{"id":3677,"date":"2014-04-30T03:39:13","date_gmt":"2014-04-30T07:39:13","guid":{"rendered":"http:\/\/pages.vassar.edu\/magnes\/?p=3677"},"modified":"2014-04-30T03:39:13","modified_gmt":"2014-04-30T07:39:13","slug":"final-data","status":"publish","type":"post","link":"https:\/\/pages.vassar.edu\/magnes\/2014\/04\/30\/final-data\/","title":{"rendered":"Final Data"},"content":{"rendered":"

My final data builds significantly from the preliminary data sets. In that set, I derived the equations for the electric field of a sphere, cylinder and the approximation of an electric dipole. Those were:<\/p>\n

  <\/span>   <\/span>\"\[ \textbf{E} = \frac{q_e}{\epsilon_0 4 \pi r^2} \textbf{$\hat{r}$} \]\"<\/p>\n

  <\/span>   <\/span>\"\[ \textbf{E} = \frac{\rho R^2 }{2r\epsilon_0} \textbf{$\hat{r}$} \]\"<\/p>\n

Where the dipole was approximated as two point charges of opposite charge (i.e. two spheres of opposite charge).<\/p>\n

I went back to mathematica and overlaid a 3-D model of a small sphere on the previously plotted vector field of the electric field of a sphere, pictured below.<\/p>\n

\"sfield\"<\/a><\/p>\n

The same was done for a cylinder, both cases allowed for it to be seen how the geometries affected the electric fields.<\/p>\n

\"cfield\"<\/a><\/p>\n

For the electric dipole, an additional approach was taken to find the electric field. The equation for the electric potential of an “ideal” dipole is given as<\/p>\n

  <\/span>   <\/span>\"\[ V = \frac{p_o \cos(\theta)}{\epsilon_0 4 \pi r^2} \]\"<\/p>\n

For the purposes of simplification in Mathematica, the equation was re-written to absorb all the constants into one term:<\/p>\n

  <\/span>   <\/span>\"\[ V = \frac{p \cos(\theta)}{r^2} \]\"<\/p>\n

Now to find an expression for the electric field, use the formula:<\/p>\n

  <\/span>   <\/span>\"\[ \textbf{E} = -\nabla V \]\"<\/p>\n

Which Mathematica can interpret and plot accordingly as:<\/p>\n

\"dipole\"<\/a><\/p>\n

For a better view of how the vectors are interacting, look at this 2-D plot of the same field:<\/p>\n

\"dipoleside\"<\/a><\/p>\n

The code for the following models can be found here: Final Data<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

My final data builds significantly from the preliminary data sets. In that set, I derived the equations for the electric field of a sphere, cylinder and the approximation of an electric dipole. Those were:         Where the dipole was approximated as two point charges of opposite charge (i.e. two spheres of opposite […]<\/p>\n","protected":false},"author":1912,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4101,54192,54190],"tags":[],"class_list":["post-3677","post","type-post","status-publish","format-standard","hentry","category-advanced-em","category-peter","category-spring-2014"],"_links":{"self":[{"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts\/3677","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\/1912"}],"replies":[{"embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/comments?post=3677"}],"version-history":[{"count":6,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts\/3677\/revisions"}],"predecessor-version":[{"id":3693,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/posts\/3677\/revisions\/3693"}],"wp:attachment":[{"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/media?parent=3677"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/categories?post=3677"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pages.vassar.edu\/magnes\/wp-json\/wp\/v2\/tags?post=3677"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}