(* 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[ 61046, 1878] NotebookOptionsPosition[ 60107, 1832] NotebookOutlinePosition[ 60449, 1847] CellTagsIndexPosition[ 60406, 1844] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Potential for an infinitesimal width plane. Take III.\ \>", "Title", CellChangeTimes->{{3.5390476089522343`*^9, 3.5390476265302396`*^9}, { 3.53905388213704*^9, 3.5390538842141585`*^9}}], Cell[TextData[{ "I\[CloseCurlyQuote]d like to attempt again to evaluate the potential for \ infinite plane distribution. The general form of our potential takes the \ form\n\n ", Cell[BoxData[ FormBox[ StyleBox[ RowBox[{ RowBox[{"\[Phi]", "(", InterpretationBox[ StyleBox["x", StripOnInput->False, FontFamily->"Helvetica", FontWeight->Bold], $CellContext`bX], ")"}], "=", RowBox[{"G", " ", "\[Rho]", " ", RowBox[{"\[Integral]", RowBox[{ FormBox[ FractionBox["1", TemplateBox[{RowBox[{ InterpretationBox[ StyleBox[ "x", StripOnInput -> False, FontFamily -> "Helvetica", FontWeight -> Bold], $CellContext`bX], "-", SuperscriptBox[ InterpretationBox[ StyleBox[ "x", StripOnInput -> False, FontFamily -> "Helvetica", FontWeight -> Bold], $CellContext`bX], "\[Prime]", MultilineFunction -> None]}]}, "Abs"]], TraditionalForm], RowBox[{"\[DifferentialD]", SuperscriptBox["V", "\[Prime]", MultilineFunction->None]}]}]}]}]}], FontSize->24], TraditionalForm]]], "\n \n We want to evaluate this with cylindrical coordinates ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ RowBox[{"r", "'"}], ",", " ", RowBox[{"\[Theta]", "'"}], ",", " ", RowBox[{"z", "'"}]}], ")"}], TraditionalForm]]], " , for a width \[Epsilon], and radius r, at distance z from the plane.\n \n", Cell[BoxData[ StyleBox[ TagBox[ RowBox[{ RowBox[{"\[Phi]", RowBox[{"(", RowBox[{"z", ",", " ", "\[Epsilon]", ",", " ", "r"}], ")"}]}], "=", " ", RowBox[{"2", " ", "\[Pi]", " ", "G", " ", "\[Sigma]", " ", FormBox[ FractionBox["1", "\[Epsilon]"], TraditionalForm], RowBox[{ SubsuperscriptBox["\[Integral]", RowBox[{ RowBox[{"r", "'"}], " ", "=", " ", "0"}], "r"], RowBox[{ RowBox[{"\[DifferentialD]", SuperscriptBox["r", "\[Prime]", MultilineFunction->None]}], RowBox[{ SubsuperscriptBox["\[Integral]", RowBox[{ RowBox[{"z", "'"}], " ", "=", " ", "0"}], "\[Epsilon]"], RowBox[{ RowBox[{"\[DifferentialD]", SuperscriptBox["z", "\[Prime]", MultilineFunction->None]}], FractionBox[ SuperscriptBox["r", "\[Prime]", MultilineFunction->None], SqrtBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"z", "-", SuperscriptBox["z", "\[Prime]", MultilineFunction->None]}], ")"}], "2"], "+", SuperscriptBox[ RowBox[{"(", SuperscriptBox["r", "\[Prime]", MultilineFunction->None], ")"}], "2"]}]]]}]}]}]}]}]}], HoldForm], FontSize->18]], "Input", CellChangeTimes->{{3.539049089473915*^9, 3.539049122975831*^9}}], "\n\nWith the assumption that we will take the limits ", Cell[BoxData[ FormBox[ RowBox[{"\[Epsilon]", " ", "\[Rule]", " ", "0"}], TraditionalForm]]], ", and ", Cell[BoxData[ FormBox[ RowBox[{"r", " ", "\[Rule]", " ", RowBox[{"\[Infinity]", ".", " "}]}], TraditionalForm]]], "With ", Cell[BoxData[ FormBox[ RowBox[{"r", "=", " ", FormBox[ FractionBox["c", SqrtBox["e"]], TraditionalForm]}], TraditionalForm]]], ", this does not convege. How about with ", Cell[BoxData[ FormBox[ RowBox[{"r", "=", " ", FormBox[ FractionBox["c", "e"], TraditionalForm]}], TraditionalForm]]], ", ?