here<\/a>. \u00a0The file named “transforming coordinates.nb” shows my attempts to use the TransformedField function to get the three Cartesian components of the general magnetic field equation, originally expressed in Spherical coordinates. \u00a0The fact that a 3D vector plot of the resulting expressions shows nothing leads me to believe something more involved is necessary to convert these expressions between Spherical and Cartesian coordinates. \u00a0I may try calculating the separate expressions for\u00a0$\\vec{B_x}$,\u00a0$\\vec{B_y}$, and\u00a0$\\vec{B_z}$ by hand, typing them all into Mathematica, and using these expressions to try to generate a 3D vector plot, but there may be easier and more informative ways of presenting the magnetic field information.<\/p>\nProfessor Magnes suggested that I try looking at only a sample of representative points near the magnetic dipole, and evaluating the magnetic field expression,<\/p>\n
$\\vec{B} = \\frac{\\mu_0 m}{4 \\pi r^3} (2 cos \\theta \\hat{r} + sin \\theta \\hat{\\theta})$<\/p>\n
at these points using the Spherical coordinates at first, and converting these points into Cartesian form before plotting a 3D vector plot. \u00a0Because of the nature of plotting vector fields in 3D in Mathematica, this may or may not prove to be less complicated. \u00a0Another option is presenting the field information in a different way, utilizing contour plots (ListContourPlot3D in Mathematica) instead of vector arrow plots. \u00a0A contour plot will show equipotential surfaces within the magnetic field instead of the arrows representing the magnitude and direction of the magnetic field at certain points. \u00a0However, the equipotential surfaces and vector arrows are intimately related: vector arrows are always perpendicular to equipotential surfaces. \u00a0Therefore, this may be an easier way to compute and show the magnetic field of a magnetic dipole. \u00a0Vector arrows can always be added on top of the contour plot to show the magnetic field in another way as well. \u00a0There are a lot of factors to play with to see what presentation style will be the most effective.<\/p>\n
The other file in the link to my Mathematica work so far contains the start of generating a list of representative points around the origin in Spherical coordinates and the start of converting them into Cartesian coordinates using the CoordinateTransform function in Mathematica. \u00a0This file is currently quite preliminary. \u00a0I still have a long way to go before I get a graph presenting some information about the magnetic field of a magnetic dipole.<\/p>\n","protected":false},"excerpt":{"rendered":"
As I began working on this modeling project in Mathematica, it quickly became apparent that even just the magnetic field of a single magnetic dipole is pretty complicated to model. \u00a0Transforming between Spherical and Cartesian coordinates via Mathematica’s built in TransformedField function is more complicated than expected, and may even not work at all. The […]<\/p>\n","protected":false},"author":1991,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4101,54054],"tags":[],"class_list":["post-3319","post","type-post","status-publish","format-standard","hentry","category-advanced-em","category-brian"],"_links":{"self":[{"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts\/3319","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\/1991"}],"replies":[{"embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/comments?post=3319"}],"version-history":[{"count":3,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts\/3319\/revisions"}],"predecessor-version":[{"id":3361,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts\/3319\/revisions\/3361"}],"wp:attachment":[{"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/media?parent=3319"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/categories?post=3319"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/tags?post=3319"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}