\"caption-attachment-22756\" style=\"width: 1131px\" class=\"wp-caption aligncenter\"><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/10\/drude_ac_4.png\"><img decoding=\"async\" class=\"wp-image-22756 size-full\" src=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/10\/drude_ac_4.png\" alt=\"\" width=\"1131\" height=\"110\" srcset=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/10\/drude_ac_4.png 1131w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/10\/drude_ac_4-300x29.png 300w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/10\/drude_ac_4-1024x100.png 1024w, https:\/\/www.antonioguilherme.web.br.com\/blog\/wp-content\/uploads\/2020\/10\/drude_ac_4-768x75.png 768w\" sizes=\"(max-width: 1131px) 100vw, 1131px\" \/><\/a><figcaption id=\"caption-attachment-22756\" class=\"wp-caption-text\">Equa\u00e7\u00e3o 6. Condutividade el\u00e9trica em fun\u00e7\u00e3o da frequ\u00eancia<\/figcaption><\/figure>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Observa-se que a condutividade depende da frequ\u00eancia da corrente e tende para o valor obtido originalmente por Drude quando a frequ\u00eancia tende a zero.<span id='easy-footnote-2-10271' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/materiais-eletricos-e-magneticos\/propriedades-eletricas\/condutividade-e-resistividade-ii\/#easy-footnote-bottom-2-10271' title=' quando s tende a zero'><sup>2<\/sup><\/a><\/span> <\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Portanto, a condutividade e resistividade el\u00e9tricas n\u00e3o s\u00e3o constantes como sugeria Ohm, mas dependem da frequ\u00eancia do campo el\u00e9trico.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Contudo, o Modelo de Drude desconsiderou os efeitos do eletromagnetismo no Modelo de Corrente Alternada. <\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">De acordo com Faraday e Maxwell, todo campo el\u00e9trico vari\u00e1vel no tempo gera um campo magn\u00e9tico proporcional tamb\u00e9m variante no tempo. <\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Por\u00e9m, pode-se desconsiderar este efeito neste caso porque ele se torna desprez\u00edvel para velocidades muito abaixo da velocidade da luz. <span id='easy-footnote-3-10271' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href='https:\/\/www.antonioguilherme.web.br.com\/blog\/materiais-eletricos-e-magneticos\/propriedades-eletricas\/condutividade-e-resistividade-ii\/#easy-footnote-bottom-3-10271' title=' Por isso, calculamos a velocidade dos el\u00e9trons nos diversos modelos para verificar esta condi\u00e7\u00e3o. '><sup>3<\/sup><\/a><\/span> <\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">Al\u00e9m disso, a Equa\u00e7\u00e3o 2 considera o campo el\u00e9trico e a for\u00e7a por ele produzida uniformes ao longo do espa\u00e7o. <\/span><span style=\"font-family: verdana, geneva, sans-serif;\">Esta aproxima\u00e7\u00e3o vale para comprimentos de onda do campo el\u00e9trico muito maiores que a dist\u00e2ncia m\u00e9dia percorrida pelos el\u00e9trons entre duas colis\u00f5es neste modelo. <\/span><span style=\"font-family: verdana, geneva, sans-serif;\">No caso dos metais, esta condi\u00e7\u00e3o vale para campos el\u00e9tricos de frequ\u00eancia inferior \u00e0 frequ\u00eancia da luz.<\/span><\/p>\n<h2 id='efeito-pelicular'  id=\"boomdevs_2\"><span style=\"font-family: verdana, geneva, sans-serif;\">Efeito Pelicular<\/span><\/h2>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">O <a href=\"https:\/\/pt.wikipedia.org\/wiki\/Efeito_pelicular\" target=\"_blank\" rel=\"noopener\">efeito pelicular<\/a> consiste no efeito da corrente el\u00e9trica vari\u00e1vel em for\u00e7ar o fluxo de corrente para a superf\u00edcie dos condutores. Esse fen\u00f4meno reduz a \u00e1rea de condu\u00e7\u00e3o da corrente provocando um aumento na resist\u00eancia el\u00e9trica do condutor.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif;\">A corrente alternada varia de acordo com a frequ\u00eancia, e a varia\u00e7\u00e3o de corrente produz campos magn\u00e9ticos de mesma frequ\u00eancia no interior e ao redor dos condutores. Consequentemente, o campo magn\u00e9tico alternado gerado produz um campo el\u00e9trico em oposi\u00e7\u00e3o \u00e0 corrente.<\/span><\/p>\n<p><span style=\"font-family: verdana, geneva, sans-serif; font-size: 48px; font-weight: bold; letter-spacing: -0.0415625em;\">Refer\u00eancias<\/span><\/p>\n<ol style=\"list-style-type: decimal;\">\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">ASHCROFT, N.W., MERMIN, N.D., <strong>Solid State Physics<\/strong>, Harcourt, 1976.