\n\nPerforming the r\[CloseCurlyQuote] integration (with ", Cell[BoxData[ FormBox[ RowBox[{"r", "=", " ", FormBox[ FractionBox["c", "e"], TraditionalForm]}], TraditionalForm]]], ") we find\n\n", Cell[BoxData[ StyleBox[ TagBox[ RowBox[{ RowBox[{"\[Phi]", RowBox[{"(", RowBox[{"z", ",", " ", "\[Epsilon]"}], ")"}]}], "=", " ", RowBox[{"2", " ", "\[Pi]", " ", "G", " ", "\[Sigma]", " ", FormBox[ FractionBox["1", "\[Epsilon]"], TraditionalForm], RowBox[{ SubsuperscriptBox["\[Integral]", RowBox[{ RowBox[{"z", "'"}], " ", "=", " ", "0"}], "\[Epsilon]"], RowBox[{ RowBox[{"(", RowBox[{ SqrtBox[ RowBox[{ FractionBox[ SuperscriptBox["c", "2"], SuperscriptBox["\[Epsilon]", "2"]], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"z", "-", RowBox[{"z", "'"}]}], ")"}], "2"]}]], "-", SqrtBox[ SuperscriptBox[ RowBox[{"(", RowBox[{"z", "-", RowBox[{"z", "'"}]}], ")"}], "2"]]}], ")"}], RowBox[{"\[DifferentialD]", SuperscriptBox["z", "\[Prime]", MultilineFunction->None]}]}]}]}]}], HoldForm], FontSize->18]], "Input", CellChangeTimes->{{3.539049089473915*^9, 3.539049122975831*^9}}], "\n\nAttempting to let ", StyleBox["Mathematica", FontSlant->"Italic"], " evaluate this takes a long time. Long enough that I aborted the attempt \ to evaluate it.\n\nInstead, first evaluating the z\[CloseCurlyQuote] integral \ we have\n\n", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"\[Phi]", "(", RowBox[{"z", ",", " ", "\[Epsilon]", ",", " ", "r"}], ")"}], "=", FractionBox[ RowBox[{"2", " ", "\[Pi]", " ", "G", " ", "\[Sigma]"}], "\[Epsilon]"]}], TraditionalForm]], FontSize->18], Cell[BoxData[ FormBox[ RowBox[{" ", RowBox[{ SubsuperscriptBox["\[Integral]", RowBox[{ RowBox[{"r", "'"}], " ", "=", " ", "0"}], RowBox[{"c", "/", "\[Epsilon]"}]], RowBox[{"(", RowBox[{ RowBox[{"log", "(", RowBox[{ SqrtBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", SuperscriptBox["r", "\[Prime]", MultilineFunction->None], ")"}], "2"], "+", SuperscriptBox["z", "2"]}]], "+", "z"}], ")"}], "-", RowBox[{"log", "(", RowBox[{ SqrtBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", SuperscriptBox["r", "\[Prime]", MultilineFunction->None], ")"}], "2"], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"z", "-", "\[Epsilon]"}], ")"}], "2"]}]], "+", "z", "-", "\[Epsilon]"}], ")"}]}], ")"}]}]}], TraditionalForm]], FontSize->18], StyleBox[" ", FontSize->18], Cell[BoxData[ RowBox[{"\[DifferentialD]", SuperscriptBox["r", "\[Prime]", MultilineFunction->None]}]], CellChangeTimes->{{3.539049089473915*^9, 3.539049122975831*^9}}, FontSize->18], "\n\nThis second integral can then be evaluated in reasonable time:\n\n", Cell[BoxData[{ FormBox[ RowBox[{"\[Phi]", "(", RowBox[{"z", ",", " ", "\[Epsilon]"}], ")"}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{"=", " ", RowBox[{ FractionBox[ RowBox[{"2", " ", "\[Pi]", " ", "G", " ", "\[Sigma]"}], SuperscriptBox["\[Epsilon]", "2"]], " ", RowBox[{"(", RowBox[{ RowBox[{"c", " ", RowBox[{"log", "(", FractionBox[ RowBox[{ SqrtBox[ RowBox[{ FractionBox[ SuperscriptBox["c", "2"], SuperscriptBox["\[Epsilon]", "2"]], "+", SuperscriptBox["z", "2"]}]], "+", "z"}], RowBox[{ SqrtBox[ RowBox[{ FractionBox[ SuperscriptBox["c", "2"], SuperscriptBox["\[Epsilon]", "2"]], "+", SuperscriptBox[ RowBox[{"(", RowBox[{"z", "-", "\[Epsilon]"}], ")"}], "2"]}]], "+", "z", "-", "\[Epsilon]"}]], ")"}]}], "+", RowBox[{"\[Epsilon]", " ", RowBox[{"(", RowBox[{ RowBox[{"z", " ", RowBox[{"log", "(", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"z", "-", "\[Epsilon]"}], ")"}], " ", RowBox[{"(", RowBox[{ SqrtBox[ RowBox[{ SuperscriptBox["c", "2"], "+", RowBox[{ SuperscriptBox["z", "2"], " ", SuperscriptBox["\[Epsilon]", "2"]}]}]], "+", "c"}], ")"}]}], "z"], ")"}]}], "+", RowBox[{ RowBox[{"(", RowBox[{"\[Epsilon]", "-", "z"}], ")"}], " ", RowBox[{"log", "(", RowBox[{ SqrtBox[ RowBox[{ SuperscriptBox["c", "2"], "+", RowBox[{ SuperscriptBox["\[Epsilon]", "2"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"z", "-", "\[Epsilon]"}], ")"}], "2"]}]}]], "+", "c"}], ")"}]}], "-", RowBox[{"\[Epsilon]", " ", RowBox[{"log", "(", RowBox[{"\[Epsilon]", " ", RowBox[{"(", RowBox[{"z", "-", "\[Epsilon]"}], ")"}]}], ")"}]}]}], ")"}]}]}], ")"}]}]}], TraditionalForm], "\[IndentingNewLine]", FormBox[ RowBox[{"=", RowBox[{"2", " ", "\[Pi]", " ", "G", " ", "\[Sigma]", " ", RowBox[{"(", RowBox[{ RowBox[{ FormBox[ FractionBox["c", SuperscriptBox["\[Epsilon]", "2"]], TraditionalForm], RowBox[{"log", "(", FormBox[ FractionBox[ RowBox[{ SqrtBox[ RowBox[{ SuperscriptBox["c", "2"], "+", RowBox[{ SuperscriptBox["z", "2"], " ", SuperscriptBox["\[Epsilon]", "2"]}]}]], "+", RowBox[{"z", " ", "\[Epsilon]"}]}], RowBox[{ SqrtBox[ RowBox[{ SuperscriptBox["c", "2"], "+", SuperscriptBox[ RowBox[{ SuperscriptBox["\[Epsilon]", "2"], "(", RowBox[{"z", "-", "\[Epsilon]"}], ")"}], "2"]}]], "+", RowBox[{"\[Epsilon]", "(", RowBox[{"z", "-", "\[Epsilon]"}], ")"}]}]], TraditionalForm], ")"}]}], "+", RowBox[{ FormBox[ FractionBox["z", "\[Epsilon]"], TraditionalForm], " ", RowBox[{"log", "(", FractionBox[ RowBox[{ RowBox[{"(", RowBox[{"z", "-", "\[Epsilon]"}], ")"}], " ", RowBox[{"(", RowBox[{ SqrtBox[ RowBox[{ SuperscriptBox["c", "2"], "+", RowBox[{ SuperscriptBox["z", "2"], " ", SuperscriptBox["\[Epsilon]", "2"]}]}]], "+", "c"}], ")"}]}], RowBox[{"z", "(", RowBox[{ SqrtBox[ RowBox[{ SuperscriptBox["c", "2"], "+", RowBox[{ SuperscriptBox["\[Epsilon]", "2"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"z", "-", "\[Epsilon]"}], ")"}], "2"]}]}]], "+", "c"}], ")"}]], ")"}]}], "+", " ", RowBox[{"log", "(", FractionBox[ RowBox[{ SqrtBox[ RowBox[{ SuperscriptBox["c", "2"], "+", RowBox[{ SuperscriptBox["\[Epsilon]", "2"], " ", SuperscriptBox[ RowBox[{"(", RowBox[{"z", "-", "\[Epsilon]"}], ")"}], "2"]}]}]], "+", "c"}], RowBox[{"\[Epsilon]", " ", RowBox[{"(", RowBox[{"z", "-", "\[Epsilon]"}], ")"}]}]], ")"}]}], ")"}]}]}], TraditionalForm]}], FontSize->24], "\n\n Does this have a limit as \[Epsilon] \[Rule] 0? 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