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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 80411, 2971]*) (*NotebookOutlinePosition[ 81105, 2996]*) (* CellTagsIndexPosition[ 81061, 2992]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Legal Notice, Disclaimer, and Book Ordering Information:", \ "Subsubsection", ImageRegion->{{0, 1}, {0, 1}}], Cell[TextData[{ "(* Legal Notice, Disclaimer, and Book Ordering Information:\n\nLast \ Modified: FIO June 8, 2002\n\n", StyleBox["Mathematica For Physics: 2nd Edition", FontColor->RGBColor[1, 0, 0]], "\n\nRobert L. Zimmerman and Fredrick I. Olness\n", "\nWebSite: \n", StyleBox[" www.physics.smu.edu/~olness\n darkwing.uoregon.edu/~phys600/\n\ \n", FontColor->RGBColor[1, 0, 0]], "MathSource Number: 0206-862", StyleBox["\n", FontColor->RGBColor[1, 0, 0]], "\nISBN 0-201-53796-6\nFor ordering information, call 1-800-282-0693\n\ Addison-Wesley Publishing Company\n", "\nCommunication with the authors:\nFredrick I. Olness: \ olness@mail.physics.smu.edu\nRobert L. Zimmerman: bob@zim.uoregon.edu\n\n\ Copyright 2002, Addison-Wesley Publishing Company, Inc. The material in\nthis \ file may be distributed freely so long as the content remains\nunchanged, \ and the copyright and reference notices are included. The\npublisher grants \ permission for the noncommercial use of these programs\nand program segments. \ All other uses require the prior written consent\nof the publisher. \n\n\ All rights are reserved with regard to the material in the\n\"Mathematica \ For Physics\" text, and may not be reproduced, stored in a\nretrieval system, \ or transmitted, in any form or by any means,\nelectronic, mechanical, \ photocopying, recording, or otherwise, without\nthe prior written permission \ of the publisher. \n\nThe programs and applications presented in this book \ have been included\nfor their instructional value. They have been tested \ with care but are\nnot guaranteed for any particular purpose. The publisher \ does not\noffer any warranties or representations, nor does it accept any\n\ liabilities with respect to the programs or applications. \n\n *)\n" }], "Input", Evaluatable->False, ImageRegion->{{0, 1}, {0, 1}}] }, Closed]], Cell["8. Electrostatics", "Title", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Introduction", "Section", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["\<\ Off[General::spell ]; Off[General::spell1]; \ \>", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True] }, Closed]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " Commands for All Sections" }], "Section", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["VEPlot", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[BoxData[ \(Needs["\"]\)], "Input", Hyphenation->False], Cell[BoxData[{ \(\(\(VEPlot[potential_, xlim_, ylim_, opts___] := Module[{plot1, plot2}, \[IndentingNewLine]plot1 = PlotGradientField[\(-potential\), xlim, ylim, ScaleFunction \[Rule] \((1 &)\), DisplayFunction \[Rule] Identity]; \[IndentingNewLine]plot2 = ContourPlot[potential, xlim, ylim, ContourShading \[Rule] False, ContourSmoothing \[Rule] True, DisplayFunction \[Rule] Identity, PlotPoints \[Rule] 50]; \[IndentingNewLine]\ Show[{plot1, plot2}, opts, DisplayFunction \[Rule] $DisplayFunction\ ]];\)\(\ \[IndentingNewLine]\) \)\), "\[IndentingNewLine]", \(VEPlot[potential_, opts___] := VEPlot[potential, {x, 1.1, \(-1.1\)}, {y, 1.1, \(-1.1\)}, opts]\)}], "Input", Hyphenation->False], Cell["Protect[VEPlot];", "Input", Hyphenation->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Example: Equipotential surface and electric field of two-point \ charges\ \>", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[BoxData[ RowBox[{ RowBox[{"dipole", "=", FormBox[\(1\/\@\(x\^2 + \((\(-y\) - 1\/2)\)\^2\) - 1\/\@\(x\^2 + \((1\/2 - y)\)\^2\)\), "TraditionalForm"]}], ";"}]], "Input", Hyphenation->False], Cell[BoxData[ \(\(\(\ \)\(VEPlot[dipole, Epilog \[Rule] {{Hue[ .3], \ Disk[{0, 1/2}, 0.1]}, {Hue[0.95], \ Disk[{0, \(-1\)/2}, 0.1]}}];\)\)\)], "Input", Hyphenation->False] }, Closed]], Cell["8.1 Point Charges, Multipoles, and Image Charges", "Section", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["Overview of Point Charges, Multipoles, and Image Charges", "Subsection", Evaluatable->False, Hyphenation->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " Commands for Section 8.1" }], "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Monopole", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["\<\ Monopole[q_:1,r0_:{0,0,0},r_:{x,y,z}]:= q/Sqrt[ Sum[(r0[[i]]-r[[i]])^2,{i,1,Length[r0]}] ]; \ \>", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Example", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["Monopole[q,{x0,y0},{x,y}]", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["Monopole[q,{x0,y0}]", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["\<\ dipole= ( Monopole[+1,{0,0,-a/2}] + Monopole[-1,{0,0,+a/2}] )\ \>", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["TrigToY", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["\<\ TrigToY[expression_,terms_:2]:= Module[{el,m,result}, result= Sum[ Integrate[ (-1)^(m) SphericalHarmonicY[el,-m,\[Theta],\[Phi]] * expression Sin[\[Theta]] ,{\[Theta],0,\[Pi]},{\[Phi],0,2 \[Pi]}] * Y[el,m,\[Theta],\[Phi]] ,{el,0,terms},{m,-el,el}]; Return[result] ];\ \>", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Example", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["eq1= Cos[\[Theta]]^2 //TrigToY", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[BoxData[ \(eq1 //. Y \[Rule] SphericalHarmonicY // Simplify\)], "Input", Hyphenation->False] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell[" TrigToP", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["\<\ TrigToP[expression_,terms_:2]:= Module[{el,result}, result= Sum[ Integrate[ LegendreP[el,Cos[\[Theta]]] expression Sin[\[Theta]] ,{\[Theta],0,\[Pi]}] * (2 el+1)/2 P[el,Cos[\[Theta]]] ,{el,0,terms}]; Return[result] ];\ \>", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Example", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[BoxData[ \(eq1 = Sin[\[Theta]]\^2 // TrigToP\)], "Input", Hyphenation->False], Cell[BoxData[ \(eq1 //. P \[Rule] LegendreP // Simplify\)], "Input", Hyphenation->False] }, Closed]] }, Closed]], Cell["\<\ Protect[Monopole,TrigToY,TrigToP]; \ \>", "Input", Hyphenation->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Problem 1: Superposition of point charges", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["Remarks and outline", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Required packages", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["\<\ Needs[\"Calculus`VectorAnalysis`\"] Needs[\"Graphics`PlotField`\"] Needs[\"Graphics`PlotField3D`\"]\ \>", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Solution", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["Clear[\"Global`*\"];", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Part a", "Subsubsection", Hyphenation->False], Cell["\<\ quad=( Monopole[2 q,{0,0, 0} ]+ Monopole[ -q,{0,0, d/2} ]+ Monopole[ -q,{0,0,-d/2} ]) \ \>", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["\<\ triangle=( Monopole[+2q, { 0,0, 0} ]+ Monopole[ -q, {+d/2,0,-d/2} ]+ Monopole[ -q, {-d/2,0,-d/2} ]) ;\ \>", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["\<\ square=( Monopole[+q,{+d/2,0, d/2} ]+ Monopole[-q,{+d/2,0,-d/2} ]+ Monopole[+q,{-d/2,0,-d/2} ]+ Monopole[-q,{-d/2,0, d/2} ]) ;\ \>", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[BoxData[ \(\(Plot3D[ square\ /. {y \[Rule] 0, q \[Rule] 1, d \[Rule] 1}\[IndentingNewLine], {x, \(-1\), 1}, {z, \(-1\), 1}\[IndentingNewLine], AxesLabel \[Rule] {"\", "\", "\<\>"}];\)\)], "Input", Hyphenation->False] }, Closed]], Cell[CellGroupData[{ Cell["Part b", "Subsubsection", Hyphenation->False], Cell[CellGroupData[{ Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(charges = \ {{Hue[0.3], \ Disk[{0, 1/2}, 0.1]}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {Hue[0.95], Disk[{0, 0}, 0.2]}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {Hue[ 0.3], \ Disk[{0, \(-1\)/2}, 0.1]}};\)\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(\(\[IndentingNewLine]\)\(p1 = PlotGradientField[\(-quad\) //. {q \[Rule] 1, d \[Rule] 1, y \[Rule] 0}\[IndentingNewLine], {x, \(-1.1\), 1.1}, {z, \(-1.1\), 1.1}\[IndentingNewLine], ScaleFunction \[Rule] \((1 &)\), Epilog \[Rule] charges];\)\)\)], "Input", Hyphenation->False] }, Closed]], Cell[BoxData[ \(\(p2 = ContourPlot[ quad //. {q \[Rule] 1, d \[Rule] 1, y \[Rule] 0}, \[IndentingNewLine]{x, \(-1.1\), 1.1}, {z, \(-1.1\), 1.1}, \[IndentingNewLine]ContourShading \[Rule] False, \[IndentingNewLine]PlotPoints \[Rule] 50, \[IndentingNewLine]DisplayFunction \[Rule] Identity\[IndentingNewLine]];\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(Show[p1, p2];\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(VEPlot[ quad //. {q \[Rule] 1, d \[Rule] 1, y \[Rule] 0}, \[IndentingNewLine]{x, \(-1.1\), 1.1}, {z, \(-1.1\), 1.1}, \[IndentingNewLine]Epilog \[Rule] charges];\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(charges = \ {{Hue[0.3], \ Disk[{\(-1\)/2, \(-1\)/2}, 0.1]}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {Hue[0.95], Disk[{0, 0}, 0.2]}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {Hue[ 0.3], \ Disk[{1/2, \(-1\)/2}, 0.1]}};\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(VEPlot[ triangle /. {q \[Rule] 1, d \[Rule] 1, y \[Rule] 0}, \[IndentingNewLine]{x, \(-1.1\), 1.1}, {z, \(-1.1\), 1.1}, \[IndentingNewLine]Epilog \[Rule] charges];\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(charges = \ {{Hue[0.95], \ Disk[{\(-1\)/2, \(-1\)/2}, 0.1]}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {Hue[0.95], Disk[{1/2, 1/2}, 0.1]}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {Hue[0.3], \ Disk[{\(-1\)/2, \ 1/2}, 0.1]}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {Hue[ 0.3], \ Disk[{\ 1/2, \(-\ 1\)/2}, 0.1]}};\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(VEPlot[ square /. {q \[Rule] 1, d \[Rule] 1, y \[Rule] 0}, {x, \(-1.1\), 1.1}, {z, \(-1.1\), 1.1}, Epilog \[Rule] charges];\)\)], "Input", Hyphenation->False] }, Closed]], Cell[CellGroupData[{ Cell["Part c", "Subsubsection", Hyphenation->False], Cell[BoxData[ \(x2rRule = Thread[{x, y, z} \[Rule] CoordinatesToCartesian[{r, \[Theta], \[Phi]}, Spherical]]\)], "Input", Hyphenation->False], Cell[BoxData[ \(seriesQuad = \ \(Series[quad\ //. x2rRule\ , {r, \[Infinity], 2}]\ // Simplify\) // Normal\)], "Input", Hyphenation->False], Cell[" seriesQuad //TrigToP //Simplify", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[BoxData[ \(seriesSquare = \ \(Series[ square\ //. x2rRule\ , {r, \[Infinity], 2}]\ \ // Simplify\)\ // Normal\)], "Input", Hyphenation->False], Cell[" seriesSquare //TrigToY //Simplify", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[BoxData[ \(seriesTriangle = \ \(Series[ triangle\ //. x2rRule\ , {r, \[Infinity], 2}]\ \ // FullSimplify\) // Normal\)], "Input", Hyphenation->False], Cell["seriesTriangle //TrigToY//Simplify ", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Part d", "Subsubsection", Hyphenation->False], Cell[BoxData[ \(\(p1 = PlotGradientField3D[ quad //. {q \[Rule] 1, d \[Rule] 1}, {x, \(-1\), 1}, {y, \(-1\), 1}, {z, \(-1\), 1}, \[IndentingNewLine]ScaleFunction \[Rule] \((1 &)\), \ \[IndentingNewLine]VectorHeads \[Rule] True, \[IndentingNewLine]PlotPoints \[Rule] 6, \[IndentingNewLine]BoxRatios \[Rule] {1, 1, 2}, \[IndentingNewLine]ViewPoint \[Rule] {1.3, \(-2.4\), 1}];\)\)], "Input", Hyphenation->False] }, Closed]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Problem 2: Point charges and grounded plane", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["Remarks and outline", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Required packages", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["\<\ Needs[\"Calculus`VectorAnalysis`\"] Needs[\"Graphics`PlotField`\"] \ \>", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Solution", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["Clear[\"Global`*\"];", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Part a", "Subsubsection", Hyphenation->False], Cell["\<\ dipole= ( Monopole[+q,{0,0,+d}] + Monopole[-q,{0,0,-d}] )\ \>", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[" dipole/.z->0", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Part b", "Subsubsection", Hyphenation->False], Cell[BoxData[ \(x2rRule = Thread[{x, y, z} \[Rule] CoordinatesToCartesian[{r, \[Theta], phi}, Spherical]]\)], "Input", Hyphenation->False], Cell[BoxData[ \(dipoleR = \ \(Series[dipole //. x2rRule\ , {r, \[Infinity], 2}]\ // Normal\) // Simplify\)], "Input", Hyphenation->False], Cell["dipoleR//TrigToP", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["\<\ efieldR= -Grad[dipoleR,Spherical[r,\[Theta],\[Phi]]]\ \>", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Part c", "Subsubsection", Hyphenation->False], Cell[BoxData[ \(charge = \(-1\)/\((4\ \[Pi])\)\ D[dipole, z] //. z \[Rule] 0\)], "Input",\ Hyphenation->False], Cell["\<\ Integrate[charge,{y,-\[Infinity],+\[Infinity]},{x,-\[Infinity],+\ \[Infinity]}]//PowerExpand\ \>", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[BoxData[ \(\(ContourPlot[ charge /. {d \[Rule] 1, q \[Rule] 1}, {y, \(-1\), 1}, {x, \(-1\), 1}];\)\)], "Input", Hyphenation->False] }, Closed]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Problem 3: Point charges and grounded sphere", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["Remarks and outline", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["Required packages", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["\<\ Needs[\"Calculus`VectorAnalysis`\"] Needs[\"Graphics`PlotField`\"] \ \>", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Solution", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["Clear[\"Global`*\"];", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Part a", "Subsubsection", Hyphenation->False], Cell[BoxData[ \(\(eq1 = {q\/\(a + d\) + q1\/\(a + d1\) == 0, q\/\(\(-a\) + d\) + q1\/\(a - d1\) == 0};\)\)], "Input", Hyphenation->False], Cell["imageRule= Solve[eq1,{d1,q1}] //ExpandAll //Flatten ", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[BoxData[ \(potential = \((Monopole[q, {0, 0, d}] + Monopole[q1, {0, 0, d1}])\) //. imageRule\)], "Input", Hyphenation->False] }, Closed]], Cell[CellGroupData[{ Cell["Part b", "Subsubsection", Hyphenation->False], Cell["\<\ x2rRule=Thread[{x,y,z}-> \ CoordinatesToCartesian[{r,\[Theta],\[Phi]},Spherical]]\ \>", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[" potentialR= potential //.x2rRule //Simplify //Expand", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[BoxData[ \(\(potentialR //. r \[Rule] a // Simplify\) // PowerExpand\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(Series[potentialR, {r, \[Infinity], 2}]\ // Normal\)\ \ // Collect[#, {\ Cos[\[Theta]], r, \ a\ }] &\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(VEPlot[ potential /. {a \[Rule] 0.2, d \[Rule] 0.4, q \[Rule] 1, y \[Rule] 0}, {x, \(-1\), 1}, {z, \(-1\), 1}, Epilog \[Rule] {\ {Hue[0.3], Disk[{0, 0.1\ }, 0.05]}, \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {Thickness[0.01], Circle[{0, 0}, 0.2]}\ \ , \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {Hue[0.95], Disk[{0, 0.4}, 0.1]}}];\)\)], "Input", Hyphenation->False] }, Closed]], Cell[CellGroupData[{ Cell["Part c", "Subsubsection", Hyphenation->False], Cell[BoxData[ RowBox[{"chargedensity", "=", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{ FormBox[\(-\(1\/\(4\ \[Pi]\)\)\), "TraditionalForm"], " ", \(D[potentialR, r]\)}], "/.", \({r \[Rule] a}\)}], "//", "Simplify"}], "//", "PowerExpand"}], "//", "Simplify"}], ")"}]}]], "Input", Hyphenation->False], Cell[BoxData[ \(totalcharge = \[IndentingNewLine]Integrate[ chargedensity\ \ 2\ \[Pi]\ a^2 Sin[\[Theta]], {\[Theta], 