(* 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[ 88437, 2700] NotebookOptionsPosition[ 85502, 2602] NotebookOutlinePosition[ 85844, 2617] CellTagsIndexPosition[ 85801, 2614] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["\<\ Potential due to cylindrial distribution\ \>", "Title", CellChangeTimes->{{3.539379651175579*^9, 3.539379661524171*^9}}], Cell[TextData[{ "Consider a cylindrical distribution of mass (or charge) as in the figure, \ with points in the cylinder given by ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ RowBox[{ TagBox["r", HoldForm], "'"}], StripOnInput->False, FontFamily->"Arial Black"], StyleBox[" ", StripOnInput->False, FontFamily->"Arial Black"], StyleBox["=", StripOnInput->False, FontFamily->"Arial Black"], StyleBox[" ", StripOnInput->False, FontFamily->"Arial Black"], RowBox[{ StyleBox["(", StripOnInput->False, FontFamily->"Arial Black"], RowBox[{ RowBox[{"r", "'"}], ",", " ", RowBox[{"\[Theta]", "'"}], ",", " ", RowBox[{"z", "'"}]}], ")"}]}], TraditionalForm]]], " coordinates, and the point of measurement of the potential measured at ", Cell[BoxData[ FormBox[ RowBox[{ StyleBox[ TagBox["r", HoldForm], StripOnInput->False, FontFamily->"Arial Black"], StyleBox[" ", StripOnInput->False, FontFamily->"Arial Black"], StyleBox["=", StripOnInput->False, FontFamily->"Arial Black"], StyleBox[" ", StripOnInput->False, FontFamily->"Arial Black"], RowBox[{ StyleBox["(", StripOnInput->False, FontFamily->"Arial Black"], RowBox[{"r", ",", " ", "0", ",", " ", "0"}], ")"}]}], TraditionalForm]]], ". Our potential, for a uniform distribution, will be proportional to\n\n", StyleBox["\[Phi](r) = \[Integral] ", FontSize->36], Cell[BoxData[ FormBox[ StyleBox[ FractionBox[ RowBox[{" ", StyleBox[ RowBox[{ TagBox[ RowBox[{"\[DifferentialD]", "V"}], HoldForm], "'"}], StripOnInput->False, FontFamily->"Arial Black"]}], TemplateBox[{RowBox[{ StyleBox[ TagBox["r", HoldForm], StripOnInput -> False, FontFamily -> "Arial Black"], "-", SuperscriptBox[ StyleBox[ TagBox["r", HoldForm], StripOnInput -> False, FontFamily -> "Arial Black"], "\[Prime]", MultilineFunction -> None]}]}, "Abs"]], FontSize->24], TraditionalForm]], EmphasizeSyntaxErrors->True, FontSize->36], StyleBox["= ", FontSize->36], Cell[BoxData[ FormBox[ TagBox[ RowBox[{ SubsuperscriptBox["\[Integral]", "0", "R"], RowBox[{ RowBox[{"r", "'"}], " ", RowBox[{"\[DifferentialD]", SuperscriptBox["r", "\[Prime]", MultilineFunction->None]}], RowBox[{ SubsuperscriptBox["\[Integral]", "0", RowBox[{"2", " ", "\[Pi]"}]], RowBox[{ RowBox[{"\[DifferentialD]", SuperscriptBox["\[Theta]", "\[Prime]", MultilineFunction->None]}], RowBox[{ SubsuperscriptBox["\[Integral]", RowBox[{"-", "L"}], "L"], FractionBox[ RowBox[{"\[DifferentialD]", RowBox[{"z", "'"}], " "}], SqrtBox[ RowBox[{ SuperscriptBox[ RowBox[{"(", RowBox[{"z", "'"}], ")"}], "2"], "+", FormBox[ SuperscriptBox[ TemplateBox[{RowBox[{"r", "-", RowBox[{ SuperscriptBox["\[ExponentialE]", RowBox[{"\[ImaginaryI]", " ", "\[Theta]"}]], " ", SuperscriptBox[ "r", "\[Prime]", MultilineFunction -> None]}]}]}, "Abs"], "2"], TraditionalForm]}]]]}]}]}]}]}], HoldForm], TraditionalForm]], FontSize->36], "\nWith\n\n", Cell[BoxData[ FormBox[ TagBox[ TagBox[ RowBox[{ TagBox[ RowBox[{ SubsuperscriptBox["\[Integral]", RowBox[{"-", "L"}], "L"], RowBox[{ FractionBox["1", SqrtBox[ RowBox[{ SuperscriptBox["z", "2"], "+", SuperscriptBox["u", "2"]}]]], RowBox[{"\[DifferentialD]", "z"}]}]}], HoldForm], "=", RowBox[{"log", "(", FractionBox[ RowBox[{"L", "+", SqrtBox[ RowBox[{ SuperscriptBox["L", "2"], "+", SuperscriptBox["u", "2"]}]]}], RowBox[{ RowBox[{"-", "L"}], "+", SqrtBox[ RowBox[{ SuperscriptBox["L", "2"], "+", SuperscriptBox["u", "2"]}]]}]], ")"}]}], HoldForm], HoldForm], TraditionalForm]], FontSize->24], "\n\nThis is found to be\n\n", Cell[BoxData[ RowBox[{ RowBox[{"\[Phi]", RowBox[{"(", "r", ")"}]}], "=", " ", RowBox[{ SubsuperscriptBox["\[Integral]", "0", "R"], RowBox[{ RowBox[{"r", "'"}], " ", RowBox[{"\[DifferentialD]", SuperscriptBox["r", "\[Prime]", MultilineFunction->None]}], RowBox[{ SubsuperscriptBox["\[Integral]", "0", RowBox[{"2", " ", "\[Pi]"}]], RowBox[{"\[DifferentialD]", SuperscriptBox["\[Theta]", "\[Prime]", MultilineFunction->None]}]}]}]}]}]], FontSize->36], StyleBox[" ", FontSize->36], Cell[BoxData[ TagBox[ TagBox[ RowBox[{"log", RowBox[{"(", FractionBox[ RowBox[{"L", "+", SqrtBox[ RowBox[{ SuperscriptBox["L", "2"], "+", SuperscriptBox[ TemplateBox[{RowBox[{"r", "-", RowBox[{ SuperscriptBox["r", "\[Prime]", MultilineFunction -> None], " ", SuperscriptBox["\[ExponentialE]", RowBox[{" ", RowBox[{"\[ImaginaryI]", " ", "\[Theta]"}]}]]}]}]}, "Abs"], "2"]}]]}], RowBox[{ RowBox[{"-", "L"}], "+", SqrtBox[ RowBox[{ SuperscriptBox["L", "2"], "+", SuperscriptBox[ TemplateBox[{RowBox[{"r", "-", RowBox[{ SuperscriptBox["r", "\[Prime]", MultilineFunction -> None], " ", SuperscriptBox["\[ExponentialE]", RowBox[{" ", RowBox[{"\[ImaginaryI]", " ", "\[Theta]"}]}]]}]}]}, "Abs"], "2"]}]]}]], ")"}]}], HoldForm], HoldForm]], CellChangeTimes->{3.5393806496636896`*^9, 3.5393807014136496`*^9}, FontSize->36], "\n\nIt is clear that we can\[CloseCurlyQuote]t evaluate this limit for ", Cell[BoxData[ FormBox[ RowBox[{"L", " ", "\[Rule]", " ", "\[Infinity]"}], TraditionalForm]]], " since that gives us ", Cell[BoxData[ FormBox[ RowBox[{"\[Infinity]", "/", "0"}], TraditionalForm]]], " in the logarithm. 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