<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">BEATY, H.W., FINK, D.G., <strong>Standard Handbook for Electrical Engineers<\/strong>, 15 edi\u00e7\u00e3o, McGraw-Hill, 2007.<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">CHELIKOWSKY, J.R., COHEN, M.L., <strong>Semiconductors: A Pillar of Pure and Applied Physics<\/strong>, Journal of Applied Physics 117, 112812 (2015).<span id='easy-footnote-4-10271' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/materiais-eletricos-e-magneticos\/propriedades-eletricas\/condutividade-e-resistividade-ii\/#easy-footnote-bottom-4-10271\" title=\" Artigo muito interessante sobre a resistividade no estudo de materiais\"><sup>4<\/sup><\/a><\/span><\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">HAYNES, W.W., <strong>Handbook of Chemistry and Physics<\/strong>, 93 edi\u00e7\u00e3o, CRC, 2012.<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">JILES, D., <strong>Introduction to the Electronic Properties of Materials<\/strong>, 2 ed., Taylor &amp; Francis, 2001. <span id='easy-footnote-5-10271' class='easy-footnote-margin-adjust'><\/span><span class='easy-footnote'><a href=\"https:\/\/www.antonioguilherme.web.br.com\/blog\/materiais-eletricos-e-magneticos\/propriedades-eletricas\/condutividade-e-resistividade-ii\/#easy-footnote-bottom-5-10271\" title=\"Scribd &amp;#8211; muito importante\"><sup>5<\/sup><\/a><\/span><\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">KASAP, S.; KOUGHIA C.; RUDA, H.E.; Electrical Conduction in Metals and Semiconductors<strong>.<\/strong> In: KASAP, S.; CAPPER, P.(Ed.); <strong>Springer Handbook of Electronic and Photonic Materials<\/strong>, <\/span><span style=\"font-family: verdana, geneva, sans-serif;\">Springer, 2017.<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">KITTEL, C., <strong>Introduction to Solid State Physics<\/strong>, 8 edi\u00e7\u00e3o, John Wiley &amp; Sons, 2005<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">NOTAROS, B.M., <strong>Eletromagnetismo<\/strong>, Pearson Education do Brasil, 2012.<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">OHRING, M., <strong>Engineering Materials Science<\/strong>, Academic Press, 1995.<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">QUINN, J.J., YI, K, <strong>Solid State Physics &#8211; Principles and Modern Applications<\/strong>, Springer-Verlag, 2009.<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">SOLYMAR, L., WALSH, D., SYMS, R.R.A., <strong>Electrical Properties of Materials<\/strong>, Oxford, 10 ed., 2019.<\/span><\/li>\n<li><span style=\"font-family: verdana, geneva, sans-serif;\">WOLKENSTEIN, T. <strong>Electrons and Crystals<\/strong>, Mir, 1985.<\/span><\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n<div class=\"ms-editor-squiggler\" style=\"color: initial; font: initial; font-feature-settings: initial; font-kerning: initial; font-optical-sizing: initial; font-variation-settings: initial; forced-color-adjust: initial; text-orientation: initial; text-rendering: initial; -webkit-font-smoothing: initial; -webkit-locale: initial; -webkit-text-orientation: initial; -webkit-writing-mode: initial; writing-mode: initial; zoom: initial; place-content: initial; place-items: initial; place-self: initial; alignment-baseline: initial; animation: initial; appearance: initial; aspect-ratio: initial; backdrop-filter: initial; backface-visibility: initial; background: initial; background-blend-mode: initial; baseline-shift: initial; block-size: initial; border-block: initial; border: initial; border-radius: initial; border-collapse: initial; border-end-end-radius: initial; border-end-start-radius: initial; border-inline: initial; border-start-end-radius: initial; border-start-start-radius: initial; inset: initial; box-shadow: initial; box-sizing: initial; break-after: initial; break-before: initial; break-inside: initial; buffered-rendering: initial; caption-side: initial; caret-color: initial; clear: initial; clip: initial; clip-path: initial; clip-rule: initial; color-interpolation: initial; color-interpolation-filters: initial; color-rendering: initial; color-scheme: initial; columns: initial; column-fill: initial; gap: initial; column-rule: initial; column-span: initial; contain: initial; contain-intrinsic-size: initial; content: initial; content-visibility: initial; 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