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Notebook[{

Cell[CellGroupData[{
Cell["\<\

Dipole Fields\
\>", "Title",
  TextAlignment->Center],

Cell[CellGroupData[{

Cell["\<\

As a simple example we will plot the potential and electric field associated \
with a dipole.  First we load in a few useful libraries\
\>", "Section"],

Cell[BoxData[{
    \(<< Graphics`PlotField3D`\), 
    \(<< Calculus`VectorAnalysis`\), 
    \(<< Graphics`PlotField`\)}], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\

Here are some distances we'll need\
\>", "Section"],

Cell[BoxData[
    \(\(d\  = \ .5; \n
    d1[x_, y_, z_]\  = \ Sqrt[\((z + d/2)\)^2\  + \ x^2\  + \ y^2]; \n
    d2[x_, y_, z_]\  = \ Sqrt[\((z - d/2)\)^2\  + \ x^2\  + \ y^2]; \n
    r[x_, y_, z_]\  = \ Sqrt[x^2\  + \ y^2\  + z^2]; \n\t\)\)], "Input"]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\

And here are the exact potential and the dipole (far-field) \
approximation\
\>", "Section"],

Cell[CellGroupData[{

Cell[BoxData[
    \(v[x_, y_, z_]\  = \(-1\)/\ d1[x, y, z]\  + \ 1/d2[x, y, z]\)], "Input"],

Cell[BoxData[
    \(1\/\@\(x\^2 + y\^2 + \((\(-0.25`\) + z)\)\^2\) - 
      1\/\@\(x\^2 + y\^2 + \((\(0.25`\[InvisibleSpace]\) + z)\)\^2\)\)], 
  "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(dipole[x_, y_, z_]\  = \ z\ d/r[x, y, z]^3\)], "Input"],

Cell[BoxData[
    FractionBox[
      RowBox[{
        StyleBox["0.5`",
          StyleBoxAutoDelete->True,
          PrintPrecision->1], " ", "z"}], 
      \(\((x\^2 + y\^2 + z\^2)\)\^\(3/2\)\)]], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell["\<\

First we plot the electric field (which is the gradient of the \
potential)\
\>", "Section"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\np1\  = \ 
      PlotGradientField[v[x, \ 1, \ z], \ {x, \ \(-5\), \ 5}, 
        \ {z, \(-5\), \ 5}, HeadWidth\  -> \ .3, \ HeadLength\  -> \ .03, \ 
        Axes\  -> \ True, \ PlotPoints\  -> \ 25]\), 
    \(PlotGradientField[dipole[x, \ 1, \ z], \ {x, \ \(-5\), \ 5}, 
      \ {z, \(-5\), \ 5}, HeadWidth\  -> \ .3, \ HeadLength\  -> \ .03, \ 
      Axes\  -> \ True, \ PlotPoints\  -> \ 25]\)}], "Input"],

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