{"id":2885,"date":"2014-04-07T16:45:05","date_gmt":"2014-04-07T20:45:05","guid":{"rendered":"http:\/\/pages.vassar.edu\/magnes\/?p=2885"},"modified":"2014-05-07T11:40:08","modified_gmt":"2014-05-07T15:40:08","slug":"project-plan-modeling-electromagnetic-fields-for-spherical-objects","status":"publish","type":"post","link":"https:\/\/pages.vassar.edu\/magnes\/2014\/04\/07\/project-plan-modeling-electromagnetic-fields-for-spherical-objects\/","title":{"rendered":"Project Plan: Modeling Electromagnetic Fields for Spherical Objects"},"content":{"rendered":"<p><strong>Sources<\/strong><\/p>\n<p>I will be utilizing\u00a0<em>Introduction to Electrodynamics<\/em>, 4th Edition,\u00a0by David J. Griffiths. Specifically, I will begin with Gauss&#8217;s Law, as defined by Griffiths on page 69:<\/p>\n<p>$ \\oint \\! \\textbf{E} \\cdot \\mathrm{d} \\textbf{a} = \\frac{1}{\\epsilon_0} Q_{enc} $<\/p>\n<p>Further, I will utilize the formula for the electric field of a point charge below (found on Griffiths page 72), which can be generalized for a spherical object:<\/p>\n<p>$\\textbf{E} = \\frac{1}{4 \\pi \\epsilon_0} \\frac{q}{r^2} \\mathbf{\\hat{r}} $<\/p>\n<p>I will additionally work with the magnetic field for the spherical object. Griffiths (page 263) gives the average magnetic field due to uniform current over a sphere as:<\/p>\n<p>$ \\textbf{B}_{ave} = \\frac{\\mu_0}{4 \\pi} \\frac{\\textbf{m}}{R^3}$<\/p>\n<p>Where\u00a0<strong>m<\/strong> is the total dipole moment of the sphere and R is the radius of the sphere.<\/p>\n<p>I will be using Mathematica 9 as my modeling tool.<\/p>\n<p><b>Plan of Action<\/b><\/p>\n<p>I will begin by using the equations above to start with modeling the electric field of a point charge. From there, I will model the electric field for a hollow spherical object. I will create a manipulatable object in Mathematica for changes in radius and charge. I will then move on to modeling the average magnetic field for a spherical object, and attempt to create a manipulatable object akin to the one for electric fields. Next, I will model the electric and magnetic fields for concentric spherical objects, with the goal of ultimately coming up with a very liberal approximation for modeling the magnetic field of the Earth, if the Earth is thought of as several concentric spheres (due to the crust, mantle, and outer\/inner cores). However, this will only occur if time permits, as will a preliminary examination of dielectrics.<\/p>\n<p><strong>Timeline<\/strong><\/p>\n<p>Week 1 (4\/6-4\/12): Work on the simplest case of a point charge, and learn to work within Mathematica<\/p>\n<p>Week 2 (4\/13-4\/19): Work to create manipulatable object for electric field of sphere, and begin working on modeling the average magnetic field for a spherical object with uniform current density<\/p>\n<p>Week 3 (4\/20-4\/26): Model electric and magnetic fields of concentric spherical objects, submit preliminary results on Tuesday on blog<\/p>\n<p>Week 4 (4\/27-5\/3): Wrap up, submit final data and conclusion on Wednesday, dielectrics if time permits<\/p>\n<p><strong>Collaborators<\/strong><\/p>\n<p>I am working with Brian Deer, who is focusing on bar magnets, and Tewa Kpulun, who is focusing on cylindrical objects. We will be meeting weekly to discuss our progress, share Mathematica-related insights, and help each other in whatever ways we can.<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Sources I will be utilizing\u00a0Introduction to Electrodynamics, 4th Edition,\u00a0by David J. Griffiths. Specifically, I will begin with Gauss&#8217;s Law, as defined by Griffiths on page 69: $ \\oint \\! \\textbf{E} \\cdot \\mathrm{d} \\textbf{a} = \\frac{1}{\\epsilon_0} Q_{enc} $ Further, I will utilize the formula for the electric field of a point charge below (found on Griffiths [&hellip;]<\/p>\n","protected":false},"author":2606,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4101,5569,54193,54190],"tags":[],"class_list":["post-2885","post","type-post","status-publish","format-standard","hentry","category-advanced-em","category-project-plan","category-ramy","category-spring-2014"],"_links":{"self":[{"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts\/2885","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/users\/2606"}],"replies":[{"embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/comments?post=2885"}],"version-history":[{"count":13,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts\/2885\/revisions"}],"predecessor-version":[{"id":3022,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts\/2885\/revisions\/3022"}],"wp:attachment":[{"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/media?parent=2885"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/categories?post=2885"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/tags?post=2885"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}