0, \[Pi]}]\ // Simplify\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(rule = {1\/\@\((a - d)\)\^2 \[Rule] 1\/\(\(-a\) + d\), \@\((a - d)\)\^2 \[Rule] \(-a\) + d};\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(totalcharge\ //. rule // PowerExpand\)\ // Simplify\)], "Input", Hyphenation->False] }, Closed]], Cell[CellGroupData[{ Cell["Part d", "Subsubsection", Hyphenation->False], Cell[BoxData[{ \(\(\(\[Theta]Min = \(+\((FindMinimum[\(+chargedensity\) /. {a \[Rule] 1, q \[Rule] 1, d \[Rule] 2}, {\[Theta], 1}] // First)\)\);\)\(\n\) \)\), "\[IndentingNewLine]", \(\(\(\[Theta]Max = \(-\((FindMinimum[\(-chargedensity\) /. {a \[Rule] 1, q \[Rule] 1, d \[Rule] 2}, {\[Theta], 1}] // First)\)\);\)\(\n\) \)\), "\[IndentingNewLine]", \(\(\(gray[\[Theta]_] = \((\((chargedensity - \ \[Theta]Min)\)/\((\[Theta]Max - \[Theta]Min)\) /. {a \[Rule] 1, q \[Rule] 1, d \[Rule] 2})\);\)\(\n\) \)\), "\[IndentingNewLine]", \({\[Theta]Min, \[Theta]Max}\)}], "Input", Hyphenation->False], Cell["\<\ radius=1; ParametricPlot3D[{radius Sin[\[Theta]] Cos[\[Phi]] ,radius Sin[\[Theta]] Sin[\[Phi]] ,radius Cos[\[Theta]] ,GrayLevel[gray[\[Theta]]]} ,{\[Theta],0,\[Pi]} ,{\[Phi],0,2 \[Pi]} ,Lighting->False ,Axes->False ,Boxed->False];\ \>", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True] }, Closed]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Problem 4: Line charge and grounded plane", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["Remarks and outline", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Required packages", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["\<\ Needs[\"Calculus`VectorAnalysis`\"] Needs[\"Graphics`PlotField`\"] \ \>", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Solution", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["Clear[\"Global`*\"];", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Part a", "Subsubsection", Hyphenation->False], Cell[BoxData[ \(\(potential = \(-q\)\ Log[x\^2 + \((y - y0)\)\^2] + q\ Log[x\^2 + \((y + y0)\)\^2];\)\)], "Input", Hyphenation->False], Cell["potential //.{y->0}", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[BoxData[ \(\(VEPlot[ potential //. {q \[Rule] 1, y0 \[Rule] 1}, {x, \(-1.1\), 1.1}, {y, \(-2\), 2}, Epilog \[Rule] {\ {Thickness[ .03], Line[{{\(-1.1\), 0}, {1.1, 0}}]\ }, \[IndentingNewLine]\ {Hue[ .35], Disk[{0, \ \(-1\)\ }, .1]}\ , \[IndentingNewLine]\ \ {Hue[ \ .95], Disk[{0, \ 1\ }, .1]}}];\)\)], "Input", Hyphenation->False], Cell["Laplacian[ potential,Cartesian[x,y,z]]//Together", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Part b", "Subsubsection", Hyphenation->False], Cell[BoxData[ RowBox[{"charge", "=", RowBox[{ RowBox[{ FormBox[\(-\(1\/\(4\ \[Pi]\)\)\), "TraditionalForm"], " ", \(D[potential, y]\)}], "/.", \({y \[Rule] 0}\)}]}]], "Input", Hyphenation->False], Cell[BoxData[ \(\(ContourPlot[ charge /. {q \[Rule] 1, y0 \[Rule] 1}, {x, \(-2\), 2}, 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AspectRatioFixed->True], Cell[BoxData[ \(eq1 \[Equal] eq2 //. {P \[Rule] LegendreP, Y \[Rule] SphericalHarmonicY} // Simplify\)], "Input", Hyphenation->False] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Required packages", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["\<\ Needs[\"Calculus`VectorAnalysis`\"] Needs[\"Graphics`PlotField`\"] \ \>", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Solution", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Part a", "Subsubsection", Hyphenation->False], Cell[BoxData[{ RowBox[{ RowBox[{"chargeDensity1", "=", FormBox[\(\(q0\ \((a - r)\)\ \((sin(\[Theta]) - \[Pi]\/4)\)\)\/a\^4\), "TraditionalForm"]}], ";"}], "\n", RowBox[{ RowBox[{"chargeDensity2", "=", FormBox[\(\(\[ExponentialE]\^\(\(a - r\)\/a\)\ q0\ \((a - 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P \[Rule] LegendreP // Simplify\)], "Input", Hyphenation->False], Cell[BoxData[ \(eField2 = \(-Grad[multi2, Spherical[r, \[Theta], \[Phi]]]\) //. P \[Rule] LegendreP // Simplify\)], "Input", Hyphenation->False], Cell[BoxData[{ \(\(eField3 = \(-Grad[multi3, Spherical[r, \[Theta], \[Phi]]]\);\)\), "\n", \(\(eField3 /. {Y \[Rule] SphericalHarmonicY} // ExpToTrig\) // Simplify\)}], "Input", Hyphenation->False] }, Closed]], Cell[CellGroupData[{ Cell["Part d", "Subsubsection", Hyphenation->False], Cell[BoxData[ \(pot1 = \ \(\(\(multi1 /. r2xRule\) /. {P \[Rule] LegendreP, Y \[Rule] SphericalHarmonicY}\) /. values\) /. {y \[Rule] 0}\ // Simplify\)], "Input", Hyphenation->False], Cell["VEPlot[pot1,{x,1.1,-1.1},{z,1.1,-1.1}];", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[BoxData[ \(pot2 = \ \(\(\(multi2 /. r2xRule\) /. {P \[Rule] LegendreP, Y \[Rule] SphericalHarmonicY}\) /. values\) /. {y \[Rule] 0}\)], "Input", Hyphenation->False], Cell["VEPlot[pot2,{x,-1.1,1.1},{z,-1.1,1.1}];", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[BoxData[ \(pot3 = \ \(\(\(\(\(multi3 /. r2xRule\) //. {P \[Rule] LegendreP, Y \[Rule] SphericalHarmonicY}\) //. values\) /. {z \[Rule] 0}\)\(\ \)\(//\)\(FullSimplify\)\(\ \)\)\)], "Input", Hyphenation->False], Cell["VEPlot[pot3];", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True] }, Closed]] }, Closed]] }, Closed]], Cell["\<\ 8.2 Laplace's Equation in Cartesian and Cylindrical Coordinates \ \ \>", "Section", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["Overview