(* Content-type: application/vnd.wolfram.cdf.text *) (*** Wolfram CDF File ***) (* http://www.wolfram.com/cdf *) (* CreatedBy='Mathematica 8.0' *) (*************************************************************************) (* *) (* The Mathematica License under which this file was created prohibits *) (* restricting third parties in receipt of this file from republishing *) (* or redistributing it by any means, including but not limited to *) (* rights management or terms of use, without the express consent of *) (* Wolfram Research, Inc. *) (* *) (*************************************************************************) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 835, 17] NotebookDataLength[ 8999, 278] NotebookOptionsPosition[ 9134, 265] NotebookOutlinePosition[ 9476, 280] CellTagsIndexPosition[ 9433, 277] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Attempt at evaluating the potential for an infinite cylinder.\ \>", "Title", CellChangeTimes->{{3.538615560700992*^9, 3.538615584600359*^9}}], Cell[TextData[{ "The potential for a radius a cylinder of length 2 L is:\n\n", StyleBox["\[Phi](R) \t= ", FontSize->14], Cell[BoxData[ FormBox[ RowBox[{"\[Integral]", StyleBox[ FractionBox[ RowBox[{"k", " ", "\[Rho]", " ", "dV"}], SqrtBox[ RowBox[{ SuperscriptBox[ TemplateBox[{RowBox[{"R", "-", RowBox[{ SuperscriptBox["\[ExponentialE]", "i\[Theta]"], " ", "r"}]}]}, "Abs"], "2"], "+", SuperscriptBox["z", "2"]}]]], FontSize->18]}], TraditionalForm]], "Input", CellChangeTimes->{{3.538616305404587*^9, 3.5386163127680078`*^9}}, FontSize->14], StyleBox["\n\t= ", FontSize->14], Cell[BoxData[ FormBox[ StyleBox[ RowBox[{"k", " ", "\[Rho]", RowBox[{ SuperscriptBox[ SubscriptBox["\[Integral]", RowBox[{"-", "L"}]], "L"], RowBox[{"dz", RowBox[{ SuperscriptBox[ SubscriptBox["\[Integral]", "0"], RowBox[{"2", " ", "\[Pi]"}]], RowBox[{"d\[Theta]", RowBox[{ SuperscriptBox[ SubscriptBox["\[Integral]", "0"], "a"], FractionBox[ RowBox[{" ", RowBox[{"r", " ", "dr"}]}], SqrtBox[ RowBox[{ RowBox[{"z", "^", "2"}], "+", RowBox[{"R", "^", "2"}], "-", RowBox[{"2", "rR", " ", SuperscriptBox[ RowBox[{"cos", "(", "\[Theta]", ")"}], "\[VeryThinSpace]"]}]}]]]}]}]}]}]}]}], FontSize->24], TraditionalForm]], FontSize->14], "\n\nEvaluating this appears non-trivial, especially if we let L tend to \ infinity. Note that the integral in the z component \ doesn\[CloseCurlyQuote]t converge if we let the L go unbounded, even if we \ take the principle value of the integral.\n\nGoogling this problem, I find:\n\ \nhttp://www.ifi.unicamp.br/~assis/J-Electrostatics-V63-p1115-1131(2005).pdf\n\ \ndecidedly non-trivial! It would take a couple of years of study to even \ try to read that paper.\n" }], "Text", CellChangeTimes->{{3.5386155929828386`*^9, 3.538616018589182*^9}, 3.5386160545812407`*^9, {3.5386160938124843`*^9, 3.538616145964467*^9}, 3.5386163203784432`*^9, {3.538616567348569*^9, 3.5386165974832926`*^9}}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"Clear", "[", RowBox[{ "ii", ",", " ", "r", ",", " ", "\[Theta]", ",", " ", "varR", ",", " ", "z", ",", " ", "varL", ",", " ", "q", ",", " ", "jj", ",", " ", "kk", ",", " ", "ll", ",", " ", "a"}], "]"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{"ii", " ", "=", " ", RowBox[{"Integrate", "[", RowBox[{ RowBox[{"1", "/", RowBox[{"Sqrt", "[", RowBox[{ RowBox[{"z", "^", "2"}], " ", "+", " ", RowBox[{"q", "^", "2"}]}], "]"}]}], ",", " ", RowBox[{"{", RowBox[{"z", ",", " ", RowBox[{"-", "varL"}], ",", " ", "varL"}], "}"}]}], "]"}]}], "\[IndentingNewLine]", RowBox[{"jj", " ", "=", " ", RowBox[{"ii", " ", "/.", " ", RowBox[{"q", " 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