{"id":283387,"date":"2016-09-22T02:00:00","date_gmt":"2016-09-21T23:00:00","guid":{"rendered":"http:\/\/www.fyysika.ee\/?guid=d8b9f81f422c70b6904bd7792ae5bf6e"},"modified":"2016-09-22T02:00:00","modified_gmt":"2016-09-21T23:00:00","slug":"the-helmholtz-theorem-and-retarded-fields","status":"publish","type":"post","link":"https:\/\/www.fyysika.ee\/?p=283387","title":{"rendered":"The Helmholtz theorem and retarded fields"},"content":{"rendered":"<p>Textbooks frequently use the Helmholtz theorem to derive expressions for electrostatic and<br \/>\nmagnetostatic fields but they do not usually apply this theorem to derive expressions for<br \/>\ntime-dependent electric and magnetic fields, even when there is no formal objection to doing so<br \/>\nbecause the proof of the theorem does not involve time derivatives but only spatial derivatives.<br \/>\nHere we address the question as to whether the Helmholtz theorem is useful in deriving expressions<br \/>\nfor the fields of Maxwell\u2019s equations. We show that when this theorem is applied to Maxwell\u2019s<br \/>\nequations we obtain instantaneous expressions of the electric and magnetic fields, which are<br \/>\nformally correct but of little practical usefulness. We then discuss two generalizations of the<br \/>\ntheorem which are shown to be useful in deriving the retarded fields.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Textbooks frequently use the Helmholtz theorem to derive expressions for electrostatic and<br \/>\nmagnetostatic fields but they do not usually apply this theorem to derive expressions for<br \/>\ntime-dependent electric and magnetic fields, even when there is no fo&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[178],"tags":[],"class_list":{"0":"post-283387","1":"post","2":"type-post","3":"status-publish","4":"format-standard","6":"category-rss-fuusikaharidus","7":"entry"},"_links":{"self":[{"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=\/wp\/v2\/posts\/283387","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=283387"}],"version-history":[{"count":0,"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=\/wp\/v2\/posts\/283387\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=283387"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=283387"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.fyysika.ee\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=283387"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}