of Cartesian and Cylindrical Coordinates", "Subsection", Hyphenation->False], Cell[CellGroupData[{ Cell["\<\ Problem 1: Separation of variables in Cartesian and cylindrical \ coordinates\ \>", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Required packages", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["Needs[\"Calculus`VectorAnalysis`\"] ", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True] }, Closed]], Cell["Solution", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["Clear[\"Global`*\"];", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Part a", "Subsubsection", Hyphenation->False], Cell["potential= Vx[x] Vy[y] Vz[z];", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["SetCoordinates[Cartesian[x,y,z]];", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[" lapEq= Laplacian[potential]", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True] }, Closed]], Cell[BoxData[ \(eq1 = lapEq\/\(Vx[x]\ Vy[y]\ Vz[z]\) // Expand\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(\(\ \)\(eq2 = eq1 /. 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b}\)}], "]"}]}], " ", "//", \(Simplify[#, Element[{my, \ mx}, Integers]] &\)}], " ", ")"}], "/.", "mzRule"}]}]], "Input", Hyphenation->False], Cell[BoxData[ RowBox[{\(pot[nx_, ny_]\), "=", RowBox[{ RowBox[{"Sum", "[", RowBox[{ RowBox[{\(A[mx, my]\), " ", FormBox[\(\(sin(\(mx\ \[Pi]\ x\)\/a)\)\ \(sin(\(my\ \[Pi]\ \ y\)\/b)\)\ \(sinh(\(mz\ \[Pi]\ z\)\/c)\)\), "TraditionalForm"]}], ",", \({mx, 1, nx}\), ",", \({my, 1, ny}\)}], "]"}], "/.", "mzRule"}]}]], "Input", Hyphenation->False], Cell[BoxData[ \(pot[1, 1]\)], "Input", Hyphenation->False] }, Closed]], Cell[CellGroupData[{ Cell["Part b", "Subsubsection", Hyphenation->False], Cell[BoxData[ \(\(values = {a \[Rule] 1, b \[Rule] 1, c \[Rule] 1, V0 \[Rule] 1, x \[Rule] 1/2};\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(Plot3D[ pot[20, 20] //. values // Evaluate, {y, 0, 1}, {z, 0, 1}];\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(VEPlot[\(pot[10, 10] //. values\) //. {x \[Rule] 1/2}\ // Evaluate, {y, 0, 1}, {z, 0, 1}];\)\)], "Input", Hyphenation->False] }, Closed]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Problem 5: Conducting cylinder with a potential on the surface \ \ \>", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["Required packages", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[" Needs[\"Graphics`ParametricPlot3D`\"]", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["Solution", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["Clear[\"Global`*\"];", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Part a", "Subsubsection", Hyphenation->False], Cell[BoxData[ RowBox[{\(A[n_]\), " ", "=", RowBox[{ RowBox[{ FormBox[\(a\^\(-n\)\/\[Pi]\), "TraditionalForm"], \((Integrate[ V0\ \ Sin[n\ \[Phi]], {\[Phi], 0, \ \[Pi]\ }]\ - Integrate[ V0\ \ Sin[n\ \[Phi]], {\[Phi], \[Pi], 2\ \[Pi]\ }]\ )\)}], "//", \(Simplify[#, Element[n, Integers]] &\)}]}]], "Input", Hyphenation->False], Cell[BoxData[ \(\(potIn[nf_] := Sum[A[n]\ Sin[n\ \[Phi]]\ r\^n, {n, 1, nf}];\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(potIn[6]\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(\(eq1\)\(=\)\(\ \)\(potIn[Infinity]\)\(\ \)\)\)], "Input", Hyphenation->False], Cell[BoxData[ RowBox[{"Vin", "=", RowBox[{ RowBox[{"eq1", "//.", " ", RowBox[{"{", FormBox[\(log(1 - aa_) \[Rule] log(\(1 - aa\)\/\(aa + 1\)) + log(aa + 1)\), "TraditionalForm"], "}"}]}], "//", "FullSimplify", " "}]}]], "Input", Hyphenation->False], Cell[BoxData[ RowBox[{\(B[\ n_]\), " ", "=", RowBox[{ RowBox[{ FormBox[\(a\^n\/\[Pi]\), "TraditionalForm"], \((Integrate[ V0\ \ Sin[n\ \[Phi]], {\[Phi], 0, \ \[Pi]\ }]\ \[IndentingNewLine]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - Integrate[ V0\ \ Sin[n\ \[Phi]], {\[Phi], \[Pi], 2\ \[Pi]\ }]\ )\)}], "//", \(Simplify[#, Element[n, Integers]] &\)}]}]], "Input", Hyphenation->False], Cell[BoxData[ \(\(potOut[nf_] := Sum[B[n]\ Sin[n\ \[Phi]]\ r\^\(-n\), {n, 1, nf}];\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(potOut[3]\)], "Input", Hyphenation->False], Cell[BoxData[ RowBox[{"Vout", "=", RowBox[{ RowBox[{\((potOut[Infinity] // Simplify)\), "//.", " ", RowBox[{"{", FormBox[\(log(1 - aa_) \[Rule] log(\(1 - aa\)\/\(aa + 1\)) + log(aa + 1)\), "TraditionalForm"], "}"}]}], "//", " ", "Simplify", " "}]}]], "Input", Hyphenation->False] }, Closed]], Cell[CellGroupData[{ Cell["Part b", "Subsubsection", Hyphenation->False], Cell[BoxData[ \(Off[Plot3D::plnc, Plot3D::gval]\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(pt1 = Plot3D[\ \(Vin /. a \[Rule] 1\) /. V0 \[Rule] 1\ // Evaluate, \[IndentingNewLine]{\[Phi], 0, 2 \[Pi]}, {r, 0, 1\ }, \[IndentingNewLine]PlotPoints \[Rule] 40, \[IndentingNewLine]DisplayFunction \[Rule] Identity];\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(pt2 = Plot3D[\ \(Vout /. a \[Rule] 1\) /. V0 \[Rule] 1\ // Evaluate, \[IndentingNewLine]{\[Phi], 0, 2 \[Pi]}, {r, 1\ , 2\ }, \[IndentingNewLine]PlotPoints \[Rule] 40, \[IndentingNewLine]DisplayFunction \[Rule] Identity];\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(Off[Graphics3D::nlist3]\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(Show[{pt1, pt2}, DisplayFunction \[Rule] $DisplayFunction];\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(Off[ParametricPlot3D::pplr]\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(pt3 = CylindricalPlot3D[\(Vin /. a \[Rule] 1\) /. V0 \[Rule] 1, {r, 0\ , 1}, {\[Phi], 0.01\ , \ \ 0.99\ \[Pi]}\ \ , \[IndentingNewLine]PlotPoints \ \[Rule] 20, DisplayFunction \[Rule] Identity];\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(pt4 = CylindricalPlot3D[\(Vout /. a \[Rule] 1\) /. V0 \[Rule] 1, {r, 1, 1.75}, {\[Phi], 0.01, 0.99\ \[Pi]}, \[IndentingNewLine]PlotPoints \[Rule] 20, DisplayFunction \[Rule] Identity];\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(Show[{pt3\ , pt4}, \ DisplayFunction \[Rule] \ $DisplayFunction\ ];\)\)], "Input", Hyphenation->False] }, Closed]] }, Closed]], Cell["8.3 Laplace's Equation in Spherical Coordinates ", "Section", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["Overview of Spherical Coordinates", "Subsubsection", Hyphenation->False], Cell[CellGroupData[{ Cell["Problem 1: A charged ring ", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Required packages", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["Needs[\"Calculus`VectorAnalysis`\"] ", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Solution", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["Clear[\"Global`*\"];", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Part a", "Subsubsection", Hyphenation->False], Cell[BoxData[ RowBox[{ RowBox[{"potZ", "=", FormBox[\(q\/\@\(a\^2 + r\^2\)\), "TraditionalForm"]}], ";"}]], "Input", Hyphenation->False], Cell[BoxData[{ \(\(potZin[n_] := \(Series[potZ, {r, 0, n}] // PowerExpand\) // Normal;\)\), "\n", \(\(potZout[n_] := \(Series[potZ, {r, \[Infinity], n}] // PowerExpand\) // Normal;\)\)}], "Input", Hyphenation->False], Cell["potZin[9]", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["potZout[9]", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True] }, Closed]], Cell[CellGroupData[{ Cell["Part b", "Subsubsection", Hyphenation->False], Cell[BoxData[ RowBox[{\(pot[s_]\), ":=", RowBox[{"Sum", "[", RowBox[{ RowBox[{ FormBox[\(\(B(n)\)\ \((r\/a)\)\^n + \((a\/r)\)\^\(n + 1\)\ \(A( n)\)\), "TraditionalForm"], \(LegendreP[n, Cos[\[Theta]]]\)}], ",", \({n, 0, s}\)}], "]"}]}]], "Input", Hyphenation->False], Cell[BoxData[ RowBox[{ RowBox[{\(Vout[s_]\), "=", RowBox[{"Sum", "[", RowBox[{ RowBox[{ FormBox[\(\((a\/r)\)\^\(n + 1\)\ \(A(n)\)\), "TraditionalForm"], \(LegendreP[n, Cos[\[Theta]]]\)}], ",", \({n, 0, s}\)}], "]"}]}], ";"}]], "Input", Hyphenation->False], Cell[BoxData[ \(coeffOut[s_] := Solve[0 \[Equal] CoefficientList[\((Vout[s] - potZout[s])\) /. \[Theta] \[Rule] 0, 1/r]\ , A\ /@ \ Range[0, s]] // Flatten\)], "Input", Hyphenation->False], Cell["coeffOut[3]", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[" potOut[s_]:=Vout[s]/. coeffOut[s] ", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[" potOut[6] ", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[BoxData[ RowBox[{\(Vin[s_]\), ":=", RowBox[{"Sum", "[", RowBox[{ RowBox[{ FormBox[\(\((r\/a)\)\^n\ \(B(n)\)\), "TraditionalForm"], \(LegendreP[n, Cos[\[Theta]]]\)}], ",", \({n, 0, s}\)}], "]"}]}]], "Input", Hyphenation->False], Cell[BoxData[ \(coeffIn[s_] := Solve[0 \[Equal] CoefficientList[\((Vin[s] - potZin[s])\) /. \[Theta] \[Rule] 0, \ r]\ , B /@ Range[0, s]] // Flatten\)], "Input", Hyphenation->False], Cell[BoxData[ \(potIn[s_] := Vin[s] /. coeffIn[s]\)], "Input", Hyphenation->False], Cell[BoxData[ \(potIn[6]\)], "Input", Hyphenation->False], Cell["\<\ SetCoordinates[Spherical[r, \[Theta], \[Phi]]];\ \>", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["electout[s_]:= -Grad[potOut[s]];", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["electin[s_]:= -Grad[potIn[s]];", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[BoxData[ \(electout[3]\)], "Input", Hyphenation->False], Cell[BoxData[ \(electin[3]\)], "Input", Hyphenation->False] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Part c", "Subsubsection", Hyphenation->False], Cell[BoxData[ \(r2xRule = {r, \[Theta], \[Phi]} \[Rule] CoordinatesFromCartesian[{x, y, z}, Spherical] // Thread\)], "Input",\ Hyphenation->False], Cell[BoxData[ \(\(values = {a \[Rule] 1, q \[Rule] 1};\)\)], "Input", Hyphenation->False], Cell[BoxData[{ \(\(potOutXYZ[x_, y_, z_\ ]\ = \(potOut[12] //. r2xRule\) //. values\ ;\)\), "\n", \(\(potInXYZ[x_, y_, z_\ ]\ = \(potIn[12] //. r2xRule\) //. values\ ;\)\ \)}], "Input", Hyphenation->False], Cell[BoxData[ RowBox[{\(ff[x_, y_, z_]\), ":=", RowBox[{"If", "[", RowBox[{ RowBox[{ RowBox[{"N", "[", FormBox[\(\@\(x\^2 + y\^2 + z\^2\)\), "TraditionalForm"], "]"}], "\[GreaterEqual]", "1"}], ",", \(potOutXYZ[x, y, z]\), ",", \(potInXYZ[x, y, z]\)}], "]"}]}]], "Input", Hyphenation->False], Cell[BoxData[ \(\(pt1 = Plot3D[ff[x, 0, z], {x, \(-2\), 2}, {z, \(-2\), 2}, PlotPoints \[Rule] 30\ , DisplayFunction \[Rule] Identity];\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(pt2 = ContourPlot[ff[x, 0, z], {x, \(-2\), 2}, {z, \(-2\), 2}, PlotPoints \[Rule] 30, \ \ ContourSmoothing \[Rule] True, DisplayFunction \[Rule] Identity];\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(Show[GraphicsArray[{pt1, pt2}]];\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(VEPlot[ ff[x, 0, z] // Evaluate, \[IndentingNewLine]{x, \(-2.1\), 2.1}, {z, \(-1.6\), 1.6}, Epilog \[Rule] {{Hue[ .3], \ Disk[{1, 0}, .2]}, \ \ {Hue[ .3], \ Disk[{\(-1\), 0\ }, .2]}}];\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(pt3 = Plot3D[ff[x, y, 0], {x, \(-2\), 2}, {y, \(-2\), 2}, PlotPoints \[Rule] 30, DisplayFunction \[Rule] Identity];\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(pt4 = ContourPlot[ ff[x, y, 0], \[IndentingNewLine]{x, \(-2\), 2}, {y, \(-2\), 2}, \[IndentingNewLine]PlotPoints \[Rule] 30, \[IndentingNewLine]ContourSmoothing \[Rule] True, Epilog \[Rule] {Hue[0.07], Thickness[ .03], Circle[{0, 0}, 1]}, \[IndentingNewLine]DisplayFunction \[Rule] Identity];\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(Show[GraphicsArray[{pt3, pt4}]];\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(VEPlot[ ff[x, y, 0], \[IndentingNewLine]{x, \(-2.1\), 2.1}, {y, \(-2.1\), 2.1}, Epilog \[Rule] {Hue[0.05], Thickness[ .008], Circle[{0, 0}, 1]}];\)\)], "Input", Hyphenation->False] }, Closed]] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Problem 2: Grounded sphere in an electric field ", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["Required packages", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[BoxData[{ \(Needs["\"]\), "\n", \(Needs["\"]\)}], "Input", Hyphenation->False], Cell["Solution", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["Clear[\"Global`*\"];", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Part a", "Subsubsection", Hyphenation->False], Cell[BoxData[ \(\(Asol = {\ A[1] \[Rule] \(-E0\)\ a, \ A[i_] \[Rule] 0}\ \ ;\)\)], "Input", Hyphenation->False], Cell[BoxData[ RowBox[{\(pot[s_]\), ":=", RowBox[{ RowBox[{"Sum", "[", RowBox[{ RowBox[{ RowBox[{"(", FormBox[\(\(A( i)\)\ \((r\/a)\)\^i + \((a\/r)\)\^\(i + 1\)\ \(B( i)\)\), "TraditionalForm"], " ", ")"}], \(LegendreP[i, Cos[\[Theta]]]\)}], ",", \({i, 0, s}\)}], "]"}], "/.", "Asol"}]}]], "Input", Hyphenation->False], Cell[BoxData[ \(\(\(Beq[ s_]\)\(:=\)\(LogicalExpand[\((pot[s] /. r \[Rule] a)\) \[Equal] 0 + O[Cos[\[Theta]]]\^\(s + 1\)]\)\(\ \)\)\)], "Input", Hyphenation->False], Cell["\<\ Beq[4] \ \>", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[BoxData[ \(\(\(\ \ \)\(Bsol[s_] := Solve[Beq[s]] // Flatten\ ;\)\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(Off[Solve::svars]\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(\(\ \)\(\(Solve[Beq[6]]\)\(//\)\(Flatten\)\(\ \)\)\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(Bsol = {B[1] \[Rule] a\ E0, B[i_] \[Rule] 0};\)\)], "Input", Hyphenation->False], Cell[BoxData[ RowBox[{"potential", " ", "=", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{"Sum", "[", RowBox[{ RowBox[{ RowBox[{"(", FormBox[\(\(A( i)\)\ \((r\/a)\)\^i + \((a\/r)\)\^\(i + 1\)\ \(B( i)\)\), "TraditionalForm"], ")"}], " ", \(LegendreP[i, Cos[\[Theta]]]\)}], ",", \({i, 0, 3}\)}], "]"}], ")"}], "/.", "Asol"}], "/.", "Bsol"}]}]], "Input", Hyphenation->False], Cell[BoxData[ \(eField = \(-Grad[potential, Spherical[r, \[Theta], \[Phi]]]\) // ExpandAll\)], "Input", Hyphenation->False] }, Closed]], Cell[CellGroupData[{ Cell["Part b", "Subsubsection", Hyphenation->False], Cell[BoxData[ \(\(r2xRule = {r, \[Theta], \[Phi]} \[Rule] CoordinatesFromCartesian[{x, y, z}, Spherical] // Thread;\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(values = {E0 \[Rule] 1, a \[Rule] 1};\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(p1 = VEPlot[\(\(potential //. r2xRule\) //. values\) //. {y \[Rule] 0}, \[IndentingNewLine]{x, \(-2.1\), 2.1}, {z, \(-2.1\), 2.1}, \[IndentingNewLine]DisplayFunction \[Rule] Identity];\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(disk = Graphics[Disk[{0, 0}, 1]];\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(Show[p1, disk, DisplayFunction \[Rule] $DisplayFunction\ ];\)\)], "Input", Hyphenation->False] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Problem 3: Sphere with an axially symmetric charge distribution \ \ \>", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Required packages", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["\<\ Needs[\"Calculus`VectorAnalysis`\"] Needs[\"Graphics`ParametricPlot3D`\"] \ \>", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True] }, Closed]], Cell["Solution", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["Clear[\"Global`*\"];", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Part a", "Subsubsection", Hyphenation->False], Cell[BoxData[ RowBox[{\(potInA[s_]\), ":=", RowBox[{"Sum", "[", RowBox[{ RowBox[{ FormBox[\(\((r\/a)\)\^i\ \(A(i)\)\), "TraditionalForm"], " ", \(LegendreP[i, Cos[\[Theta]]]\)}], ",", \({i, 0, s}\)}], "]"}]}]], "Input", Hyphenation->False], Cell[BoxData[ RowBox[{\(potOutA[s_]\), ":=", RowBox[{"Sum", "[", RowBox[{ RowBox[{\(A[i]\), " ", FormBox[\(\((a\/r)\)\^\(i + 1\)\), "TraditionalForm"], \(LegendreP[i, Cos[\[Theta]]]\)}], ",", \({i, 0, s}\)}], "]"}]}]], "Input", Hyphenation->False], Cell[BoxData[ RowBox[{\(chargeDensity[s_]\), ":=", RowBox[{ RowBox[{ FormBox[\(1\/\(4\ \[Pi]\)\), "TraditionalForm"], \((D[potInA[s], r] - D[potOutA[s], r])\)}], "//.", \({r \[Rule] a}\)}]}]], "Input", Hyphenation->False], Cell[BoxData[ \(chargeDensity[2] // Apart\)], "Input", Hyphenation->False], Cell[BoxData[ \(aEquation[s_, Qdensity_] := LogicalExpand[ chargeDensity[s] \[Equal] Qdensity + O[Cos[\[Theta]]]\^\(s + 1\)]\)], "Input", Hyphenation->False], Cell[BoxData[ \(aRule[s_, Qdensity_] := Solve[aEquation[s, Qdensity], Array[A, s + 1, 0]] // Flatten\)], "Input",\ Hyphenation->False], Cell[BoxData[ RowBox[{"aRule", "[", RowBox[{"4", ",", FormBox[\(\(Q\ \(\(cos\^2\)(\[Theta])\)\)\/a\^2\), "TraditionalForm"]}], "]"}]], "Input", Hyphenation->False], Cell[BoxData[ RowBox[{\(potInA[4]\), "/.", RowBox[{"aRule", "[", RowBox[{"4", ",", FormBox[\(\(Q\ \(\(cos\^2\)(\[Theta])\)\)\/a\^2\), "TraditionalForm"]}], "]"}]}]], "Input", Hyphenation->False], Cell[BoxData[ RowBox[{\(potOutA[4]\), "/.", RowBox[{"aRule", "[", RowBox[{"4", ",", FormBox[\(\(Q\ \(\(cos\^2\)(\[Theta])\)\)\/a\^2\), "TraditionalForm"]}], "]"}]}]], "Input", Hyphenation->False] }, Closed]], Cell[CellGroupData[{ Cell["Part b", "Subsubsection", Hyphenation->False], Cell[BoxData[ RowBox[{"surfaceQ", "=", RowBox[{ FormBox[\(\(Q\ \((\(\(cos\^2\)(\[Theta])\)\/2 + cos(\[Theta]))\)\)\/a\^2\), "TraditionalForm"], "//", "Expand"}]}]], "Input", Hyphenation->False], Cell["potOutQ= potOutA[8] //. aRule[8,surfaceQ]", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["potInQ= potInA[8]//.aRule[8,surfaceQ]", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[BoxData[ \(\(values = {Q \[Rule] 1, a \[Rule] 1};\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(p1 = CylindricalPlot3D[ potInQ //. values, {r, 0, 1}, {\[Theta], 0, 2\ \[Pi]}, DisplayFunction \[Rule] Identity];\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(p2 = CylindricalPlot3D[ potOutQ //. values, {r, 1, 3}, {\[Theta], 0, 2 \[Pi]}, DisplayFunction \[Rule] Identity];\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(Show[p1, p2, \[IndentingNewLine]AxesLabel \[Rule] {a, b, c}, \[IndentingNewLine]ViewPoint \[Rule] {\(-1.5\), \(-2\), 2}, \[IndentingNewLine]BoxRatios \[Rule] {1, 1, 1}, DisplayFunction \[Rule] $DisplayFunction];\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(r2xRule = {r, \[Theta], \[Phi]} \[Rule] CoordinatesFromCartesian[{x, y, z}, Spherical] // Thread;\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(\( (*for\ 0 < r < 1*) \)\(\[IndentingNewLine]\)\(eq1[y_, z_] = \(potInQ /. r2xRule\) /. {Q \[Rule] 1, a \[Rule] 1, x \[Rule] 0} // Simplify;\)\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(\( (*\ for\ 1 < r\ *) \)\(\n\)\(eq2[y_, z_] = \(potOutQ /. r2xRule\) /. {Q \[Rule] 1, a \[Rule] 1, x \[Rule] 0}\ // Simplify;\)\)\)], "Input", Hyphenation->False], Cell[BoxData[ RowBox[{\(pot[y_, z_]\), ":=", RowBox[{"If", "[", RowBox[{ FormBox[\(y\^2 + z\^2 < 1\), "TraditionalForm"], ",", \(eq1[y, z]\), ",", \(eq2[y, z]\)}], "]"}]}]], "Input", Hyphenation->False], Cell[BoxData[ \(\(circle = {Thickness[0.02], Circle[{0, 0}, 1]}\ // Graphics;\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(Show[{VEPlot[pot[y, z], {y, \(-2\), 2}, {z, \(-2\), 2}, DisplayFunction \[Rule] Identity], \[IndentingNewLine]circle}, \ \[IndentingNewLine]DisplayFunction \[Rule] $DisplayFunction];\)\)], "Input", Hyphenation->False] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Problem 4: Sphere with a given axially symmetric potential ", \ "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["Required packages", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["\<\ Needs[\"Calculus`VectorAnalysis`\"] Needs[\"Graphics`ParametricPlot3D`\"] \ \>", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["Solution", "Subsubsection", Evaluatable->False, InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell["Clear[\"Global`*\"];", "Input", InitializationCell->False, Hyphenation->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell["Part a", "Subsubsection", Hyphenation->False], Cell[BoxData[ RowBox[{\(potIn[V_, s_]\), ":=", RowBox[{"Sum", "[", RowBox[{ RowBox[{ FormBox[\(\((r\/r0)\)\^n\ \(B(V, n)\)\), "TraditionalForm"], " ", \(P[n, Cos[\[Theta]]]\)}], ",", \({n, 0, s}\)}], "]"}]}]], "Input", Hyphenation->False], Cell[BoxData[ RowBox[{\(potOut[V_, s_]\), ":=", RowBox[{"Sum", "[", RowBox[{ RowBox[{ FormBox[\(\((r0\/r)\)\^\(n + 1\)\ \(B(V, n)\)\), "TraditionalForm"], " ", \(P[n, Cos[\[Theta]]]\)}], ",", \({n, 0, s}\)}], "]"}]}]], "Input", Hyphenation->False], Cell[BoxData[ RowBox[{\(B[V_, n_]\), ":=", RowBox[{ FormBox[\(1\/2\ \((2\ n + 1)\)\), "TraditionalForm"], " ", \(Integrate[ V\ LegendreP[n, Cos[\[Theta]]]\ Sin[\[Theta]], {\[Theta], 0, \[Pi]}]\)}]}]], "Input", Hyphenation->False] }, Closed]], Cell[CellGroupData[{ Cell["Part b", "Subsubsection", Hyphenation->False], Cell[BoxData[ \(pOut = potOut[V0\ Cos[\[Theta]]\^2, 4]\)], "Input", Hyphenation->False], Cell[BoxData[ \(pIn = potIn[V0\ Cos[\[Theta]]\^2, 4]\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(values = {r0 \[Rule] 1, V0 \[Rule] 1, P \[Rule] LegendreP};\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(p1 = CylindricalPlot3D[pIn /. values, {r, 0, 1}, {\[Theta], 0, 2 \[Pi]}, DisplayFunction \[Rule] Identity];\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(p2 = CylindricalPlot3D[pOut /. values, {r, 1, 3}, {\[Theta], 0, 2 \[Pi]}, DisplayFunction \[Rule] Identity];\)\)], "Input", Hyphenation->False], Cell[BoxData[ \(\(Show[p1, p2, BoxRatios \[Rule] {1, 1, 1}, ViewPoint \[Rule] {\(-1.5\), \(-2\), 2}, DisplayFunction \[Rule] $DisplayFunction];\)\)], "Input", Hyphenation->False] }, Closed]], Cell[CellGroupData[{ Cell["Part 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